../src/ch-1.tex:5:The most basic and important theorem concerning polynomials
../src/ch-1.tex:7:tells us that every polynomial factors completely over the complex
../src/ch-1.tex:9:relationships between the location of the zeros of a polynomial and
../src/ch-1.tex:11:relationships between the zeros of a polynomial and the zeros of its
../src/ch-1.tex:14:of polynomials. Highlights of this chapter include the Fundamental
../src/ch-1.tex:21:\titleb {1.1 Polynomials and Rational Functions}
../src/ch-1.tex:22:The focus for this book is the polynomial of a single variable.
../src/ch-1.tex:23:This is an extended notion of the polynomial, as we will see later,
../src/ch-1.tex:24:but the most important examples are the algebraic and trigonometric polynomials, which
../src/ch-1.tex:27:polynomials of degree at most $n$ with complex coefficients is
../src/ch-1.tex:36:When we restrict our attention to polynomials with real
../src/ch-1.tex:50:The set of trigonometric polynomials ${\Cal T}^c_n$ is defined by
../src/ch-1.tex:53:A real trigonometric polynomial of degree at most $n$ is an element of
../src/ch-1.tex:55:all real trigonometric polynomials of degree at most
../src/ch-1.tex:61:and so many properties of trigonometric polynomials reduce to the
../src/ch-1.tex:62:study of algebraic polynomials of twice the degree on the
../src/ch-1.tex:67:of Algebra. It says that a polynomial of exact degree $n$ (that
../src/ch-1.tex:79:is a polynomial of degree $5$ with a zero of multiplicity
../src/ch-1.tex:81:The polynomial
../src/ch-1.tex:92:\noindent The importance of the solution of polynomial equations in the
../src/ch-1.tex:106:finding the zeros of a polynomial of degree at least 5, in general, by a formula
../src/ch-1.tex:118:the quadratic polynomial $x^2 + bx + c$ has zeros at $${{-b-{\sqrt{b^2 - 4c}}}\over2}\,, \qquad
../src/ch-1.tex:121:\noindent {\bf b] \enskip Cubic Equations.} \enskip Verify that the cubic polynomial $x^3 + bx
../src/ch-1.tex:134:\noindent {\bf c]} \enskip Show that an arbitrary cubic polynomial,
../src/ch-1.tex:136:into a cubic polynomial as in part b] by a transformation $x \mapsto ex + f$.
../src/ch-1.tex:138:\noindent {\bf d]} \enskip Observe that if the polynomial $x^3 + bx + c$ has three distinct real
../src/ch-1.tex:145:polynomial $x^4 + ax^3 + bx^2 + cx + d$ has zeros at
../src/ch-1.tex:177:A {\it polynomial of $n$ variables} is a function that is a polynomial in each of its
../src/ch-1.tex:178:variables. A {\it symmetric polynomial of $n$ variables} is a polynomial of $n$ variables
../src/ch-1.tex:182:symmetric polynomial in $n$ variables (with integer coefficients)
../src/ch-1.tex:183:may be written uniquely as a polynomial (with integer
../src/ch-1.tex:187:{\it Hint:} For a symmetric polynomial $f$ in $n$ variables, let
../src/ch-1.tex:199:index. This gives a (partial) well ordering of symmetric polynomials
../src/ch-1.tex:200:in $n$ variables, that is, every set of symmetric polynomials in $n$
../src/ch-1.tex:262:polynomials for each of the above inequalities. That is, for
../src/ch-1.tex:301:So every polynomial of degree $n$ can be evaluated by using at most $n$
../src/ch-1.tex:322:and is called the {\it Lagrange interpolation polynomial}.
../src/ch-1.tex:325:real, then this unique interpolation polynomial is in ${\Cal P}_n.$
../src/ch-1.tex:339:Lagrange interpolation polynomial satisfying
../src/ch-1.tex:360:unique $p \in {\Cal P}^c_{n}$, called the {\it Hermite interpolation polynomial}, so that
../src/ch-1.tex:362:If all the $z_i$ and $y_{i,j}$ are real, then this unique interpolation polynomial
../src/ch-1.tex:370:polynomial satisfying
../src/ch-1.tex:382:Polynomial interpolation and related topics are studied thoroughly in Davis [75];
../src/ch-1.tex:390:\subhead{E.9 \enskip Factorization of Trigonometric Polynomials} \endsubhead
../src/ch-1.tex:403:\subhead {E.10 \enskip Newton Interpolation and Integer-Valued Polynomials} \endsubhead \enskip Let
../src/ch-1.tex:413:is a polynomial of degree $k$ that takes
../src/ch-1.tex:430:characterizes such polynomials.
../src/ch-1.tex:450:\proclaim {Theorem 1.2.1} The polynomial
../src/ch-1.tex:483:The exact relationship between the coefficients of a polynomial
../src/ch-1.tex:525:Then the polynomial $h \in {\Cal P}_n^c$ of the form
../src/ch-1.tex:544:real polynomial factors completely into real linear or
../src/ch-1.tex:558:polynomial, $u$ and $v$, have the property that the curves
../src/ch-1.tex:570:The ``geometry of polynomials'' is extensively studied in Marden
../src/ch-1.tex:632:\noindent {\bf a]} \enskip Suppose $p$ is a polynomial of degree
../src/ch-1.tex:634:polynomial $q$ of degree $n-1$ such that
../src/ch-1.tex:636:{\it Hint:} Consider the usual division algorithm for polynomials. \qed
../src/ch-1.tex:638:\noindent {\bf b]} \enskip A polynomial of degree $n$
../src/ch-1.tex:642:remaining content is that every nonconstant polynomial has at
../src/ch-1.tex:646:mostly for polynomials on circles. The point of this exercise is
../src/ch-1.tex:650:\subhead {E.3 \enskip Polynomial Complex Analysis}\endsubhead
../src/ch-1.tex:653:formula for polynomials on circles.
../src/ch-1.tex:674:polynomials $f$ and $g$ given by their factorizations, and for circles $\gamma$.
../src/ch-1.tex:687:can be sharpened for polynomials as follows. If $p, q \in {\Cal P}_n^c$ and
../src/ch-1.tex:691:Equivalently, a polynomial $p \in {\Cal P}_n^c$ is either identically $0$ or
../src/ch-1.tex:696:{\it Every nonconstant polynomial has at least one
../src/ch-1.tex:749:\subhead {E.7 \enskip The Number of Positive Zeros of a Polynomial} \endsubhead
../src/ch-1.tex:784:of the derivative of a polynomial to the zeros of the
../src/ch-1.tex:785:polynomial is variously attributed to Gauss, Lucas,
../src/ch-1.tex:822:polynomials formulated by Jensen. We need to
../src/ch-1.tex:824:polynomial $p \in {\Cal P}_n$. For $p \in {\Cal P}_n$ the
../src/ch-1.tex:1027:\subhead {E.6 \enskip Positive Zeros of M\"untz Polynomials}\endsubhead \enskip
../src/ch-1.tex:1038:\subhead {E.7 \enskip Apolar Polynomials and Szeg\H o's Theorem} \endsubhead \enskip
../src/ch-1.tex:1039:Two polynomials
../src/ch-1.tex:1052:{\it Suppose that $f$ and $g$ are apolar polynomials.
../src/ch-1.tex:1103:So the polynomial
../src/ch-1.tex:1116:Show that the polynomial $q$ defined by $q(x) : = \int^x_0 p(t) \,dt$
../src/ch-1.tex:1198:\endsubhead \enskip For a polynomial
../src/ch-1.tex:1254:\subhead{E.13 \enskip Fej\'er's Theorem on the Zeros of M\"untz Polynomials}
../src/ch-1.tex:1281:It follows from Lucas' theorem that if $q$ is a polynomial with
../src/ch-2.tex:1:\titlea{2 Some Special Polynomials}
../src/ch-2.tex:3:Chebyshev polynomials are introduced and their central role in
../src/ch-2.tex:6:our primary interest is in orthogonal polynomials
../src/ch-2.tex:9:third section of this chapter is concerned with orthogonal polynomials; it
../src/ch-2.tex:10:introduces the most classical of these. These polynomials satisfy many
../src/ch-2.tex:11:extremal properties, similar to those of the Chebyshev polynomials, but
../src/ch-2.tex:14:deals with polynomials with positive coefficients in various bases.
../src/ch-2.tex:17:\titleb{2.1 Chebyshev Polynomials}
../src/ch-2.tex:18:The ubiquitous Chebyshev polynomials lie at the heart
../src/ch-2.tex:26:The Chebyshev polynomials are defined
../src/ch-2.tex:34:left for the reader (see E.1). The $n$th Chebyshev polynomial
../src/ch-2.tex:45:n$.) The Chebyshev polynomial $T_n$ satisfies the following
../src/ch-2.tex:79:\noindent The Chebyshev polynomials $T_n$ are named after
../src/ch-2.tex:84:polynomials may be found in Rivlin [90]. Throughout
../src/ch-2.tex:85:later sections of this book the Chebyshev polynomials
../src/ch-2.tex:87:properties of the Chebyshev polynomials.
../src/ch-2.tex:95:length is attained by the $n$th Chebyshev polynomial
../src/ch-2.tex:138:defines a polynomial from ${\Cal P}_{n-1}$ on ${\Bbb R}$ having at least
../src/ch-2.tex:209:Chebyshev polynomial shifted to the interval $[0,1].$ Suppose
../src/ch-2.tex:228:polynomials of degree $n$ and for all positive
../src/ch-2.tex:254:polynomial $p_n$ of degree exactly $n$ so that
../src/ch-2.tex:298:\subhead {E.6 \enskip Trigonometric Polynomials of Longest Arc
../src/ch-2.tex:334:\subhead {E.7 \enskip Monic Polynomials with Minimal Norm on an Interval}\endsubhead
../src/ch-2.tex:337:{\bf a]} \enskip The unique monic polynomial
../src/ch-2.tex:346:{\bf b]} \enskip Let $0 < a < b$. Find all monic polynomials
../src/ch-2.tex:358:\subhead {E.8 \enskip Lower Bound for the Norm of Polynomials on the Unit Disk}
../src/ch-2.tex:363:Thus $z^n$ plays the role of the $n$th Chebyshev polynomial
../src/ch-2.tex:374:basic properties, such as irreducibility of cyclotomic polynomials
../src/ch-2.tex:414:\subhead {E.10 \enskip Chebyshev Polynomials of the
../src/ch-2.tex:415:Second Kind}\endsubhead Let the {\it Chebyshev polynomials of the second kind}
../src/ch-2.tex:468:for $E.$ If the points $z_i$ are the $n$th Fekete points for $E,$ then the polynomial
../src/ch-2.tex:470:is called an $n$th (monic) {\it Fekete polynomial} for $E.$
../src/ch-2.tex:489:$n$th Fekete polynomial for $E.$ Let
../src/ch-2.tex:618:The following section specializes the discussion to polynomials.
