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A4

Inequalities for Generalized Polynomials in L_{p}

Overview

Many inequalities for generalized polynomials are given in this appendix. Of particular interest are the extensions of virtually all the basic inequalities to L_{p} spaces. The principal tool is a generalized version of Remez’s inequality.

Inequalities for Generalized Polynomials in Lp

Generalized (nonnegative) polynomials are defined by (A.4.1) and (A.4.3). The basic inequalities of Chapter 5 are extended to these functions by replacing the degree with the generalized degree. The crucial observation is that Remez’s inequality extends naturally to this setting. This Remez inequality then plays a central role in the extensions of the other inequalities. These generalizations allow for a simple general treatment of L_{p} inequalities, which is one main feature of this appendix.

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

Inequalities for Generalized Polynomials in L_{p} 393

(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}. Throughout this section we will study generalized nonnegative polynomials restricted to the real line. If the exponents r_{j} in (A.4.1) or (A.4.3) are even integers, then f is a nonnegative algebraic or trigonometric polynomial, respectively. Note that the classes GAP_{N} and GTP_{N} are not linear spaces. Note also that if f ∈ GAP_{N} or f ∈ GTP_{N} is of the form (A.4.1) or (A.4.3), respectively, with all r_{j} ≥ 1, then the one-sided derivatives of f exist at every x ∈ ℝ with the same modulus, hence |f′(x)| is well-defined for every x ∈ ℝ. We use the notation |f′(x)| for f ∈ GAP_{N} or f ∈ GTP_{N} and x ∈ ℝ throughout this section with this understanding. If f ∈ GAP_{N} is of the form (A.4.1) or f ∈ GTP_{N} is of the form (A.4.3), where the zeros z_{j} ∈ ℂ, j = 1, 2, …, m, are distinct, then r_{j} is called the multiplicity of z_{j} in f. Our intention in this section is to extend most of the classical inequalities of Section 5.1 to generalized nonnegative polynomials. In addition, we prove Nikolskii-type inequalities for GAP_{N} and GTP_{N}.

Theorem A.4.1 (Remez-Type Inequality for GAP_{N}). The inequality

∥f∥_{[−1,1]} ≤ (\frac{\sqrt{2} + \sqrt{s}}{\sqrt{2} − \sqrt{s}})^{N}

holds for every f ∈ GAP_{N} and s ∈ (0, 2) satisfying

m({x ∈ [−1, 1] : f(x) ≤ 1}) ≥ 2 − s .

E.5 shows that this inequality is sharp. Note that if 0 < s ≤ 1, then

(\frac{\sqrt{2} + \sqrt{s}}{\sqrt{2} − \sqrt{s}})^{N} ≤ exp(5N\sqrt{s}).

Throughout this section, as before, K := ℝ (mod 2π).

394 A4. Inequalities for Generalized Polynomials in L_{p}

Theorem A.4.2 (Remez-Type Inequality for GTP_{N}). The inequality

∥f∥_{K} ≤ exp (N (s + \frac{7}{4}s^{2})) ≤ exp(4Ns)

holds for every f ∈ GTP_{N} and s ∈ (0, π/2] satisfying

(A.4.5)  m({x ∈ [−π, π) : f(x) ≤ 1}) ≥ 2π − s .

The inequality

∥f∥_{K} ≤ (\frac{\sqrt{2} + \sqrt{σ}}{\sqrt{2} − \sqrt{σ}} )^{N} , σ = 1 − cos(s/2)

holds for every even f ∈ GTP_{N} and s ∈ (0, 2π) satisfying (A.4.5).

We do not discuss what happens when s ∈ (π/2, 2π) in the general case because the case when s ∈ (0, π/2] is satisfactory for our needs.

Proof of Theorem A.4.1. First assume that f ∈ GAP_{N} is of the form (A.4.1) with rational exponents r_{j} = q_{j}/q, where q_{j}, q ∈ ℕ. Let k ∈ ℕ be an integer. Then (restricted to ℝ) p := f^{2kq} ∈ P_{2kqN} and

m ({x ∈ [−1, 1] : |p(x)| ≤ 1}) ≥ 2 − s .

Hence Theorem 5.1.1 yields

∥f∥_{[−1,1]} = ∥p∥_{[−1,1]}^{1/(2kq)} ≤ (T_{2kqN} (\frac{2 + s}{2 − s}))^{1/(2kq)} .

