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Notation

Definitions of the more commonly used spaces are given. The equation numbers here are the same as the equation numbers in the text.

Throughout the book, unless otherwise stated, spans should be assumed to be real. Likewise, in function spaces, unless otherwise stated, the functions should be assumed to be real valued.

The Basic Spaces.

(1.1.1)  P_{n}^{c} := {p : p(z) = ∑_{k=0}^{n} a_{k}z^{k}, a_{k} ∈ ℂ} .

(1.1.2)  P_{n} := {p : p(z) = ∑_{k=0}^{n} a_{k}z^{k}, a_{k} ∈ ℝ} .

(1.1.3)  R_{m,n}^{c} := \{\frac{p}{q} : p ∈ P_{m}^{c}, q ∈ P_{n}^{c}\} .

(1.1.4)  R_{m,n} := \{\frac{p}{q} : p ∈ P_{m}, q ∈ P_{n}\} .

468 Notation

(1.1.5) T_{n}^{c} : = {t : t(θ) := ∑_{k=−n}^{n} a_{k}e^{ikθ}, a_{k} ∈ ℂ}

= {t : t(z) = a_{0} + ∑_{k=1}^{n}(a_{k} cos kz + b_{k} sin kz) , a_{k}, b_{k} ∈ ℂ} .

T_{n} : = {t : t(z) = a_{0} + ∑_{k=1}^{n}(a_{k} cos kz + b_{k} sin kz) , a_{k}, b_{k} ∈ ℝ} .

Müntz Spaces.

(3.4.2)  M_{n}(Λ) := span{x^{λ_{0}}, x^{λ_{1}}, …, x^{λ_{n}}} .

(3.4.3)  M(Λ) := ⋃_{n=0}^{∞} M_{n}(Λ) = span{x^{λ_{0}}, x^{λ_{1}}, … } .

Associated with a sequence (λ_{i})_{i=0}^{∞} with Re(λ_{i}) > −1/2 for each i, the nth Müntz-Legendre polynomial is:

(3.4.5)  L_{n}(x) := L_{n}{λ_{0}, …, λ_{n}}(x)

:= \frac{1}{2πi} ∫_{Γ} ∏_{k=0}^{n−1} \frac{t + \overline{λ}_{k} + 1}{t − λ_{k}} \frac{x^{t} dt}{t − λ_{n}} .

For distinct λ_{i},

(3.4.6)  L_{n}{λ_{0}, …, λ_{n}}(x) = ∑_{k=0}^{n} c_{k,n}x^{λ_{k}},  x ∈ (0, ∞)

with

c_{k,n} := \frac{∏_{j=0}^{n−1}(λ_{k} + \overline{λ}_{j} + 1)}{∏_{j=0,j≠k}^{n}(λ_{k} − λ_{j})} .

Also,

(3.4.8)  L_{n}^{∗} := (1 + λ_{n} + \overline{λ}_{n})^{1/2}L_{n}

is the nth orthonormal Müntz-Legendre polynomial.

Notation 469

Rational Spaces. K := ℝ (mod 2π) , D := {z ∈ ℂ : |z| < 1} , ∂D := {z ∈ ℂ : |z| = 1} .

(3.5.3)  P_{n}(a_{1}, a_{2}, …, a_{n}) := \{\frac{p(x)}{∏_{k=1}^{n} |x − a_{k}|} : p ∈ P_{n}\} .
(3.5.4)  T_{n}(a_{1}, a_{2}, …, a_{n}) := \{\frac{t(θ)}{∏_{k=1}^{n} |cos θ − a_{k}|} : t ∈ T_{n}\} .

Also

T_{n}(a_{1}, a_{2}, …, a_{2n}; K) := T_{n}(a_{1}, a_{2}, …, a_{2n})

:= \{\frac{t(θ)}{∏_{k=1}^{2n} | sin((θ − a_{k})/2)|} : t ∈ T_{n}\}

and

T_{n}^{c}(a_{1}, a_{2}, …, a_{2n}; K) := \{\frac{t(θ)}{∏_{k=1}^{2n} sin((θ − a_{k})/2)} : t ∈ T_{n}^{c}\}

on K with a_{1}, a_{2}, …, a_{2n} ∈ ℂ \ ℝ;

P_{n}(a_{1}, a_{2}, …, a_{n}; [−1, 1]) := \{\frac{p(x)}{∏_{k=1}^{n} |x − a_{k}|} : p ∈ P_{n}\}

and

P_{n}^{c}(a_{1}, a_{2}, …, a_{n}; [−1, 1]) := \{\frac{p(x)}{∏_{k=1}^{n}(x − a_{k})} : p ∈ P_{n}^{c}\}

on [−1, 1] with a_{1}, a_{2}, …, a_{n} ∈ ℂ \ [−1, 1];

