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Index

Abel, Niels Henrik, 3, 61 Algorithms, 356–371

evaluation of x^{n}, 363 fast Fourier transform, 359 fast polynomial division, 362 fast polynomial expansion, 363 fast polynomial multiplication,

361 for Chebyshev polynomials, 371 for counting zeros, 364–370 for polynomial evaluations, 363 for reversion of power series, 362 interpolation, 360 Newton’s method, 362, 364–367 Remez, 371 zero finding for polynomials,

366–370

Alternation set, 93 Alternation theorem, 94 Apolar polynomials, 23, 24, 25 Arc length of algebraic polynomials,

31

Arc length of trigonometric polynomials, 35

Berman’s formula, 166 Bernstein factor, 145, 150, 152,

322–328

Bernstein polynomials, 163–164 Bernstein-Szegő inequality, 231, 245,

259, 321–323 for entire functions of exponential

type, 245 for rational functions on [−1, 1],

322 for rational functions on K, 322 for rational functions on ℝ, 323 for trigonometric polynomials, 232

Bernstein-type inequality bounded, 178, 182, 213–214 for Chebyshev spaces, 206 for constrained polynomials,

420–447 for entire functions of exponential

type, 245 for exponential sums, 289 for generalized polynomials,

392–416 for generalized polynomials in L_{p},

401–417 for higher derivatives, 258 for nondense Müntz spaces, 213,

474 Index

310 for polynomials, 232–233, 390 for polynomials in L_{p}, 235, 390,

401–417 for products of Müntz spaces, 317 for rational functions on [−1, 1],

323, 327 for rational functions on D, 324 for rational functions on K, 322,

327 for rational functions on ℝ, 323,

329 for self-reciprocal polynomials,

339 for trigonometric polynomials, 232 unbounded, 154, 206–217 weighted, 257

Bessel’s inequality, 46 Best approximation, 94

by rationals, 99 to x^{λ}, a problem of Lorentz, 108 to x^{n}, 99 uniqueness, 98

Binomial coefficient, 62 Blaschke product, 190, 324 Blumenthal’s theorem, 78 Bombieri’s norm, 274 Bounded Bernstein-type inequality;

see Bernstein-type inequality

Bounded Chebyshev-type inequality; see Chebyshev-type inequality

Bounded linear functionals, 50 Budan-Fourier theorem, 369

Cardano, Girolamo, 3 Cartan’s lemma, 350 Cauchy determinant, 106 Cauchy indices, 367 Cauchy’s integral formula, 14 Cauchy-Schwarz inequality, 42

for sequences, 46 Cesàro means, 165 Chebyshev constants, 39 Chebyshev, P.L., 31 Chebyshev polynomials

algorithms for, 371 best approximation to x^{n}, 30 composition

characterization, 33 explicit formulas, 30, 32 in Chebyshev spaces, 114–125 in rational spaces; see Chebyshev

rationals orthogonality, 32 reducibility, 36 second kind, 37 three-term recursion, 32 trigonometric on subintervals, 235 uniqueness, 118

Chebyshev rationals, 139–153 and orthogonality, 147 derivative formulas, 146 in algebraic rational spaces, 142 in trigonometric rational spaces,

143 of the first and second kind, 141 on the real line, 151 partial fraction representation,

144

Chebyshev space, 92 dimension on the circle, 100 functions with prescribed sign

changes, 100

Chebyshev system, 91–100

extended complete, 97 Chebyshev’s inequality, 235, 390 Chebyshev-type inequality

bounded, 179, 182 explicit bounds via Paley-Weiner

theorem, 196 for entire functions of exponential

type, 245

Christoffel function, 74

for Müntz spaces, 195 Christoffel numbers, 67 Christoffel-Darboux formula, 60 Coefficient bounds

for polynomials in special bases,

124 in nondense rational spaces, 153 of Markov, 248

Comparison theorem, 103, 120, 122,

183

Completeness, 48, 79 Complexity concerns, 356–371 Consecutive zeros of p′, 26 Constrained polynomials, 417–447

Bernstein-type inequality, 420, 425, 427 L_{p} inequalities, 422 Markov-type inequality, 417–447 Nikolskii-type, 444 Remez-type, 443–445 Schur-type, 436–437

Cotes numbers, 67 Cubic equations, 4

d’Alembert, Jean le Rond, 13 de la Vallée Poussin theorem, 99 Denseness, 154–226

of Markov spaces, 206–217 of Müntz polynomials, 171–205 of Müntz rationals, 218–226 of polynomials, 154–170