Since by E.4,

(A.4.6)  lim_{k→∞} (T_{2kqN} (\frac{2 + s}{2 − s}))^{1/(2kq)} = (\frac{\sqrt{2} + \sqrt{s}}{\sqrt{2} − \sqrt{s}})^{N} ,

the theorem is proved. The case when the exponents r_{j} > 0 are arbitrary real numbers can be easily reduced to the already proved rational case by a straightforward density argument. ⊓ ⊔

Theorem A.4.2 follows from Theorem 5.1.2 in exactly the same way that Theorem A.4.1 follows from Theorem 5.1.1; see E.6.

Theorem A.4.3 (Nikolskii-Type Inequality for GTP_{N}). Let χ be a nonnegative nondecreasing function defined on [0, ∞) such that χ(x)/x is nonincreasing on [0, ∞). Then there is an absolute constant c_{1} > 0 such that

∥χ(f)∥_{L_{p}(K)} ≤ (c_{1}(1 + qN))^{1/q−1/p}∥χ(f)∥_{L_{q}(K)}

for every f ∈ GTP_{N} and 0 < q < p ≤ ∞. If χ(x) = x, then c_{1} ≤ e(4π)^{−1}.

Theorem A.4.4 (Nikolskii-Type Inequality for GAP_{N}). Let χ be a nonnegative nondecreasing function defined on [0, ∞) such that χ(x)/x is nonincreasing on [0, ∞). Then there is an absolute constant c_{2} > 0 such that

∥χ(f)∥_{L_{p}[−1,1]} ≤ (c_{2}(2 + qN))^{2/q−2/p}∥χ(f)∥_{L_{q}[−1,1]}

for every f ∈ GAP_{N} and 0 < q < p ≤ ∞. If χ(x) = x, then c_{2} ≤ e^{2}(2π)^{−1}.

In the proof of the second part of Theorem A.4.4 we will need the following Schur-type inequality, which is interesting in its own right.

Theorem A.4.5 (Schur-Type Inequality for GAP_{N}). The inequality

∥f∥_{[−1,1]}^{q} ≤ e(1 + qN)∥\sqrt{1 − x^{2}}f^{q}(x)∥_{[−1,1]}

holds for every f ∈ GAP_{N} and q > 0.

According to Theorem 5.1.9 (Schur’s inequality), if N ∈ ℕ, f ∈ P_{N}, and q ∈ ℕ, then the constant e in the above inequality can be replaced by 1.

It is sufficient to prove Theorems A.4.3 and A.4.4 when p = ∞, and then a simple argument gives the required results for arbitrary exponents 0 < q < p < ∞. To see this, say, in the trigonometric case, assume that there is a constant C_{N} such that

∥χ(f)∥_{K} ≤ C_{N}^{1/q}∥χ(f)∥_{L_{q}(K)}

for every f ∈ GTP_{N} and 0 < q < ∞. Then

∥χ(f)∥_{L_{p}(K)}^{p} = ∥(χ(f))^{p−q+q}∥_{L_{1}(K)}
≤ ∥χ(f)∥_{K}^{p−q}∥χ(f)∥_{L_{q}(K)}^{q}
≤ C_{N}^{p/q−1}∥χ(f)∥_{L_{q}(K)}^{p−q}∥χ(f)∥_{L_{q}(K)}^{q} ,

and therefore

∥χ(f)∥_{L_{p}(K)} ≤ C_{N}^{1/q−1/p}∥χ(f)∥_{L_{q}(K)}

for every f ∈ GTP_{N} and 0 < q < p ≤ ∞.

Proof of Theorem A.4.3 (when p = ∞). Since χ is nonnegative and nondecreasing and (χ(x)/x)^{q} is nonincreasing on [0, ∞), we have

(χ(f(θ)))^{q} ≥ exp(−qNs)∥χ(f)∥_{K}^{q}

whenever

f(θ) ≥ exp(−Ns)∥f∥_{K} .