../src/ch-2.tex:844:orthogonal polynomials, is old and far-reaching. As we
../src/ch-2.tex:846:with the classical orthogonal polynomials including Chebyshev,
../src/ch-2.tex:855:``Orthogonal Polynomials.'' Of course, orthogonal
../src/ch-2.tex:856:polynomials are intimately connected to Fourier series
../src/ch-2.tex:1234:complex trigonometric polynomials is dense in
../src/ch-2.tex:1324:\subhead{E.9 \enskip Denseness of Polynomials in $L_2(\mu)$ on ${\Bbb R}$ }\endsubhead
../src/ch-2.tex:1335:polynomials is dense in $C^* [-\pi, \pi]$ (see
../src/ch-2.tex:1367:with some $r >0.$ Show that the set ${\Cal P}^c$ of all complex algebraic polynomials
../src/ch-2.tex:1415:\titleb{2.3 Orthogonal Polynomials}
../src/ch-2.tex:1416:The classical orthogonal polynomials arise on
../src/ch-2.tex:1421:\infty)$. The main examples we consider are the {\it Jacobi polynomials}
../src/ch-2.tex:1427:\beta = -1/2$ the Jacobi polynomials are the {\it Chebyshev polynomials of
../src/ch-2.tex:1432:{\it Chebyshev polynomials of the second kind},
../src/ch-2.tex:1437:{\it Legendre polynomials},
../src/ch-2.tex:1440:{\it Laguerre polynomials} are
../src/ch-2.tex:1442:The {\it Hermite polynomials} are
../src/ch-2.tex:1447:polynomials with a standard normalization; see the exercises.
../src/ch-2.tex:1449:orthonormality. All of these much studied polynomials arise
../src/ch-2.tex:1452:the special properties of these classical orthogonal polynomials
../src/ch-2.tex:1465:\proclaim{Theorem 2.3.1 (Existence and Uniqueness of Orthonormal Polynomials)} \enskip
../src/ch-2.tex:1467:polynomials $(p_n)^\infty_{n=0}$ with the following
../src/ch-2.tex:1484:polynomials} associated with an $m$-distribution $\alpha$.
../src/ch-2.tex:1486:polynomials} associated with an $m$-distribution $\alpha$ if
../src/ch-2.tex:1488:where $(p_n)^\infty_{n=0}$ is the sequence of orthonormal polynomials
../src/ch-2.tex:1501:sequence of orthogonal (orthonormal) polynomials
../src/ch-2.tex:1510:polynomials from general orthogonal systems is
../src/ch-2.tex:1516:polynomials with respect to an $m$-distribution
../src/ch-2.tex:1535:for any two polynomials $p$ and $q$. Since
../src/ch-2.tex:1547:Here the lead coefficient of the left-hand side polynomial is $\gamma_n,$ while the
../src/ch-2.tex:1548:lead coefficient of the right-hand side polynomial is $d_{n+1}
../src/ch-2.tex:1596:polynomials associated with an $m$-distribution
../src/ch-2.tex:1603:of orthogonal polynomials by Szeg\H o [82, vol. III], writes:
../src/ch-2.tex:1606:``The classical orthogonal polynomials
../src/ch-2.tex:1610:work on the Laguerre polynomials $L_n^0(x)$. Abel's
../src/ch-2.tex:1612:first published work on these polynomials that uses
../src/ch-2.tex:1614:polynomials were studied extensively by Laplace in
../src/ch-2.tex:1616:real contribution to these polynomials was to
../src/ch-2.tex:1617:introduce Hermite polynomials in several
../src/ch-2.tex:1619:relation for Legendre polynomials.''\par
../src/ch-2.tex:1623:diverse ways in which these polynomials can arise.
../src/ch-2.tex:1626:orthogonal polynomials. In particular, Askey and Ismail [84], Chihara [78], Erd\'elyi et al.
../src/ch-2.tex:1630:polynomials. The connections linking orthogonal polynomials, the
../src/ch-2.tex:1636:polynomials associated with an $m$-distribution $\alpha$.
../src/ch-2.tex:1653:$(p_n)^\infty_{n=0}$ be the sequence of orthonormal polynomials associated with an
../src/ch-2.tex:1673:$(p_n)^\infty_{n=0}$ be the sequence of orthonormal polynomials
../src/ch-2.tex:1712:the classical orthogonal polynomials. Proofs are available in Szeg\H o
../src/ch-2.tex:1715:\subhead {E.5 \enskip Jacobi Polynomials}\endsubhead \enskip
../src/ch-2.tex:1723:Then $(P^{(\alpha, \beta)}_n)^\infty_{n =0}$ is a sequence of orthogonal polynomials
../src/ch-2.tex:1733:In the rest of the exercise, the polynomials $P^{(\alpha, \beta)}_n$ are
../src/ch-2.tex:1784:defined. The Legendre polynomials
../src/ch-2.tex:1791:The Chebyshev polynomials
../src/ch-2.tex:1796:{\it polynomials} $C_n^{(\alpha)}$ are defined by
../src/ch-2.tex:1801:In terms of $C^{(\alpha)}_n,$ the Chebyshev polynomials of the
../src/ch-2.tex:1806:\subhead {E.6 \enskip Hermite Polynomials}\endsubhead \enskip
../src/ch-2.tex:1811:Then $(H_n)^\infty_{n=0}$ is a sequence of orthogonal polynomials
../src/ch-2.tex:1820:In the rest of the exercise, the polynomials $H_n$ are as in part a].
../src/ch-2.tex:1845:\subhead {E.7 \enskip Laguerre Polynomials}\endsubhead \enskip Let $\alpha \in (-1,\infty)$.
../src/ch-2.tex:1851:is a sequence of orthogonal polynomials on $[0, \infty)$ associated
../src/ch-2.tex:1860:In the rest of the exercise, the polynomials $L^{(\alpha)}_n$ are
../src/ch-2.tex:1898:be the sequence of orthonormal polynomials associated with
../src/ch-2.tex:2025:the sequence of orthonormal polynomials associated with an
../src/ch-2.tex:2046:A polynomial $p$ is called {\it nonnegative} if it takes nonnegative values on
../src/ch-2.tex:2050:holds for every nonnegative polynomial $p \in {\Cal P}_n$ of the
../src/ch-2.tex:2229:$(b_n)^\infty_{n=0} \subset {\Bbb R},$ the polynomials
../src/ch-2.tex:2241:{\bf a]} \enskip Show that the polynomials $p_n$ are of the form
../src/ch-2.tex:2302:orthonormal polynomials associated with $\alpha$.
../src/ch-2.tex:2319:for polynomials $q \in {\Cal P}^c_{n-1}$ satisfying
../src/ch-2.tex:2332:polynomial $p_n$.
../src/ch-2.tex:2352:$m$-distribution with associated orthonormal polynomials
../src/ch-2.tex:2366:interpolation) to find polynomials $P \in {\Cal P}_{2n-1}$ and
../src/ch-2.tex:2391:\subhead {E.15 \enskip Orthonormal Polynomials as Determinants} \endsubhead \enskip
../src/ch-2.tex:2407:{\bf b]} \enskip Show that the orthonormal polynomials $p_n$ associated
../src/ch-2.tex:2419:of orthonormal polynomials associated with $\alpha$ as in
../src/ch-2.tex:2420:Theorem 2.3.2. Show that the monic orthogonal polynomials
../src/ch-2.tex:2435:recursion for the sequence of $(p_n)_{n=0}^\infty$ of orthonormal polynomials associated with an
../src/ch-2.tex:2491:There is an analogous theory of orthogonal polynomials
../src/ch-2.tex:2503:the $n$th {\it monic} orthogonal polynomial on $[-a,a]$ associated
../src/ch-2.tex:2510:Suppose the sequence of polynomials
../src/ch-2.tex:2528:\subhead {E.18 \enskip Completeness of Orthogonal Polynomials} \endsubhead
../src/ch-2.tex:2531:polynomials associated with an $m$-distribution $\alpha$.
../src/ch-2.tex:2540:\subhead {E.19 \enskip Bounds for Jacobi Polynomials}\endsubhead \enskip
../src/ch-2.tex:2549:hold, where $(p_n)_{n=0}^\infty$ is the sequence of orthonormal Jacobi polynomials
../src/ch-2.tex:2557:\titleb {2.4 Polynomials with Nonnegative
../src/ch-2.tex:2559:A quadratic polynomial $x^2+\alpha x+ \beta$ with real coefficients has
../src/ch-2.tex:2572:P}^+_n$ the set of polynomials in ${\Cal P}_n,$ that have all
../src/ch-2.tex:2577:where $s$ and $t$ are both polynomials with all nonnegative
../src/ch-2.tex:2580:Since a polynomial $p$ that is real-valued on the positive real axis has
../src/ch-2.tex:2593:\demo {Proof} The quadratic polynomial $x^2 -
../src/ch-2.tex:2697:It suffices to prove this result for quadratic polynomials; this is left
../src/ch-2.tex:2715:quadratic polynomial with zeros forming a pair of conjugate zeros of $p$
../src/ch-2.tex:2721:\noindent Polynomials with all nonnegative coefficients have a number of
../src/ch-2.tex:2725:polynomials; see E.2. So a Weierstrass-type theorem does not hold for
../src/ch-2.tex:2726:these polynomials. This is quite different
../src/ch-2.tex:2727:from approximation by polynomials of the form
../src/ch-2.tex:2729:Since every polynomial that is strictly positive on $(-1,1)$
../src/ch-2.tex:2737:fractions of polynomials with all nonnegative coefficients form a
../src/ch-2.tex:2741:polynomials with all nonnegative coefficients.