Derivatives of Markov systems, 112 Descartes’ rule of signs, 22, 102 Descartes system, 100–113

examples, 103 lexicographic properties, 103

Divide and conquer, 358 Division of polynomials, 15, 362

Elementary symmetric function, 5 Eneström-Kakeya theorem, 12 Erdős inequality, 431 Erdős-Turán inequality, 435 Euler, Leonhard, 13 Evaluation of x^{n}, 363 Exponential sums; see Müntz

polynomials a problem of Lorentz, 291 Bernstein-type inequality, 291 Markov-type inequality, 294 Nikolskii-type inequality, 289 Turán’s inequality, 295 with nonnegative exponents, 294

Exponential type, 196, 245 Extended complete Chebyshev system,

97

Factor inequalities, 260–274

via Mahler’s measure, 271–273 Factorization, 10, 36 Fast Fourier transform, 359 Favards theorem, 73 Fejér gap, 27–28 Fejér operators, 164 Fejér’s theorem, 165 Fekete point, 38 Fekete polynomial, 38 Ferrari, Ludovico, 3 Ferro, Scipione del, 3 Fourier coefficient, 53 Fourier series, 53 Fundamental theorem of algebra, 3

Gamma function, 63 Gauss, Carl Friedrich, 13 Gaussian hypergeometric series, 62 Gauss-Jacobi quadrature, 67, 75 Gauss-Lucas theorem, 18 Gegenbauer polynomials, 65 Generalized polynomials

Bernstein-type inequality, 399,

407 L_{p} inequalities, 401–407 Markov-type inequality, 399, 407 Nikolskii-type inequality, 394–395 Remez-type inequality, 393–394,

414 Schur-type inequality, 395 weighted inequalites, 407

Girard, Albert, 13 Grace’s complex version of Rolle’s

theorem, 25

Grace’s theorem, 18 Gram’s lemma, 176 Gram-Schmidt, 44

Haar space, 92 Haar system, 92 Halley’s method, 365 Hardy space, 189 Helly’s convergence theorem, 71 Helly’s selection theorem, 71 Hermite interpolation, 9 Hermite polynomials, 57

explicit formulas, 65 Hilbert space, 42 Hölder’s inequality, 17, 49 Horner’s rule, 8 Hypergeometric differential equation,

63

Hypergeometric functions, 62

Identity theorem, 15 Inequalities Bernstein-Szegő inequality; see

Bernstein-Szegő inequality Bernstein-type; see Bernstein-type

inequality Bessel’s; see Bessel’s inequality bounded Bernstein-type; see

bounded Bernstein-type inequality

bounded Chebyshev-type; see bounded Chebyshev-type inequality

Cauchy-Schwarz; see Cauchy-

Schwarz inequality Chebyshev-type; see Chebyshev-

type inequality for factors; see factor inequalities for Müntz polynomials; see Müntz

polynomials Hölder’s; see Hölder’s inequality Lax-type; see Lax Markov-type; see Markov-type

inequality metric; see metric inequalities Minkowski’s; see Minkowski’s

inequality Nikolskii-type; see Nikolskii-type

inequality Remez-type; see Remez-type

inequality Russak’s; see Russak Schur-type; see Schur-type

inequality Triangle; see Triangle inequality unbounded Bernstein-type; see

unbounded Bernstein-type inequality

Inner product, 41 Inner product space, 41 Integer-valued polynomials, 10 Interpolation

Hermite; see Hermite interpolation Lagrange; see Lagrange

interpolation Newton; see Newton interpolation

Jacobi polynomials, 57, 63

explicit formulas, 63 Jensen circles, 19 Jensen’s formula, 187 Jensen’s inequality, 414 Jensen’s theorem, 19

Kernel function, 47, 132 Kolmogorov’s inequality, 285 Korovkin’s theorems, 163

Lacunary spaces, 308

quasi-Chebyshev polynomials, 316 Lagrange interpolation, 8 Laguerre polynomials, 57, 66, 130

explicit formulas, 66 Laguerre’s theorem, 20 Lax-type inequality, 438

for rationals, 329 Malik’s extension, 438 on a half-plane, 338

Legendre polynomials, 57 Lemniscates of constant modulus,

352–353

Lexicographic properties, 116 for Müntz polynomials, 120, 314 for Müntz-Legendre polynomials,