Hence, by E.7 b], we can deduce that

m (\{θ ∈ [−π, π) : (χ(f(θ)))^{q} ≥ exp(−qNs)∥χ(f)∥_{K}^{q}\}) ≥ \frac{s}{4}

for every f ∈ GTP_{N} and s ∈ (0, 2π). Choosing s := (1 + qN)^{−1}, we get

m ({θ ∈ [−π, π) : (χ(f(θ)))^{q} ≥ e^{−1}∥χ(f)∥_{K}^{q}}) ≥ (4(1 + qN))^{−1} .

Hence, integrating only on the subset I of K where

(χ(f(θ)))^{q} ≥ e^{−1}∥χ(f)∥_{K}^{q} ,

we conclude that

∥χ(f)∥_{K}^{q} ≤ 4e(1 + qN) ∫_{I}(χ(f(θ)))^{q} dθ
≤ 4e(1 + qN)∥χ(f)∥_{L_{q}(K)}^{q} ,

and the first part of the theorem is proved.

Now we turn to the second statement. Let

D := {z ∈ ℂ : |z| < 1}  and  ∂D := {z ∈ ℂ : |z| = 1} .

If h is analytic in the open unit disk D and continuous on the closed unit disk \overline{D}, then by Cauchy’s integral formula we have

(1 − |rz|^{2})h(rz) = \frac{1}{2πi} ∫_{∂D} h(u)\frac{1 − r\overline{z}u}{u − rz} du

whenever z ∈ \overline{D} and r ∈ [0, 1). Note that u ∈ ∂D and z ∈ ∂D imply |1 − r\overline{z}u| = |u − rz| for all r ∈ [0, 1). Hence, if P ∈ P_{n}^{c} and 0 < q < ∞, then

(1 − r^{2})|P^{∗}(rz)|^{q} ≤ \frac{1}{2π} ∫_{∂D} |P^{∗}(u)|^{q} |du|

whenever z ∈ ∂D and r ∈ [0, 1], where P^{∗} is obtained from the factorization of P by replacing each factor (z − α) of P with |α| < 1 by (1 − \overline{α}z). Since

\frac{1}{2}(1 + r)|z − σ| ≤ |rz − σ| , |σ| > 1 , z ∈ ∂D , r ∈ [0, 1] ,

we have

(1 − r^{2}) (\frac{1}{2}(1 + r))^{q deg(P)} |P (z)|^{q} ≤ \frac{1}{2π} ∫_{∂D} |P^{∗}(u)|^{q} |du|

whenever z ∈ ∂D and r ∈ [0, 1]. Maximizing the left-hand side for r ∈ [0, 1] and using the fact that |P^{∗}(z)| = |P (z)| for z ∈ ∂D, we conclude that

|P (z)|^{q} ≤ \frac{(2 + q deg(P))e}{8π} ∫_{−π}^{π} |P (e^{iθ})|^{q} dθ , z ∈ ∂D .

Hence, by E.8,

∥R∥_{K}^{q} ≤ \frac{(1 + qn)e}{4π} ∫_{−π}^{π} |R(θ)|^{q} dθ

for every R ∈ T_{n}. If f ∈ GTP_{N} is of the form (A.4.3) with rational exponents r_{j} = α_{j}/α, where α_{j}, α ∈ ℕ, then on applying the above inequality to R := f^{2α} ∈ T_{2αN} with q replaced by q/(2α), we conclude that

∥f∥_{K}^{q} ≤ \frac{1 + qN}{4π} ∫_{−π}^{π} f(θ)^{q} dθ

and the second statement of the theorem is proved. The case when the exponents r_{j} > 0, j = 1, 2, …, m, are arbitrary real numbers can be reduced to the already proved rational case by a straightforward density argument. ⊓ ⊔

Proof of Theorem A.4.5. Let P ∈ P_{n} and

M := ∥\sqrt{1 − x^{2}} |P (x)|^{q}∥_{[−1,1]} .

By E.8 c], there exists an R ∈ P_{2n} such that |R(e^{iθ})| = |P (cos θ)|, θ ∈ ℝ. We define R^{∗} ∈ P_{2n}^{c} from the factorization of R by replacing all the factors (z − α) of R with |α| < 1 by (1 − \overline{α}z). Note that |1 − e^{2iθ}| = 2| sin θ| and |R(e^{iθ})| = |R^{∗}(e^{iθ})| for all θ ∈ ℝ. Hence the maximum principle yields that

|1 − (rz)^{2}||R^{∗}(rz)|^{q} ≤ max_{z∈∂D} |1 − z^{2}||R^{∗}(z)|^{q}
= max_{θ∈ℝ} 2 | sin θ||P (cos θ)|^{q} = 2M .