../src/ch-2.tex:2743:Various inequalities for polynomials of the form $(2.4.2)$ are
../src/ch-2.tex:2760:and the zero of the quadratic polynomial. \qed
../src/ch-2.tex:2838:\subhead {E.2 \enskip Polynomials with
../src/ch-2.tex:2846:{\bf b]} \enskip If $(p_n)^\infty_{n=1}$ is a sequence of polynomials with
../src/ch-2.tex:2866:Polynomials and Sums of Squares}\endsubhead
../src/ch-2.tex:2913:\subhead {E.4 \enskip Lorentz Degree of Polynomials} \endsubhead \enskip Given a
../src/ch-2.tex:2914:polynomial $p \in {\Cal P}_n,$ let $d=d(p)$ be the minimal
../src/ch-2.tex:2915:nonnegative integer for which the polynomial $p$ is of the form
../src/ch-2.tex:2918:the {\it Lorentz degree} of the polynomial $p.$
../src/ch-2.tex:2945:polynomial $p \in {\Cal P}_2 \setminus {\Cal P}_1$ lie on the ellipse
../src/ch-2.tex:2970:any two polynomials $p$ and $q.$
../src/ch-2.tex:2981:\noindent {\bf i]} \enskip Let $p$ be a polynomial. Show that
../src/ch-2.tex:3077:This shows that polynomials $p$ with the property $d(p) = \deg (p)$
../src/ch-2.tex:3082:\subhead {E.5 \enskip \enskip Lorentz Degree of Trigonometric Polynomials}
../src/ch-2.tex:3084:polynomial $t \in {\Cal T}_n,$ let $d = d_{\omega}(t)$ be the
../src/ch-2.tex:3145:polynomials $t_1$ and $t_2.$
../src/ch-2.tex:3158:\noindent {\bf g]} \enskip Let $p$ be a trigonometric polynomial
../src/ch-3.tex:6:prototype is the space ${\Cal P}_n$ of real algebraic polynomials of
../src/ch-3.tex:8:polynomials many basic properties. The first section is an
../src/ch-3.tex:14:the {\it Chebyshev ``polynomials''} associated with Chebyshev spaces. These associated
../src/ch-3.tex:15:Chebyshev polynomials, which equioscillate like
../src/ch-3.tex:16:the usual Chebyshev polynomials, are
../src/ch-3.tex:23:Chebyshev ``polynomials'' associated with the Chebyshev spaces
../src/ch-3.tex:33:From an approximation theoretic point of view an essential property that polynomials
../src/ch-3.tex:36:points. This is equivalent to the fact that a polynomial
../src/ch-3.tex:559:of polynomials. There are two additional types of
../src/ch-3.tex:664:polynomials is presented in the exercises. The
../src/ch-3.tex:749:{\it Hint:} The determinant is a polynomial in $x_0, x_1, \ldots,
../src/ch-3.tex:859:polynomials of the same degree, $m-1,$ in each variable
../src/ch-3.tex:1071:M\"untz Polynomials.}
../src/ch-3.tex:1156:\subhead {E.7 \enskip Descartes' Rule of Signs for Polynomials}\endsubhead
../src/ch-3.tex:1264:\titleb{3.3 Chebyshev Polynomials in Chebyshev Spaces}
../src/ch-3.tex:1268:polynomial}
../src/ch-3.tex:1299:polynomials; see E.7 of Section 2.1.
../src/ch-3.tex:1302:Chebyshev polynomials $T_n$ for $H_n$ on $A$ encode much of the
../src/ch-3.tex:1305:Chebyshev polynomials.
../src/ch-3.tex:1309:Chebyshev polynomials}
../src/ch-3.tex:1318:of the zeros of the associated Chebyshev polynomials; see Section 4.1.
../src/ch-3.tex:1321:the Chebyshev polynomials is the following:
../src/ch-3.tex:1326:$[a,b]$ with associated Chebyshev polynomial
../src/ch-3.tex:1363:polynomials
../src/ch-3.tex:1393:Chebyshev polynomials
../src/ch-3.tex:1411:denote the associated Chebyshev polynomials.
../src/ch-3.tex:1460:polynomials $T_n$ and $S_n$ to make the proof of the above statement transparent.)
../src/ch-3.tex:1472:is called an $L_p$ Chebyshev polynomial for $H_n$ on $A.$
../src/ch-3.tex:1474:of these $L_p$ Chebyshev polynomials
../src/ch-3.tex:1480:\subhead {E.1 \enskip Existence and Uniqueness of Chebyshev Polynomials} \endsubhead \enskip
../src/ch-3.tex:1486:{\bf a] \enskip Existence of Chebyshev Polynomials.} \enskip Show that there exists a
../src/ch-3.tex:1521:{\bf c] \enskip Uniqueness of Chebyshev Polynomials.} \enskip
../src/ch-3.tex:1522:Show that the Chebyshev polynomials
../src/ch-3.tex:1531:\subhead {E.2 \enskip More on Chebyshev Polynomials} \endsubhead \enskip
../src/ch-3.tex:1535:with associated Chebyshev polynomial denoted by $T_n: = T_n \{f_0, \ldots, f_n; [a,b]\}.$
../src/ch-3.tex:1553:Chebyshev system on $[a,b].$ (So this applies to ordinary polynomials on
../src/ch-3.tex:1557:system on $[a,b]$ with associated Chebyshev polynomial
../src/ch-3.tex:1568:with associated Chebyshev polynomial $$T_n: = T_n\{f_0, f_1, \ldots, f_n; [a,b]\}$$
../src/ch-3.tex:1596:Denote the associated Chebyshev polynomials for
../src/ch-3.tex:1713:Denote the associated Chebyshev polynomials for
../src/ch-3.tex:1783:and Kro\'o and Szabados [94]. Various coefficient estimates for polynomials are discussed in
../src/ch-3.tex:1784:Milovanovi\'c, Mitrinovi\'c, and Rassias [94]. An estimate for the coefficients of polynomials
../src/ch-3.tex:1786:such polynomials is studied in Baishanski [83].
../src/ch-3.tex:1789:\subhead{E.6 \enskip Coefficient Bounds for Polynomials in a Special
../src/ch-3.tex:1794:for every polynomial $p$ of the form
../src/ch-3.tex:1805:polynomial
../src/ch-3.tex:1816:\subhead {E.7 \enskip On the Zeros of the Chebyshev Polynomials for M\"untz Spaces} \endsubhead \enskip
../src/ch-3.tex:1819:Denote the associated Chebyshev polynomials for $H_n$ on $[0,1]$ by
../src/ch-3.tex:1850:Polynomials}
../src/ch-3.tex:1855:particular, we explicitly construct orthogonal ``polynomials'' for this system.
../src/ch-3.tex:1871:x^{\lambda_n} \}$ is called a {\it M\"untz polynomial} or a
../src/ch-3.tex:1872:$\Lambda$-{\it polynomial}. We denote the set of all such
../src/ch-3.tex:1873:polynomials by $M_n(\Lambda),$ that is,
../src/ch-3.tex:1886:$\Lambda$-polynomials
../src/ch-3.tex:1889:define the orthogonal $\Lambda$-polynomials with respect to
../src/ch-3.tex:1890:Lebesgue measure. We call these {\it M\"untz-Legendre polynomials}.
../src/ch-3.tex:1896:\noindent {\bf Definition 3.4.1 (M\"untz-Legendre Polynomials).}
../src/ch-3.tex:1898:We define the $n$th {\it M\"untz-Legendre polynomial} on $(0,\infty)$ by
../src/ch-3.tex:1931:$\Lambda$ polynomial provided the numbers $\lambda_i$ are distinct.
../src/ch-3.tex:1945:denote the $n$th M\"untz-Legendre polynomial $L_n\{\lambda_0,
../src/ch-3.tex:1953:the same we recover the Laguerre polynomials; see E.1.
../src/ch-3.tex:2019:{\it orthonormal M\"untz-Legendre polynomials}.
../src/ch-3.tex:2022:polynomials (see E.2). Let $$p_n(x) = \sum^n_{k=0} {\frac{x^{\lambda_k}}
../src/ch-3.tex:2089:polynomials at $1$ can now all be calculated.
../src/ch-3.tex:2122:\noindent M\"untz polynomials are just exponential polynomials $\sum a_k
../src/ch-3.tex:2131:Various properties of the M\"untz-Legendre polynomials
../src/ch-3.tex:2136:Theorem 3.3.4 to the associated Chebyshev polynomials on $[a,b]$ to deduce
../src/ch-3.tex:2139:M\"untz-Legendre polynomials in E.7.
../src/ch-3.tex:2141:\subhead {E.1 \enskip Laguerre Polynomials} \endsubhead
../src/ch-3.tex:2148:Laguerre polynomial orthonormal with respect to the weight
../src/ch-3.tex:2168:is an orthonormal sequence of polynomials on $[0,\infty)$
../src/ch-3.tex:2239:\subhead {E.7 \enskip On the Zeros of M\"untz-Legendre Polynomials}
../src/ch-3.tex:2337:$(\lambda_{k,i})^\infty_{i=1}$ such that the M\"untz-Legendre polynomials
../src/ch-3.tex:2446:polynomial with respect to the weight $e^{-x}$ on $[0,\infty),$
../src/ch-3.tex:2454:\subhead {E.9 \enskip The Order of the Zero at $1$ of Certain Polynomials} \endsubhead
../src/ch-3.tex:2456:of a polynomial whose coefficients are bounded in modulus by the leading coefficient.
../src/ch-3.tex:2465:If $p$ has a zero at $1$ of multiplicity $m$, then for every polynomial $f$
../src/ch-3.tex:2468:We construct a polynomial $f$ of degree at most
../src/ch-3.tex:2474:Let $T_\nu$ be the $\nu$th Chebyshev polynomial defined by (2.1.1). Let $k \in {\Bbb N},$ and let
../src/ch-3.tex:2492:\noindent {\bf b]} \enskip For every $n \in {\Bbb N},$ there exists a polynomial
../src/ch-3.tex:2502:Then $L_n$ is a polynomial of degree $n^2$ with a zero of multiplicity at least
../src/ch-3.tex:2516:is a polynomial of degree $2n^2$ with a zero of order $n$ at $1.$ Also
../src/ch-3.tex:2522:\noindent {\bf c]} \enskip For every $n \in {\Bbb N},$ there exists a polynomial
../src/ch-3.tex:2528:\titleb{3.5 Chebyshev Polynomials in Rational Spaces}
../src/ch-3.tex:2530:There are very few situations where Chebyshev polynomials can
../src/ch-3.tex:2534:However, the explicit formulas for the Chebyshev polynomials
../src/ch-3.tex:2549:nonuniqueness of the Chebyshev polynomials. Note that ordinary
../src/ch-3.tex:2550:polynomials arise as a limiting case of the span of system $(3.5.2)$ on letting
../src/ch-3.tex:2579:We can construct Chebyshev polynomials of the first and second
../src/ch-3.tex:2602:The {\it Chebyshev polynomials of the first kind} for the spaces
../src/ch-3.tex:2612:The {\it Chebyshev polynomials of the second kind}
../src/ch-3.tex:2623:Chebyshev polynomials preserve many of the elementary properties
../src/ch-3.tex:2625:polynomials. This is the content of the next three results.