136 for sinh systems, 122

Liouville’s theorem, 15 Logarithmic capacity, 38 Lorentz degree, 82

for polynomials, 86

for trigonometric polynomials, 89 Lorentz’s problem, 108, 291 L_{p} norm, 6, 48, 471 l_{p} norm, 6, 471 Lucas’ theorem, 18

Mahler’s measure, 271 Mairhuber theorem, 98 Markov system, 100

closure of nondense, 211

derivative of, 112 Markov-Stieltjes inequality, 76 Markov-type inequality

for P_{n}, 233 for P_{n}^{c}, 255 for constrained polynomials,

417–447 for constrained polynomials in L_{p},

422, 428–429 for exponential sums, 276–280,

294–295 for generalized polynomials,

399–407, 445 for generalized polynomials in L_{p},

401–407 for higher derivatives, 248–260 for monotone polynomials, 439–

441 for Müntz polynomials, 276–279,

287–288 for Müntz polynomials in L_{p},

279–280 for nonnegative polynomials, 420,

439 for rational functions, 336 for self-reciprocal polynomials,

339 for trigonometric polynomials on

subintervals, 242–245 in the complex plane, 235 weighted, 442–443

Maximum principal, 15 m-distribution, 57 Mergelyan’s theorem, 170 Mesh of zeros, 155 Metric inequalities, 344–355

for polynomials, 345–346 for rational functions, 347–349

Minkowski’s inequality in L_{p}, p ≥ 1, 49 in L_{p}, p ≤ 1, 52

Moment, 57 Moment problem, 70 Monotone operator theorem, 163 Multiplication of polynomials, 361 Müntz polynomials

bounded Bernstein-type inequality, 178, 182, 213, 310, 317

bound for smallest zero, 313 lexicographic properties of zeros,

116, 120 Newman’s inequality, 276, 301 Newman’s inequality in L_{p}, 279 Newman’s problem, 317 Nikolskii-type inequality, 281, 298,

317 positive zeros, 22 Remez-type inequality, 304, 307,

316 where the sup norm lives, 301

Müntz rationals, 218–226

denseness, 218 Müntz space, 125–126

lexicographic properties of zeros,

120 products of, 222, 316 quotients of, 218–226

Müntz system, 125–136 closure, 171–205 nondense, 303–319

Müntz-Legendre polynomials, 125–138 definition, 126 differential recursion, 129 global estimate of zeros, 136 integral recursion, 132 lexicographic properties of zeros,

133–136 orthogonality, 127, 132 orthonormality, 128 Rodrigues-type formula, 128, 131 zeros of, 133

Müntz’s theorem, 171–205 another proof, 176, 192 closure of span in, 178, 181, 185 in C[0, 1], 171 in C[a, b], 180, 184 in L_{2}[0, 1], 171 in L_{p}[0, 1], 172 in L_{p}[a, b], 186 in L_{p}(w), 311 on sets of positive measure, 303

Newman’s conjecture on denseness of products, 316 on denseness of quotients, 220,

223

Newman’s inequality, 275–279 an improvement, 287 for Müntz polynomials, 276 for Müntz polynomials in L_{p}, 279 on positive intervals, 301

Newton’s identities, 5 Newton Interpolation, 10 Newton’s method, 362, 364–367

for x^{1/2}, 365 in many variables, 366 Nikolskii-type inequality

for constrained polynomials, 444 for exponential sums, 289 for generalized polynomials, 394,

395 for Müntz polynomials, 281, 298,

317 for products of Müntz spaces, 317

Nondense Müntz spaces, 303–319 Nonnegative polynomials, 70, 85, 417,

420

Nonnegative trigonometric

polynomials, 85, 409

Norms, 471 L_{p}; see L_{p} norm l_{p}; see l_{p} norm supremum; see supremum norm

Orthogonal collection, 43 Orthogonal functions, 41–56 Orthogonal polynomials, 57–79

as continued fractions, 79 as determinants, 76 characterization of compact

support, 77 Gegenbauer; see Gegenbauer

polynomials Hermite; see Hermite polynomials interlacing of zeros, 61 Jacobi; see Jacobi polynomials Laguerre; see Laguerre

polynomials Legendre; see Legendre

polynomials Müntz-Legendre; see Müntz-

Legendre polynomials simple real zeros, 61 ultraspherical; see Ultraspherical

polynomials

Orthogonal rational functions, 147 Orthonormal set, 43

Paley-Weiner theorem, 196 Parallelogram law, 42 Parseval’s identity, 48 Partial fraction decomposition, 7, 144 Pellet’s theorem, 16 Polar derivative, 20 Polynomials as sums of squares, 85, 348 Bernstein; see Bernstein