By E.9 we have

|R^{∗}(z)| ≤ (\frac{2}{1 + r} )^{2n} |R^{∗}(rz)| , z ∈ ∂D , r ∈ [0, 1] .

Hence

|R^{∗}(z)|^{q} ≤ \frac{2^{2nq}}{(1 + r)^{2nq}} \frac{1}{1 − r^{2}} 2M , z ∈ ∂D , r ∈ [0, 1] ,

where the minimum on [0, 1] of the right-hand side is taken at

r := \frac{qn}{1 + qn} .

Estimating the right-hand side at this value of r, we get

∥P ∥_{[−1,1]}^{q} = max_{z∈∂D} |R^{∗}(z)|^{q} ≤ e(1 + qn)∥\sqrt{1 − x^{2}} |P (x)|^{q}∥_{[−1,1]} .

If f ∈ GAP_{N} is of the form (A.4.1) with rational exponents r_{j} = α_{j}/α, α_{j}, α ∈ ℕ, then applying the above inequality to P = f^{2α} ∈ P_{2αN} with q replaced by q/(2α), we get

∥f∥_{[−1,1]}^{q} ≤ e(1 + qN)∥\sqrt{1 − x^{2}} f^{q}(x)∥_{[−1,1]} ,

and the theorem is proved. The case when the exponents r_{j} > 0 are arbitrary real numbers can be easily reduced to the already proved rational case once again by a standard approximation. ⊓ ⊔

Proof of Theorem A.4.4 (when p = ∞). Since χ is nonnegative and nondecreasing and (χ(x)/x)^{q} is nonincreasing on [0, ∞), we have

(χ(f(x)))^{q} ≥ exp(−qN\sqrt{s})∥χ(f)∥_{[−1,1]}^{q}

whenever

f(x) ≥ exp(−N\sqrt{s})∥f∥_{[−1,1]} .

So by E.7 a] we can deduce that

m (\{x ∈ [−1, 1] : (χ(f(x)))^{q} ≥ exp(−qN\sqrt{s})∥χ(f)∥_{[−1,1]}^{q}\}) ≥ \frac{s}{8}

for all s ∈ (0, 2). Choosing s := (1 + qN)^{−2}, we obtain

m (\{x ∈ [−1, 1] : (χ(f(x)))^{q} ≥ e^{−1}∥χ(f)∥_{[−1,1]}\}) ≥ \frac{1}{8(1 + qN)^{2}} .

Integrating on the subset I of [−1, 1] where

(χ(f(x)))^{q} ≥ e^{−1}∥χ(f)∥_{[−1,1]} ,

we conclude that

∥χ(f)∥_{[−1,1]}^{q} ≤ 8e(1 + qN)^{2} ∫_{I}(χ(f(x)))^{q} dx
≤ 8e(1 + qN)^{2}∥χ(f)∥_{L_{q}[−1,1]}^{q} .

Thus the first part of the theorem is proved.

To show that the given constant works in the case that χ(x) = x we use another method. Let h ∈ GAP_{M}. Then by E.10, g(θ) = h(cos θ) ∈ GTP_{M}. On using the substitution x = cos θ, from Theorem A.4.3 we get

∥h∥_{[−1,1]} ≤ e(2π)^{−1}(1 + qM) ∫_{−1}^{1} |h(x)|^{q}(1 − x^{2})^{−1/2} dx .

If f ∈ GAP_{N}, then h(x) = f(x)(1 − x^{2})^{1/(2q)} ∈ GAP_{M} with M = N + q^{−1}, so an application of the above inequality yields

(A.4.7)  ∥\sqrt{1 − x^{2}}f^{q}(x)∥_{[−1,1]} ≤ e(2π)^{−1}(2 + qN) ∫_{−1}^{1} f^{q}(x) dx .

(Note that the weaker assumption h ∈ GAP_{M} instead of f ∈ GAP_{N} already implies (A.4.7).)

Now a combination of Theorem A.4.5 and inequality (A.4.7) gives that if χ(x) = x, then the inequality of Theorem A.4.4 holds with c_{2} := e^{2}(2π)^{−1}.