../src/ch-3.tex:2627:\proclaim{Theorem 3.5.1 (Chebyshev Polynomials of the First and Second
../src/ch-3.tex:2652:\demo{ Proof} Observe that there are polynomials $p_1
../src/ch-3.tex:2693:\proclaim{Theorem 3.5.2 (Chebyshev Polynomials in Algebraic Rational Spaces)} \enskip Given
../src/ch-3.tex:2716:property of the Chebyshev polynomials, which also extends to certain linear
../src/ch-3.tex:2717:combinations of Chebyshev polynomials. In the trigonometric polynomial case
../src/ch-3.tex:2721:Chebyshev polynomials for ${\Cal T}_n (a_1, a_2, \ldots, a_n)$ and
../src/ch-3.tex:2724:\proclaim{Theorem 3.5.3 (Chebyshev Polynomials in Trigonometric Rational Spaces)} \quad
../src/ch-3.tex:2788:trigonometric polynomial $t \in {\Cal T}_{2n}$ such that
../src/ch-3.tex:2803:$U_n$ be the Chebyshev polynomials of the first and second kinds
../src/ch-3.tex:2845:Various further properties of these Chebyshev polynomials for
../src/ch-3.tex:2850:be made explicit in terms of the Chebyshev polynomials. Various
../src/ch-3.tex:3004:{\bf a]} \enskip Show that there is a polynomial $q_{2n} \in {\Cal P}^c_{2n}$
../src/ch-3.tex:3088:\subhead{E.5 \enskip Chebyshev Polynomials for ${\Cal P}_n (a_1, a_2,\ldots, a_n)$
../src/ch-4.tex:7:Chebyshev polynomials. This is the
../src/ch-4.tex:22:Much of the utility of polynomials stems from the fact that all
../src/ch-4.tex:30:we define, as in Section 3.3, the Chebyshev polynomials
../src/ch-4.tex:40:For a sequence $(T_n)_{n=0}^\infty $ of Chebyshev polynomials associated with a fixed Markov
../src/ch-4.tex:50:associated Chebyshev polynomials.
../src/ch-4.tex:57:where $M_n$ is the mesh of the associated Chebyshev polynomials. \endproclaim
../src/ch-4.tex:99:Chebyshev polynomial $T_n.$ Suppose each $g_i \in C^1[a, b]$ and
../src/ch-4.tex:206:\proclaim{Corollary 4.1.3} The polynomials are dense in
../src/ch-4.tex:211:associated Chebyshev polynomials are just the usual Chebyshev
../src/ch-4.tex:212:polynomials $T_n$ (see Section 2.1) and
../src/ch-4.tex:255:incomplete polynomials, where the zeros of the Chebyshev polynomials
../src/ch-4.tex:262:approximated by polynomials with real coefficients.
../src/ch-4.tex:266:approximated by polynomials with complex coefficients.}
../src/ch-4.tex:289:uniformly approximated by polynomials on $[-1,1].$
../src/ch-4.tex:324:can be uniformly approximated by polynomials with complex coefficients.}
../src/ch-4.tex:348:${\Cal P} := \cup^\infty_{n=0} {\Cal P}_n$ of all polynomials with real coefficients is a
../src/ch-4.tex:352:\noindent{\bf b]} \enskip Observe that the real polynomials in $x^2$
../src/ch-4.tex:362:\demo {Proof} If $f \in {\Cal A},$ then $p(f) \in {\Cal A}$ for any polynomial
../src/ch-4.tex:415:convergence of special polynomials, such as the Bernstein
../src/ch-4.tex:416:polynomials.
../src/ch-4.tex:466:\subhead {E.4 \enskip Bernstein Polynomials} \endsubhead \enskip
../src/ch-4.tex:467:The $n$th {\it Bernstein polynomial} for a function
../src/ch-4.tex:482:For more on Bernstein polynomials, see Lorentz [86b].
../src/ch-4.tex:536:of all real trigonometric polynomials is dense in $C(K),$ the set of
../src/ch-4.tex:539:polynomials is dense in $C(K),$ the set of all
../src/ch-4.tex:588:{\bf g] \enskip The Norm of Operators that Preserve Trigonometric Polynomials.} \enskip
../src/ch-4.tex:598:\subhead {E.6 \enskip Polynomials in $x^{\lambda_n}$} \endsubhead \enskip
../src/ch-4.tex:608:Chebyshev polynomial for ${\Cal P}_n(\lambda_n)$ on
../src/ch-4.tex:620:Chebyshev polynomial of degree $n$ as defined by (2.1.1).
../src/ch-4.tex:698:of all real algebraic polynomials are dense in $L_p[a,b].$}
../src/ch-4.tex:705:\subhead{E.8 \enskip Density of Polynomials with Integer Coefficients}\endsubhead
../src/ch-4.tex:710:polynomial $p$ with integer coefficients such that
../src/ch-4.tex:725:Note that $\widetilde{B}_n(f)$ is a polynomial with integer
../src/ch-4.tex:734:contain an integer. Show that polynomials with integer
../src/ch-4.tex:744:a polynomial $p$ with integer coefficients such that
../src/ch-4.tex:748:polynomials with integer coefficients.
../src/ch-4.tex:750:The existence of such a polynomial $p$ follows from the identity
../src/ch-4.tex:760:polynomials with
../src/ch-4.tex:775:$2 \pi.$ Then the set ${\Cal P}^c$ of all polynomials with complex coefficients is
../src/ch-4.tex:788:can deduce that there are monic polynomials
../src/ch-4.tex:912:and study $L^*_n$, the $n$th orthonormal M\"untz-Legendre polynomial
../src/ch-4.tex:999:of M\"untz polynomials from a given M\"untz space
../src/ch-4.tex:1311:{\bf b] \enskip The Closure of M\"untz Polynomials.} \enskip Let $f \in C[0,1]$,
../src/ch-4.tex:1312:and suppose there exist M\"untz polynomials $p_n \in \text{span}\{x^{\lambda_i}\}^\infty_{i=-\infty}$ of the form
../src/ch-4.tex:1966:orthonormal M\"untz-Legendre polynomials on $[0,1].$
../src/ch-4.tex:2093:Chebyshev polynomial
../src/ch-4.tex:2452:\subhead {E.22 \enskip The Zeros of the Chebyshev Polynomials in Nondense M\"untz
../src/ch-4.tex:2457:be the Chebyshev polynomials for $\text{span}\{x^{\lambda_0}, \ldots, x^{\lambda_n}\}$
../src/ch-4.tex:2465:property of the zeros of the Chebyshev polynomials $T_n$. \qed
../src/ch-4.tex:2472:associated Chebyshev polynomials. The principal result of this section is a characterization of
../src/ch-4.tex:2494:Chebyshev polynomial is close to extremal for
../src/ch-4.tex:2504:polynomial. Then
../src/ch-4.tex:2564:associated Chebyshev polynomials,
../src/ch-4.tex:2590:The Chebyshev polynomials $T_n$ (of the first kind) and $U_n$ (of the second kind) for
../src/ch-4.tex:2643:associated Chebyshev polynomials $(T_n)$ has a subsequence $(T_{n_i})$ with no
../src/ch-4.tex:2822:be the Chebyshev polynomial for
../src/ch-4.tex:2839:$0 < n_1 < n_2 < \cdots$ such that none of the Chebyshev polynomials
../src/ch-4.tex:2889:of associated Chebyshev polynomials on $[a,b]$.
../src/ch-4.tex:2906:be the $n$th Chebyshev polynomial on $[0,2].$
../src/ch-4.tex:2945:be the Chebyshev polynomial for $\text{span}\{\varphi_0, \varphi_1, \ldots, \varphi_n\}$
../src/ch-4.tex:3051:polynomial approximations, the existence of $\epsilon$-zoomers
../src/ch-4.tex:3117:the M\"untz polynomials themselves associated with $(\lambda_i)^\infty_{i=0}$
../src/ch-4.tex:3154:``extra'' multiplication of M\"untz polynomials should not carry the
../src/ch-4.tex:3156:it is shown that products $pq$ of M\"untz polynomials from nondense
../src/ch-5.tex:4:The classical inequalities for algebraic and trigonometric polynomials
../src/ch-5.tex:8:is concerned with the size of factors of polynomials.
../src/ch-5.tex:12:\titleb{5.1 Classical Polynomial Inequalities}
../src/ch-5.tex:20:generalized nonnegative polynomials (discussed in Appendix 4)
../src/ch-5.tex:24:nonnegative polynomials, where simple density arguments do not
../src/ch-5.tex:32:polynomial of degree $n$ defined by $(2.1.1)$. Equality holds if and only
../src/ch-5.tex:37:establishes that an extremal polynomial is of the required form.
../src/ch-5.tex:60:$p^*(\zeta) = \|p^*\|$. Then the polynomials
../src/ch-5.tex:72:$q_j \in {\Cal P}_n(s)$, $j=1,2$, are extremal polynomials attaining their
../src/ch-5.tex:100:The polynomial $q(x) := p_1(x+h)p_2(x)$ with $0 < h < \zeta_2-\zeta_3$
../src/ch-5.tex:118:By E.2, among all polynomials $p \in {\Cal P}_n$ with $\|p\| \leq 1$, the
../src/ch-5.tex:119:Chebyshev polynomial $T_n$ increases fastest for $x >1.$ Hence,
../src/ch-5.tex:120:by a linear transformation, we see that the four polynomials
../src/ch-5.tex:122:polynomials. In particular,
../src/ch-5.tex:126:trigonometric polynomials. Throughout this section, as before, $K: = {\Bbb R}$
../src/ch-5.tex:147:$m(\Omega) \leq s.$ Then the polynomial $p \in {\Cal P}_n$ defined by
../src/ch-5.tex:162:The polynomial
../src/ch-5.tex:192:polynomial in terms of its maximum modulus on the period $K.$
../src/ch-5.tex:219:contradiction. To find all the extremal polynomials, see the
../src/ch-5.tex:244:The above corollary implies the following algebraic polynomial version of
../src/ch-5.tex:260:get the algebraic polynomial case of Bernstein's inequality
../src/ch-5.tex:289:polynomial
../src/ch-5.tex:302:polynomial of a $p \in {\Cal P}_{n-1}$ with nodes $x_1, x_2,\ldots,
../src/ch-5.tex:327: versions of these basic polynomial inequalities also appear in the literature.