polynomials Chebyshev; see Chebyshev

polynomials Gegenbauer; see Gegenbauer

polynomials generalized; see generalized

polynomials growth in the complex plane, 239 Hermite; see Hermite polynomials in x^{λ_{n}}, 167 integer valued; see Integer valued

polynomials Jacobi; see Jacobi polynomials Laguerre; see Laguerre

polynomials Legendre; see Legendre

polynomials Müntz; see Müntz polynomial Müntz-Legendre; see Müntz-

Legendre polynomials number of real roots, 17, 137 symmetric, 5 trigonometric; see Trigonometric

polynomial Ultraspherical; see Ultraspherical

polynomials with integer coefficients, 169 with nonnegative coefficients,

79–90, 417 with real roots, 345, 347–348

Products of Müntz spaces, 222, 316

Quadratic equations, 4 Quartic equations, 4 Quasi-Chebyshev polynomials, 316,

342

Railway track theorem, 98 Rakhmanov’s theorem, 78 Rational functions

algebraic, 139 Chebyshev polynomials of, 139–

153 coefficient bounds, 153 inequalities; see inequalities trigonometric, 139

Rational spaces of algebraic rational functions,

139–153, 320–321 of trigonometric rational

functions, 139–153, 320– 321

Recursive bounds, 359 Remez’s algorithm, 371 Remez-type inequality

for algebraic polynomials, 228 for constrained polynomials, 443,

445 for generalized polynomials, 393,

394 for generalized polynomials in L_{p},

401–402 for Müntz spaces, 307 for nondense Müntz spaces, 304 for products of Müntz spaces, 316 pointwise, 414 for trigonometric polynomials, 230

Reproducing kernel, 47, 132 Reversion of power series, 362 Riemann-Lebesgue lemma, 54 Riesz representation theorem, 50 Riesz’s identity, 390 Riesz’s lemma, 237 Riesz-Fischer theorem, 50 Rising factorial, 62 Rolle’s theorem, 25 Rouché’s theorem, 14, 16 Russak’s inequalities, 336

Salem numbers, 6 Schur’s theorem, 17 Schur-type inequality

for algebraic polynomials, 233 for constrained polynomials,

436–437 for generalized polynomials, 395 for rational functions, 337 for trigonometric polynomials, 238

Self-reciprocal polynomials, 339

quasi-Chebyshev polynomials, 342 Somorjai’s theorem, 218 Space

Chebyshev; see Chebyshev space Descartes; see Descartes space Haar; see Haar space Müntz; see Müntz space

rational; see rational space Stieltjes’ theorem, 78 Stone-Weierstrass theorem, 161 Sums of squares of polynomials, 85,

348

Supremum norm, 6, 29, 471 Symmetric function, 5 Symmetric polynomial, 5 Szegő’s inequality, 391 Szegő’s theorem, 23, 235

Tartaglia, Niccolo, 3 Tchebychev; see Chebyshev Three-term recursion, 59 Totally positive kernels, 110 Transfinite diameter, 38 Triangle inequality, 42 Trigonometric polynomial, 2 Trigonometric polynomials of longest

arc length, 35

Turán’s inequality, 434 Turán’s inequality for exponential

sums, 295

Ultraspherical polynomials, 65 Unbounded Bernstein-type inequality,

206–217 characterization of denseness, 207

Unicity theorem, 15

Variation diminishing property, 111 Vandermonde determinant, 38, 103 Videnskii’s inequalities, 242–245

Walsh’s two circle theorem, 20 Weierstrass’ theorem, 154–170

for Markov systems, 155 for polynomials, 159 for polynomials in x^{λ}, 167 for polynomials with integer coefficients, 169

for trigonometric polynomials, 165 in L_{p}, 169 on arcs, 170 Stone-Weierstrass theorem, 161

Wronskian, 22

Zeros, 11–18 algorithms for finding, 364–367 complexity of, 367 counting by winding number, 370 in a disk, 369 in an interval, 368 in Chebyshev spaces, 99 localizing, 367 maximum number at one, 137 maximum number of positive, 17 of Chebyshev polynomials, 34,

116, 120, 122 of derivatives of polynomials,

18–28 of integrals of polynomials, 24 of Müntz polynomials, 120 of Müntz-Legendre polynomials,

133–136 of orthogonal polynomials, 61

Zolotarev, 35 Zoomers, 218