P_{n}^{c}(a_{1}, a_{2}, …, a_{n}; ∂D) := \{\frac{p(z)}{∏_{k=1}^{n}(z − a_{k})} : p ∈ P_{n}^{c}\}

on ∂D with a_{1}, a_{2}, …, a_{n} ∈ ℂ \ ∂D; and

P_{n}(a_{1}, a_{2}, …, a_{n}; ℝ) := \{\frac{p(x)}{∏_{k=1}^{n} |x − a_{k}|} : p ∈ P_{n}\}

and

P_{n}^{c}(a_{1}, a_{2}, …, a_{n}; ℝ) := \{\frac{p(z)}{∏_{k=1}^{n}(z − a_{k})} : p ∈ P_{n}^{c}\}

on ℝ with a_{1}, a_{2}, …, a_{n} ∈ ℂ \ ℝ.

Generalized Polynomials.

The function

(A.4.1)  f(z) = |ω| ∏_{j=1}^{m} |z − z_{j}|^{r_{j}}

with 0 < r_{j} ∈ ℝ, z_{j} ∈ ℂ, and 0 ≠ ω ∈ ℂ is called a generalized nonnegative (algebraic) polynomial of (generalized) degree

(A.4.2)  N := ∑_{j=1}^{m} r_{j} .

The set of all generalized nonnegative algebraic polynomials of degree at most N is denoted by GAP_{N}.

The function

(A.4.3)  f(z) = |ω| ∏_{j=1}^{m} | sin((z − z_{j})/2)|^{r_{j}}

with 0 < r_{j} ∈ ℝ, z_{j} ∈ ℂ, and 0 ≠ ω ∈ ℂ is called a generalized nonnegative trigonometric polynomial of degree

(A.4.4)  N := \frac{1}{2} ∑_{j=1}^{m} r_{j} .

The set of all generalized nonnegative trigonometric polynomials of degree at most N is denoted by GTP_{N}.

Constrained Polynomials.

The following classes of polynomials with constraints appear in Appendix 5:

P_{n,k} := {p ∈ P_{n} : p has at most k zeros in D} , 0 ≤ k ≤ n ,

B_{n}(a, b) := {p ∈ P_{n} : p(x) = ± ∑_{j=0}^{n} α_{j}(b−x)^{j}(x−a)^{n−j} , α_{j} ≥ 0} , a < b ,

P̃_{n,k}(a, b) := {p = hq : h ∈ B_{n−k}(a, b) , q ∈ P_{k}} , 0 ≤ k ≤ n .

Function Spaces.

The uniform or supremum norm of a complex-valued function f defined on a set A is defined by

∥f∥_{A} := sup_{x∈A} |f(x)| .

The space of all real-valued continuous functions on a topological space A equipped with the uniform norm is denoted by C(A). If A := [a, b] is equipped with the usual metric topology, then the notation C[a, b] := C(A) is used.

Let (X, μ) be a measure space (μ nonnegative) and p ∈ (0, ∞]. The space L_{p}(μ) is defined as the collection of equivalence classes of real-valued measurable functions for which ∥f∥_{L_{p}(μ)} < ∞, where

∥f∥_{L_{p}(μ)} := (∫_{X} |f|^{p} dμ)^{1/p} , p ∈ (0, ∞)

and

∥f∥_{L_{∞}(μ)} := sup{α ∈ ℝ : μ({x ∈ X : |f(x)| > α}) > 0} < ∞ .

The equivalence classes are defined by the equivalence relation f ∼ g if f = g μ-almost everywhere on X. When we write L_{p}[a, b] we always mean L_{p}(μ), where μ is the Lebesgue measure on X = [a, b]. The notations L_{p}(a, b), L_{p}[a, b), and L_{p}(a, b] are also used analogously to L_{p}[a, b]. Again, it is always our understanding that the space L_{p}(μ) is equipped with the L_{p}(μ) norm.

Sometimes C(A) denotes the space of all complex-valued continuous functions defined on A equipped with the uniform norm. Similarly, L_{p}(μ) may denote the space all complex-valued continuous functions for which ∥f∥_{L_{p}(μ)} < ∞. This should always be clear from the context; many times, but not always, the reader is reminded if C[a, b] or L_{p}(μ) is complex.