../src/ch-5.tex:345:Chebyshev polynomial of degree $n$ as in (2.1.1).
../src/ch-5.tex:354:Throughout the exercises $T_n$ denotes the Chebyshev polynomial of degree $n$
../src/ch-5.tex:376:with $\lambda := T_n(y) p(y)^{-1} \in [-1,1],$ the polynomial
../src/ch-5.tex:382:\subhead {E.3 \enskip Trigonometric Chebyshev Polynomials on Subintervals of $K$}\endsubhead
../src/ch-5.tex:459:Chebyshev polynomial $T_n,$ that is,
../src/ch-5.tex:468:is the Chebyshev polynomial of the second kind defined in E.10 of Section 2.1.
../src/ch-5.tex:485:$m$ zeros by a Hermite interpolation polynomial of degree at
../src/ch-5.tex:487:the identically zero polynomial. The formula for the remainder term
../src/ch-5.tex:488:of the Hermite interpolation polynomial and
../src/ch-5.tex:531:and only if $p = T_n$, where $T_n$ is the Chebyshev polynomial of
../src/ch-5.tex:540:\subhead {E.14 \enskip A Markov-Type Inequality for Trigonometric Polynomials on
../src/ch-5.tex:608:\subhead {E.17 \enskip Growth of Polynomials in the Complex Plane} \endsubhead \enskip Let
../src/ch-5.tex:749:for trigonometric polynomials on an interval shorter than the period.
../src/ch-5.tex:781:is the Chebyshev polynomial for ${\Cal T}_n$ on
../src/ch-5.tex:880:known for $E_{\tau}.$ Since a trigonometric polynomial of degree
../src/ch-5.tex:883:polynomials. More on various inequalities for entire functions
../src/ch-5.tex:957:\subhead{E.22 \enskip The Interval where the Sup Norm of a Weighted Polynomial Lives}
../src/ch-5.tex:988:explicit form (2.1.1) of the Chebyshev polynomial $T_n$, we can deduce that
../src/ch-5.tex:1027:Let $T_n$ be the Chebyshev polynomial of degree $n$ defined by (2.1.1).
../src/ch-5.tex:1084:of the above. E.2 d] extends the following result to polynomials with
../src/ch-5.tex:1091:where $T_n $ is the Chebyshev polynomial of
../src/ch-5.tex:1152:Let $h(z)$ be another polynomial with the same leading coefficient
../src/ch-5.tex:1210:imposed on the interpolating polynomial $q$ (with $n$ replaced
../src/ch-5.tex:1293:\subhead {E.1 \enskip A Property of Chebyshev Polynomials} \endsubhead \enskip
../src/ch-5.tex:1319:\noindent {\bf b]} \enskip Show that the Chebyshev polynomials
../src/ch-5.tex:1417:is solved by the Chebyshev polynomial $T_n,$ hence
../src/ch-5.tex:1568:polynomial $T_n$ of degree $n$ defined by (2.1.1), and the factor $q \in {\Cal P}^c_m$ is chosen
../src/ch-5.tex:1604:suppose $p=qr$ for some polynomials $q$ and $r.$ Then
../src/ch-5.tex:1609:and equality holds when $p$ is the Chebyshev polynomial $T_n$ of degree $n$, and the
../src/ch-5.tex:1623:We proceed to show that there are extremal polynomials $q \in {\Cal P}^c_m$ and
../src/ch-5.tex:1624:$r \in {\Cal P}^c_{n-m}$ such that $p:= qr$ is the Chebyshev polynomial $T_n$ of degree
../src/ch-5.tex:1638:First we show that there exist extremal polynomials
../src/ch-5.tex:1657:and $\tilde r \in {\Cal P}^c_{n-m}$ are extremal polynomials for
../src/ch-5.tex:1661:are extremal polynomials for which (5.3.5) holds. Then
../src/ch-5.tex:1662:there are extremal polynomials $\tilde{q} \in {\Cal P}^c_m$ and
../src/ch-5.tex:1672:polynomials for which (5.3.5) holds, and all the zeros of both $\tilde{q}$ and $\tilde{r}$ are
../src/ch-5.tex:1677:extremal polynomials having only real zeros for
../src/ch-5.tex:1679:extremal polynomials $\tilde{q} \in {\Cal P}^c_m$ and $\tilde{r} \in
../src/ch-5.tex:1690:extremal polynomials having all their zeros in $[-1,1]$ for
../src/ch-5.tex:1694:are extremal polynomials having all their zeros in $[-1,1]$ for which
../src/ch-5.tex:1700:which are also extremal polynomials having all
../src/ch-5.tex:1703:It is now clear that if $q$ and $r$ are extremal polynomials with the above
../src/ch-5.tex:1707:$q(\alpha) = r(\beta) =0,$ then the polynomials
../src/ch-5.tex:1718:So now we have extremal polynomials $q \in {\Cal P}^c_m$ and $r \in {\Cal P}^c_{n-m}$ of the form
../src/ch-5.tex:1742:polynomial $\pm T_n$ defined by (2.1.1) since $ \pm T_n$ are the only
../src/ch-5.tex:1743:polynomials of degree at most $n$ that equioscillate $n+1$ times on $[-1,1]$ with uniform norm $1.$
../src/ch-5.tex:1753:the polynomials
../src/ch-5.tex:1784:polynomials
../src/ch-5.tex:1810:for every $\beta >0.$ Equality holds if $p$ is the Chebyshev polynomial
../src/ch-5.tex:1853:Equality holds if $p$ is the Chebyshev polynomial $T_{n, \beta}$ of degree
../src/ch-5.tex:1871:for every $\beta >0$. Equality holds if $p$ is the Chebyshev polynomial
../src/ch-5.tex:2014:extremal polynomials $p \in {\Cal P}^c_n$ and $q \in {\Cal P}^c_m$ such that all
../src/ch-5.tex:2021:the Chebyshev polynomial on $[- \beta, \beta]$ normalized to be
../src/ch-5.tex:2041:\subhead {E.3 \enskip A Version of Theorem 5.3.10 for Complex Polynomials} \endsubhead \enskip
../src/ch-6.tex:8:polynomials and exponential sums. The second section provides inequalities,
../src/ch-6.tex:15:inequality for M\"untz polynomials. This simplification allows us to prove
../src/ch-6.tex:17:of Section 3.4 on orthonormal M\"untz-Legendre polynomials,
../src/ch-6.tex:18:we prove an $L_2$ version of Newman's inequality for M\"untz polynomials with
../src/ch-6.tex:20:Nikolskii-type inequalities for M\"untz polynomials are studied.
../src/ch-6.tex:21:The exercises treat a number of other inequalities for M\"untz polynomials
../src/ch-6.tex:318:M\"untz-Legendre polynomials on $[0,1].$ Using the recurrence
../src/ch-6.tex:459:polynomial on $[0,1].$ Show that if there exists a $q \in
../src/ch-6.tex:497:{\it Hint:} Show that there is an extremal polynomial ${\widetilde p}$
../src/ch-6.tex:538:{\it Outline.} Let $Q_n$ be the Chebyshev polynomial $T_n$ transformed linearly from
../src/ch-6.tex:746:where $T_{2n-1}$ denotes the Chebyshev polynomial of degree $2n-1$ defined by
../src/ch-6.tex:768:$n \in {\Bbb N}$ is odd. Let $T_n$ be the Chebyshev polynomial of degree $n$ defined
../src/ch-6.tex:823:M\"untz polynomials $0 \neq p$ of the form
../src/ch-6.tex:830:be the Chebyshev polynomial for $M_n(\Lambda_\epsilon)$ on $[a,b].$ Use E.3 b] and
../src/ch-6.tex:949:\subhead{E.7 \enskip Nikolskii-Type Inequality for M\"untz Polynomials}\endsubhead \enskip Suppose that
../src/ch-6.tex:987:M\"untz-Legendre polynomial on $[0,1].$ Let
../src/ch-6.tex:1033:\subhead{E.9 \enskip On the Interval Where the Sup Norm of a M\"untz Polynomial Lives}\endsubhead
../src/ch-6.tex:1425:the Chebyshev polynomials
../src/ch-6.tex:1581:\subhead {E.2 \enskip On the Smallest Zero of Chebyshev Polynomials in Nondense
../src/ch-6.tex:1617:\subhead {E.4 \enskip A Lexicographic Property of Chebyshev Polynomials in
../src/ch-6.tex:1675:\noindent {\bf a]} \enskip Show that the Chebyshev polynomials
../src/ch-6.tex:1728:The following exercise constructs quasi-Chebyshev polynomials
../src/ch-6.tex:1731:\subhead {E.7 \enskip Quasi-Chebyshev Polynomials in Very Lacunary M\"untz Spaces} \endsubhead
../src/ch-6.tex:1864:\subhead {E.10 \enskip Polynomials in $x^{\lambda_n}$} \endsubhead \enskip
../src/ch-7.tex:9:as are inequalities for self-reciprocal polynomials.
../src/ch-7.tex:11:polynomials and rational functions.
../src/ch-7.tex:16:Sharp extensions of most of the polynomial inequalities of Section 5.1 are established
../src/ch-7.tex:23:polynomials are presented in the exercises.
../src/ch-7.tex:51:The Chebyshev polynomials $\tilde{T}_n$, $\tilde{U}_n$, and
../src/ch-7.tex:88:for the Chebyshev polynomials for these classes ${\Cal T}_n (a_1, a_2, \ldots, a_{2n} ; K).$
../src/ch-7.tex:147:To formulate our next theorem we introduce some notation. For a polynomial
../src/ch-7.tex:316:Bernstein's classical polynomial inequalities discussed in Section 5.1
../src/ch-7.tex:400:is a real trigonometric polynomial of degree at most $n$ (see E.5 a]). Also
../src/ch-7.tex:456:Bernstein-type inequality on the unit disk for polynomials $p \in
../src/ch-7.tex:459:for such polynomials.
../src/ch-7.tex:492:one of the Chebyshev polynomials for ${\Cal T}_{n,a}$ defined in Theorem 3.5.3
../src/ch-7.tex:574:trigonometric polynomial of degree at most $n.$
../src/ch-7.tex:708:and $T_n$ is the Chebyshev polynomial for ${\Cal P}_n(a_1, a_2, \ldots, a_n; [-1,1])$
../src/ch-7.tex:736:Chebyshev polynomial (of the second kind) for ${\Cal P}_n(a_1, a_2, \ldots, a_n; [-1,1])$
../src/ch-7.tex:801:Let $\text{\rm SR}^c_n$ denote the set of all self-reciprocal polynomials $p \in {\Cal P}^c_n$ satisfying
../src/ch-7.tex:803:Let $\text{SR}_n$ denote the set of all real self-reciprocal polynomials
../src/ch-7.tex:805:For a polynomial $p \in {\Cal P}^c_n$
../src/ch-7.tex:811:Let $\text{\rm ASR}^c_n$ denote the set of all antiself-reciprocal polynomials
../src/ch-7.tex:814:Let $\text{ASR}_n$ denote the set of all real antiself-reciprocal polynomials
../src/ch-7.tex:816:Let $\text{\rm ASR}_n : =\text{\rm ASR}^c_n \cap {\Cal P}_n$. For a polynomial $p \in {\Cal P}^c_n$
../src/ch-7.tex:916:\subhead{E.12 \enskip Quasi-Chebyshev Polynomials for $\text{\rm SR}_{2n}$ and $\text{\rm ASR}_{2n}$
../src/ch-7.tex:1015:By using the quasi Chebyshev polynomials for $\text{SR}_{2n}$ and $\text{ASR}_{2n}$,
../src/ch-7.tex:1032:extension from the polynomial case to the rational case.
../src/ch-7.tex:1080:In order to extend Theorem 7.2.1 to arbitrary polynomials we
../src/ch-7.tex:1101:By Lemma 7.2.3 we can find polynomials $q \in {\Cal P}_{2n}$ and $s \in {\Cal P}_{2n}$ such that
../src/ch-7.tex:1167:of a monic polynomial $p \in {\Cal P}^c_n$ of the form
../src/ch-7.tex:1171:problems concerning the lemniscate of monic polynomials. One in particular,
../src/ch-7.tex:1173:that for monic polynomials $p \in{\Cal P}^c_n,$ the length of the boundary of
../src/ch-7.tex:1178:of $E(p)$ for a monic polynomial $p \in{\Cal P}^c_n$ is always at least $2$.
../src/ch-7.tex:1182:\subhead {E.1 \enskip Polynomials as Sums of Polynomials
../src/ch-7.tex:1197:{\it Hint:} Let $p$ be a polynomial of degree $2n$ that is
../src/ch-7.tex:1199:polynomial for the Chebyshev system
../src/ch-7.tex:1205:are the required polynomials. Use a similar construction for part b]. \qed
../src/ch-7.tex:1209:denote the polynomial of degree at most $n$ with nonnegative
../src/ch-7.tex:1293:with equality only for the Chebyshev polynomial of degree $n$
../src/ch-7.tex:1298:5.1, show that the Chebyshev polynomial transformed to an interval of length $4$
../src/ch-7.tex:1401:that $L = {\Bbb R}.$ Now observe that there is a polynomial $P(x, y)$ of
../src/ch-7.tex:1492:Let $p \in {\Cal P}^c_n$ be an arbitrary monic polynomial of degree $n,$ that is,
../src/ch-7.tex:1525:since $p \in {\Cal P}^c_n$ is a monic polynomial of degree $n,$
../src/ch-7.tex:1555:Suppose $p$ is a monic polynomial with complex coefficients. As before, let
../src/ch-a1.tex:5:polynomials and rational functions and discusses some of the complexity issues.
../src/ch-a1.tex:9:\titleb{Algorithms and Computational Concerns} Polynomials
../src/ch-a1.tex:13:finite polynomial or rational approximation or truncation. This often
../src/ch-a1.tex:50:polynomials of degree $n,$ the measure will often be $n.$ The
../src/ch-a1.tex:54:required to evaluate the polynomial at a point. (So, for example,
../src/ch-a1.tex:64:polynomial ring). Our cases are fairly simple, and the measures
../src/ch-a1.tex:77:recursively. For example, addition of two polynomials of degree
../src/ch-a1.tex:78:at most $2n$ reduces to two additions of polynomials of degree at most $n$, plus
../src/ch-a1.tex:88:functions. Here $n$ is the maximum degree of the polynomials $p$ and
../src/ch-a1.tex:104:These are the complexity functions for polynomial
../src/ch-a1.tex:105:addition, polynomial multiplication, polynomial evaluation at a
../src/ch-a1.tex:106:single point, and polynomial evaluation at $n$ points, respectively. The
../src/ch-a1.tex:164:the unique polynomial
../src/ch-a1.tex:171:coefficients of a polynomial $p_n$ of degree at most $n,$ calculate the
../src/ch-a1.tex:200:where $r$ and $q$ are both polynomials of degree at most $2^{m-1} -1.$
../src/ch-a1.tex:209:additions and multiplications required to evaluate a polynomial
../src/ch-a1.tex:249:As an application we construct a fast polynomial multiplication.
../src/ch-a1.tex:251:\noindent {\bf b] \enskip Fast Polynomial Multiplication.} \enskip Suppose the polynomials
../src/ch-a1.tex:273:polynomial multiplication. The only known lower bound is the trivial
../src/ch-a1.tex:274:one $O(n).$ In parallel (on a PRAM) polynomial multiplication
../src/ch-a1.tex:283:\noindent {\bf a] \enskip Fast Polynomial Division.} \enskip For polynomials $p$ of
../src/ch-a1.tex:285:polynomials $u$ and $r$ with $\deg(r) < \deg(q)$ \linebreak such that
../src/ch-a1.tex:309:performed by using an FFT-based polynomial multiplication and it needs
../src/ch-a1.tex:327:\noindent {\bf c] \enskip Fast Polynomial Expansion.} \enskip Given
../src/ch-a1.tex:338:\noindent {\bf d] \enskip Fast Polynomial Expansion at Arbitrary Points.}
../src/ch-a1.tex:339:\enskip Given a polynomial $p$ of degree at most $n,$ and $n+1$ distinct
../src/ch-a1.tex:400:Much further material on the complexity of polynomial operations
../src/ch-a1.tex:459:problem for polynomials, but in a general setting
../src/ch-a1.tex:512:For a polynomial $f,$
../src/ch-a1.tex:519:\noindent {\bf b] \enskip Finding All Zeros of a Polynomial.}
../src/ch-a1.tex:532:root of $p,$ where $p$ is a polynomial of degree $n$ with $n$
../src/ch-a1.tex:553:computes a zero of a polynomial with an error $<2^{-b}$ in $O(n\log b \log n)$
../src/ch-a1.tex:586:be nonzero polynomials. Define polynomials $p_0, p_1, \ldots, p_m$ and
../src/ch-a1.tex:600:\noindent {\bf c]} \enskip Let $p_0, p_1, \ldots, p_m$ be the polynomials generated
../src/ch-a1.tex:609:\noindent {\bf d]} \enskip Show that for real polynomials $p_0$
../src/ch-a1.tex:614:polynomials $p_i$ are generated by the Euclidean algorithm as in part b]. (As before, by a sign
../src/ch-a1.tex:621:if $p_i(x)=0$ for some $i.$ By continuity of the polynomials $p_i,$ and by part c],
../src/ch-a1.tex:636:$p$ is a real polynomial and $p(a)p(b) \ne 0.$ Then $I^b_a (p^\prime/p)$
../src/ch-a1.tex:651:real polynomial on the real line equals
../src/ch-a1.tex:659:\enskip Suppose that the monic polynomial $p \in {\Cal P}^c_n$ has (exactly) $k$ real
../src/ch-a1.tex:731:\subhead {E.8 \enskip Computing General Chebyshev Polynomials}
../src/ch-a1.tex:735:compute the associated Chebyshev polynomial $T_n$? That is, how does one find the
../src/ch-a2.tex:22:and related orthogonal polynomials; see, for example, Borwein [91a], [92] or
../src/ch-a2.tex:34:This leads to orthogonal functions that generalize the Legendre polynomials
../src/ch-a2.tex:37:with polynomials $p_n, q_n \in {\Cal P}_n$ of degree $n.$ Legendre polynomials are
../src/ch-a2.tex:122:immediately gives that $B_n(x)$ has a zero at 1, so $B_n(x)/(1-x)$ is a polynomial,
../src/ch-a2.tex:138:{\bf c]} \enskip $B_n(x)/(1-x)$ is a polynomial with all
../src/ch-a2.tex:165:the polynomial $d_nB_n(x)$ has integer coefficients.
../src/ch-a2.tex:185:The first term of the last expression is a polynomial multiple of $\log(xy),$ and the
../src/ch-a2.tex:186:last term is a polynomial. Both have degree $n-1$
../src/ch-a2.tex:191:One $d_n$ arises from each of the two integrations of a polynomial
../src/ch-a2.tex:226:$p_n$ and $q_n$ are polynomials of degree $n.$
../src/ch-a2.tex:405:be the $n$th Legendre polynomial on $[0,1].$
../src/ch-a3.tex:319:the extremal polynomial for the given inequality with the help of Theorem A.3.3.
../src/ch-a3.tex:347:where $T_n$ is the Chebyshev polynomial of degree $n$ defined
../src/ch-a3.tex:393:{\bf i]} \enskip Find all extremal polynomials in parts a] to e] and h].
../src/ch-a4.tex:1:\titlea{A4 Inequalities for Generalized Polynomials in $L_p$}
../src/ch-a4.tex:4:Many inequalities for generalized polynomials are given
../src/ch-a4.tex:10:\titleb{Inequalities for Generalized Polynomials in $L_p$}
../src/ch-a4.tex:12:Generalized (nonnegative) polynomials are defined by (A.4.1) and (A.4.3).
../src/ch-a4.tex:25:(algebraic) polynomial of (generalized) degree
../src/ch-a4.tex:27:The set of all generalized nonnegative algebraic polynomials of
../src/ch-a4.tex:35:trigonometric polynomial of degree
../src/ch-a4.tex:37:The set of all generalized nonnegative trigonometric polynomials of
../src/ch-a4.tex:40:we will study generalized nonnegative polynomials
../src/ch-a4.tex:43:algebraic or trigonometric polynomial, respectively. Note that the classes
../src/ch-a4.tex:57:Section 5.1 to generalized nonnegative polynomials. In
../src/ch-a4.tex:307:inequalities for generalized nonnegative polynomials.
../src/ch-a4.tex:349:polynomial of degree $2n$ defined by (2.1.1). By E.11 there exists an absolute constant
../src/ch-a4.tex:408:generalized nonnegative polynomials in $L_p.$ In the proofs we
../src/ch-a4.tex:412:inequalities for generalized nonnegative polynomials. First we
../src/ch-a4.tex:596:polynomials $f \in {\Cal P}_n$ and for arbitrary $q \in (0,\infty),$
../src/ch-a4.tex:598:An early version of Markov's inequality in $L_p$ for ordinary polynomials is
../src/ch-a4.tex:604:for orthonormal polynomials and
../src/ch-a4.tex:606:functions. Further applications in the theory of orthogonal polynomials
../src/ch-a4.tex:613:polynomials was first established by Zygmund [77]
../src/ch-a4.tex:652:$f$ and $w$ are generalized nonnegative polynomials. In these
../src/ch-a4.tex:679:\subhead {E.1 \enskip Another Representation of Generalized Nonnegative Polynomials}\endsubhead
../src/ch-a4.tex:709:\subhead {E.3 \enskip Generalized Nonnegative Polynomials with Rational Exponents} \endsubhead
../src/ch-a4.tex:719:polynomial $T_n.$
../src/ch-a4.tex:763:\subhead {E.8 \enskip Nonnegative Trigonometric Polynomials} \endsubhead \enskip Part a]
../src/ch-a4.tex:808:\noindent{\it Hint:} Use the explicit formula (2.1.1) for the Chebyshev polynomial $T_n$. \qed
../src/ch-a4.tex:898:{\bf a]} \enskip Show that there exists a sequence of polynomials
../src/ch-a4.tex:906:{\it Hint:} Study the Chebyshev polynomials $T_n$ transformed linearly from
../src/ch-a4.tex:911:polynomials $t_n \in {\Cal T}_n$ and an absolute constant $c >0$ such that
../src/ch-a4.tex:924:trigonometric polynomials $t_n \in {\Cal T}_n$ that shows the
../src/ch-a4.tex:932:a sequence of polynomials $Q_{n,p} \in {\Cal P}_n$ which shows the
../src/ch-a4.tex:936:Legendre polynomial on $[-1,1]$ (see E.5 of Section 2.3), and let
../src/ch-a4.tex:967:exists a sequence of polynomials $Q_{n,q} \in {\Cal P}_n$
../src/ch-a4.tex:989:there exists a sequence of trigonometric polynomials $t_{n,q} \in {\Cal T}_n$
../src/ch-a4.tex:1039:where $T_{2n}$ is the Chebyshev polynomial of degree $2n$ defined by (2.1.1),
../src/ch-a4.tex:1095:the polynomials
../src/ch-a4.tex:1097:where $T_n$ is the Chebyshev polynomial of degree
../src/ch-a5.tex:1:\titlea{A5 Inequalities for Polynomials \linebreak with Constraints}
../src/ch-a5.tex:2:\titlearunning{A5 Inequalities for Polynomials with Constraints}
../src/ch-a5.tex:4:This appendix deals with inequalities for constrained polynomials.
../src/ch-a5.tex:9:\titleb{Inequalities for Polynomials with Constraints}
../src/ch-a5.tex:10:%Inequalities for polynomials with constraints on their zeros or coefficients are studied in this section. The
../src/ch-a5.tex:11:%following classes of constrained polynomials are of special
../src/ch-a5.tex:169:polynomials
../src/ch-a5.tex:259:A Markov-type inequality for polynomials having
../src/ch-a5.tex:265:Polynomials with At Most $k$ Zeros in $D_r$)} Let $k \in {\Bbb N}$ and $r \in (0,1].$
../src/ch-a5.tex:284:inequality for every $p \in {\Cal P}_{n,0}, \enskip n \geq 2.$ The polynomials
../src/ch-a5.tex:312:gave polynomials $p_{n.k} \in {\Cal P}_{n,k}$ with only real
../src/ch-a5.tex:351:Polynomials} \endsubhead \enskip
../src/ch-a5.tex:385:Show that there exist polynomials $p_{n,k} \in
../src/ch-a5.tex:423:there exists a polynomial $Q \in {\Cal P}_{n,k}$ such that
../src/ch-a5.tex:429:show that the uniform limit of a sequence of polynomials from
../src/ch-a5.tex:439:polynomial
../src/ch-a5.tex:447:for every polynomial ${\Cal P}_{n,k}$ having all its zeros in
../src/ch-a5.tex:462:for every polynomial $p \in {\Cal P}_n$ having only real zeros
../src/ch-a5.tex:480:extremal polynomial of part a], and let
../src/ch-a5.tex:525:\ldots,n$, and $\delta \in (0,1],$ there exists a polynomial
../src/ch-a5.tex:531:$\pm1.$ Show that there is a polynomial $Q \in {\Cal P}_{n,k}$
../src/ch-a5.tex:545:Show that there exist polynomials $p_{n,k} \in {\Cal P}_{n,k}$ and
../src/ch-a5.tex:661:\noindent {\bf a]} \enskip Show that there exist polynomials
../src/ch-a5.tex:788:there exists a polynomial $Q \in {\Cal P}_{n,0}$ such that
../src/ch-a5.tex:809:[-1,1],$ there are polynomials $p_{n,m,y} \in {\Cal B}_n(-1,1)$
../src/ch-a5.tex:826:where $Q_{n,j,m}$ is a polynomial of degree $m$ with only real
../src/ch-a5.tex:857:Polynomials $p_{n,m,y}$
../src/ch-a5.tex:1119:\subhead {E.17 \enskip Markov-Type Inequality for Nonnegative Polynomials}
../src/ch-a5.tex:1128:\subhead {E.18 \enskip Markov's Inequality for Monotone Polynomials}
../src/ch-a5.tex:1130:inequality for monotone polynomials is not essentially better than for arbitrary
../src/ch-a5.tex:1131:polynomials. He proved that if $n$ is odd, then
../src/ch-a5.tex:1169:Show that there is an extremal polynomial ${\widetilde p}$ for the above inequality
../src/ch-a5.tex:1215:Weighted polynomial inequalities and their applications are beyond the scope of this
../src/ch-a5.tex:1225:polynomial approximation of the weight function is typically far more complicated.
../src/ch-a5.tex:1228:\endsubhead \enskip Part a] presents a simple weighted polynomial inequality
../src/ch-a5.tex:1263:classes of weight functions and constrained polynomials. \qed
../src/ch-a5.tex:1265:\subhead {E.20 \enskip Inequalities for Generalized Polynomials
../src/ch-a5.tex:1271:The polynomials $T_{n,k} \in {\Cal P}_{n,k}$ are defined by
../src/ch-a5.tex:1277:$[-1,1],$ and, to be precise, $T_{n,k}$ denotes the polynomial defined
../src/ch-a5.tex:1398:corresponding polynomial inequality for the classes
../src/ch-a5.tex:1445:{\bf b]} \enskip The polynomial $P(z) := z^n -1$ shows the sharpness
../src/ch-a5.tex:1454:The polynomials $p_n(x):=(x+1)^n$ show that up to the absolute constant
../src/ch-a5.tex:1456:the ``right'' Markov factor on $[-1,1]$ for polynomials of degree at most $n$
../src/ch-a5.tex:1466:Show also that there exist polynomials $p_n \in {\Cal P}_n^c$
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../src/ch-bib.tex:1383:\ref \by Olivier, P.F., \& A.O. Watt \paper Polynomials
../src/ch-bib.tex:1395:\ref \by Osval'd, P. \paper Some inequalities for trigonometric polynomials in
../src/ch-bib.tex:1402:and polynomials \jour SIAM Rev. \yr 1992 \vol 34 \pages 225--262
../src/ch-bib.tex:1406:\ref \by Peherstorfer, F. \paper On Tchebycheff polynomials on disjoint
../src/ch-bib.tex:1446:\ref \by Pommerenke, Ch. \paper On some metric properties of polynomials with real zeros
../src/ch-bib.tex:1452:\ref \by Pommerenke, Ch. \paper On the derivative of a polynomial
../src/ch-bib.tex:1458:\paper On some metric properties of complex polynomials \jour Michigan Math. J.
../src/ch-bib.tex:1463:\ref \by Pommerenke, Ch. \paper On some metric properties of polynomials
../src/ch-bib.tex:1474:\ref \by Protapov, M.K. \paper Some inequalities for polynomials and their
../src/ch-bib.tex:1480:\ref \by Rahman, Q.I. \paper On extremal properties of the derivatives of polynomials
../src/ch-bib.tex:1486:polynomials and rational functions \jour Amer. J. Math. \vol 113 \yr 1991
../src/ch-bib.tex:1498:\ref \by Rahman, Q.I., \& G. Schmeisser \paper Some inequalities for polynomials with
../src/ch-bib.tex:1531:\ref \by Reznick, B. \paper An inequality for products of polynomials
../src/ch-bib.tex:1549:\ref \by Rivlin, T.J. \book Chebyshev Polynomials, {\rm 2nd ed.}
../src/ch-bib.tex:1559:\ref \by Rogosinski, W.W. \paper Extremum problems for polynomials
../src/ch-bib.tex:1560:and trigonometric polynomials \jour J. London Math. Soc. \publ \publaddr \vol 29
../src/ch-bib.tex:1587:mean estimates for algebraic and trigonometric polynomials with restricted zeros
../src/ch-bib.tex:1598:polynomials \jour Internat. J. Math. Math. Sci. \vol 1 \yr 1978 \pages 407--420
../src/ch-bib.tex:1603:polynomials \jour Math. Z. \vol 177 \yr 1981 \pages 297--314
../src/ch-bib.tex:1614:V. Markoff for derivatives of polynomials \jour Bull. Amer. Math. Soc. \publ \publaddr \vol 44
../src/ch-bib.tex:1619:\ref \by Scheick, J.T. \paper Inequalities for derivatives of polynomials
../src/ch-bib.tex:1630:multiplication and division of polynomials with complex coefficients
../src/ch-bib.tex:1668:\ref \by Stahl, H., \& V. Totik \book General Orthogonal Polynomials
../src/ch-bib.tex:1691:derivative of a polynomial with real zeros \jour in: Functional Analysis and Approximation,
../src/ch-bib.tex:1697:polynomials having real zeros \jour in: Approximation Theory
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../src/ch-bib.tex:1822:Enestr\"om-Kakeya functional for polynomials
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../src/ch-bib.tex:1841:inequalities for derivatives of trigonometric polynomials on an interval shorter than the period
../src/ch-bib.tex:1847:\ref \by Walker, P. \paper Separation of the zeros of polynomials \jour
../src/ch-bib.tex:1882:\ref \by Werner, W. \book On the Simultaneous Determination of Polynomial
../src/ch-gloss.tex:43:the $n$th M\"untz-Legendre polynomial is:
../src/ch-gloss.tex:58:is the $n$th orthonormal M\"untz-Legendre polynomial.
../src/ch-gloss.tex:101:\subhead {Generalized Polynomials}\endsubhead
../src/ch-gloss.tex:108:(algebraic) polynomial of (generalized) degree
../src/ch-gloss.tex:110:The set of all generalized nonnegative algebraic polynomials of
../src/ch-gloss.tex:118:trigonometric polynomial of degree
../src/ch-gloss.tex:120:The set of all generalized nonnegative trigonometric polynomials of
../src/ch-gloss.tex:123:\subhead {Constrained Polynomials}\endsubhead
../src/ch-gloss.tex:126:The following classes of polynomials with constraints appear in Appendix 5:
../src/ch-index.tex:109:\sub fast polynomial division, 362
../src/ch-index.tex:110:\sub fast polynomial expansion, 363
../src/ch-index.tex:111:\sub fast polynomial multiplication, 361
../src/ch-index.tex:112:\sub for Chebyshev polynomials, 371
../src/ch-index.tex:114:\sub for polynomial evaluations, 363
../src/ch-index.tex:119:\sub zero finding for polynomials, 366--370
../src/ch-index.tex:122:Apolar polynomials, 23, 24, 25
../src/ch-index.tex:123:Arc length of algebraic polynomials, 31
../src/ch-index.tex:124:Arc length of trigonometric polynomials, 35
../src/ch-index.tex:128:Bernstein polynomials, 163--164
../src/ch-index.tex:134:\sub for trigonometric polynomials, 232
../src/ch-index.tex:138:\sub for constrained polynomials, 420--447
../src/ch-index.tex:141:\sub for generalized polynomials, 392--416
../src/ch-index.tex:142:\sub for generalized polynomials in $L_p$, 401--417
../src/ch-index.tex:145:\sub for polynomials, 232--233, 390
../src/ch-index.tex:146:\sub for polynomials in $L_p$, 235, 390, 401--417
../src/ch-index.tex:152:\sub for self-reciprocal polynomials, 339
../src/ch-index.tex:153:\sub for trigonometric polynomials, 232
../src/ch-index.tex:181:Chebyshev polynomials
../src/ch-index.tex:217:\sub for polynomials in special bases, 124
../src/ch-index.tex:224:Constrained polynomials, 417--447
../src/ch-index.tex:238:\sub of M\"untz polynomials, 171--205
../src/ch-index.tex:240:\sub of polynomials, 154--170
../src/ch-index.tex:247:Division of polynomials, 15, 362
../src/ch-index.tex:255:Exponential sums; {\it see} M\"untz polynomials
../src/ch-index.tex:274:Fekete polynomial, 38
../src/ch-index.tex:286:Gegenbauer polynomials, 65
../src/ch-index.tex:287:Generalized polynomials
../src/ch-index.tex:308:Hermite polynomials, 57
../src/ch-index.tex:326:\sub for M\"untz polynomials; {\it see} M\"untz polynomials
../src/ch-index.tex:340:Integer-valued polynomials, 10
../src/ch-index.tex:346:Jacobi polynomials, 57, 63
../src/ch-index.tex:358:\sub quasi-Chebyshev polynomials, 316
../src/ch-index.tex:360:Laguerre polynomials, 57, 66, 130
../src/ch-index.tex:367:Legendre polynomials, 57
../src/ch-index.tex:370:\sub for M\"untz polynomials, 120, 314
../src/ch-index.tex:371:\sub for M\"untz-Legendre polynomials, 136
../src/ch-index.tex:376:\sub for polynomials, 86
../src/ch-index.tex:377:\sub for trigonometric polynomials, 89
../src/ch-index.tex:392:\sub for constrained polynomials, 417--447
../src/ch-index.tex:393:\sub for constrained polynomials in $L_p$, 422, 428--429
../src/ch-index.tex:395:\sub for generalized polynomials, 399--407, 445
../src/ch-index.tex:396:\sub for generalized polynomials in $L_p$, 401--407
../src/ch-index.tex:398:\sub for monotone polynomials, 439--441
../src/ch-index.tex:399:\sub for M\"untz polynomials, 276--279, 287--288
../src/ch-index.tex:400:\sub for M\"untz polynomials in $L_p$, 279--280
../src/ch-index.tex:401:\sub for nonnegative polynomials, 420, 439
../src/ch-index.tex:403:\sub for self-reciprocal polynomials, 339
../src/ch-index.tex:404:\sub for trigonometric polynomials on subintervals, 242--245
../src/ch-index.tex:412:\sub for polynomials, 345--346
../src/ch-index.tex:420:Multiplication of polynomials, 361
../src/ch-index.tex:421:M\"untz polynomials
../src/ch-index.tex:441:M\"untz-Legendre polynomials, 125--138
../src/ch-index.tex:467:\sub for M\"untz polynomials, 276
../src/ch-index.tex:468:\sub for M\"untz polynomials in $L_p$, 279
../src/ch-index.tex:476:\sub for constrained polynomials, 444
../src/ch-index.tex:478:\sub for generalized polynomials, 394, 395
../src/ch-index.tex:479:\sub for M\"untz polynomials, 281, 298, 317
../src/ch-index.tex:482:Nonnegative polynomials, 70, 85, 417, 420
../src/ch-index.tex:483:Nonnegative trigonometric polynomials, 85, 409
../src/ch-index.tex:491:Orthogonal polynomials, 57--79
../src/ch-index.tex:495:\sub Gegenbauer; {\it see} Gegenbauer polynomials
../src/ch-index.tex:496:\sub Hermite; {\it see} Hermite polynomials
../src/ch-index.tex:498:\sub Jacobi; {\it see} Jacobi polynomials
../src/ch-index.tex:499:\sub Laguerre; {\it see} Laguerre polynomials
../src/ch-index.tex:500:\sub Legendre; {\it see} Legendre polynomials
../src/ch-index.tex:501:\sub M\"untz-Legendre; {\it see} M\"untz-Legendre polynomials
../src/ch-index.tex:503:\sub ultraspherical; {\it see} Ultraspherical polynomials
../src/ch-index.tex:513:Polynomials
../src/ch-index.tex:515:\sub Bernstein; {\it see} Bernstein polynomials
../src/ch-index.tex:516:\sub Chebyshev; {\it see} Chebyshev polynomials
../src/ch-index.tex:517:\sub Gegenbauer; {\it see} Gegenbauer polynomials
../src/ch-index.tex:518:\sub generalized; {\it see} generalized polynomials
../src/ch-index.tex:520:\sub Hermite; {\it see} Hermite polynomials
../src/ch-index.tex:522:\sub integer valued; {\it see} Integer valued polynomials
../src/ch-index.tex:523:\sub Jacobi; {\it see} Jacobi polynomials
../src/ch-index.tex:524:\sub Laguerre; {\it see} Laguerre polynomials
../src/ch-index.tex:525:\sub Legendre; {\it see} Legendre polynomials
../src/ch-index.tex:526:\sub M\"untz; {\it see} M\"untz polynomial
../src/ch-index.tex:527:\sub M\"untz-Legendre; {\it see} M\"untz-Legendre polynomials
../src/ch-index.tex:530:\sub trigonometric; {\it see} Trigonometric polynomial
../src/ch-index.tex:531:\sub Ultraspherical; {\it see} Ultraspherical polynomials
../src/ch-index.tex:539:Quasi-Chebyshev polynomials, 316, 342
../src/ch-index.tex:545:\sub Chebyshev polynomials of, 139--153
../src/ch-index.tex:555:\sub for algebraic polynomials, 228
../src/ch-index.tex:556:\sub for constrained polynomials, 443, 445
../src/ch-index.tex:557:\sub for generalized polynomials, 393, 394
../src/ch-index.tex:558:\sub for generalized polynomials in $L_p$, 401--402
../src/ch-index.tex:563:\sub for trigonometric polynomials, 230
../src/ch-index.tex:579:\sub for algebraic polynomials, 233
../src/ch-index.tex:580:\sub for constrained polynomials, 436--437
../src/ch-index.tex:581:\sub for generalized polynomials, 395
../src/ch-index.tex:583:\sub for trigonometric polynomials, 238
../src/ch-index.tex:584:Self-reciprocal polynomials, 339
../src/ch-index.tex:585:\sub quasi-Chebyshev polynomials, 342
../src/ch-index.tex:595:Sums of squares of polynomials, 85, 348
../src/ch-index.tex:598:Symmetric polynomial, 5
../src/ch-index.tex:608:Trigonometric polynomial, 2
../src/ch-index.tex:609:Trigonometric polynomials of longest arc length, 35
../src/ch-index.tex:613:Ultraspherical polynomials, 65
../src/ch-index.tex:625:\sub for polynomials, 159
../src/ch-index.tex:626:\sub for polynomials in $x^{\lambda}$, 167
../src/ch-index.tex:627:\sub for polynomials with integer coefficients, 169
../src/ch-index.tex:628:\sub for trigonometric polynomials, 165
../src/ch-index.tex:644:\sub of Chebyshev polynomials, 34, 116, 120, 122
../src/ch-index.tex:645:\sub of derivatives of polynomials, 18--28
../src/ch-index.tex:646:\sub of integrals of polynomials, 24
../src/ch-index.tex:647:\sub of M\"untz polynomials, 120
../src/ch-index.tex:648:\sub of M\"untz-Legendre polynomials, 133--136
../src/ch-index.tex:649:\sub of orthogonal polynomials, 61