⊓ ⊔

Now we prove extensions (up to multiplicative absolute constants) of Markov’s and Bernstein’s inequalities for generalized nonnegative polynomials.

Theorem A.4.6 (Bernstein-Type Inequality for GTP_{N}). There exists an absolute constant c_{3} > 0 such that

∥f′∥_{K} ≤ c_{3}N∥f∥_{K}

for every f ∈ GTP_{N} of the form (A.4.3) with each r_{j} ≥ 1.

Theorem A.4.7 (Bernstein-Type Inequality for GAP_{N}). The inequality

|f′(x)| ≤ \frac{c_{3}N}{\sqrt{1 − x^{2}}} ∥f∥_{[−1,1]} , x ∈ (−1, 1) ,

holds for every f ∈ GAP_{N} of the form (A.4.1) with each r_{j} ≥ 1, where c_{3} is as in Theorem A.4.6.

Theorem A.4.8 (Markov-Type Inequality for GAP_{N}). There exists an absolute constant c_{4} > 0 such that

∥f′∥_{[−1,1]} ≤ c_{4}N^{2}∥f∥_{[−1,1]}

for every f ∈ GAP_{N} of the form (A.4.1) with each r_{j} ≥ 1.

To prove Theorem A.4.6 we need the following lemma.

Lemma A.4.9. Suppose g ∈ GTP_{N} is of the form (A.4.3) with each z_{j} ∈ ℝ, and suppose at least one of any two adjacent (in K) zeros has multiplicity at least 1. Then there exists an absolute constant c_{5} > 0 such that for every such g there is an interval I ⊂ K of length at least c_{5}N^{−1} for which

min_{θ∈I} g(θ) ≥ e^{−1}∥g∥_{K} .

Proof. Take a g ∈ GTP_{N} satisfying the hypothesis of the lemma. Because of the periodicity of g we may assume that

(A.4.8)  g(π) = ∥g∥_{K} .

Define

(A.4.9)  Q_{n,ω}(θ) := T_{2n} (\frac{sin(θ/2)}{sin(ω/2)})

with

(A.4.10)  n := ⌊N⌋  and  ω := π − (3N)^{−1} ,

where T_{2n} is the Chebyshev polynomial of degree 2n defined by (2.1.1). By E.11 there exists an absolute constant c_{6} > 1 such that Q_{n,ω}(π) ≥ c_{6}. Introduce the set

A := {θ ∈ [π − (3N)^{−1}, π + (3N)^{−1}] : g(θ) ≥ e^{−1}g(π)} .

We study h := g|Q_{n,ω}| ∈ GTP_{2N}. The inequality Q_{n,ω}(π) ≥ c_{6} and assumption (A.4.8) yield

h(θ) ≤ g(θ) ≤ c_{6}^{−1}Q_{n,ω}(π)g(π) = c_{6}^{−1}∥h∥_{K}

for all θ ∈ [−ω, ω] = [−π + (3N)^{−1}, π − (3N)^{−1}]. Further, the definition of the set A, the fact that ∥Q_{n,ω}∥_{K} = Q_{n,ω}(π), and (A.4.8) imply that

h(θ) ≤ e^{−1}g(π)Q_{n,ω}(π) = e^{−1}∥h∥_{K}

for all θ ∈ [π − (3N)^{−1}, π + (3N)^{−1}] \ A. From the last two inequalities we conclude that

h(θ) ≤ c_{7}^{−1}∥h∥_{[−1,1]} for all θ ∈ [−π, π] \ A ,

where c_{7} := min{c_{6}, e} > 1 is an absolute constant. Therefore, by E.7 b]

m(A) ≥ c_{8}N^{−1} with c_{8} := \frac{1}{17} log c_{7} > 0 .

Since g ∈ GTP_{N} is of the form (A.4.3) with each z_{j} ∈ ℝ, and at least one of any two adjacent zeros of g has multiplicity at least 1, E.12 and assumption (8.1.8) imply that g cannot have two or more distinct zeros in [π − (3N)^{−1}, π + (3N)^{−1}]. Hence A is the union of at most two intervals. Therefore there exists an interval I ⊂ A such that m(I) ≥ c_{8}(2N)^{−1}, and the lemma is proved. ⊓ ⊔ Proof of Theorem A.4.6. Let f ∈ GTP_{N} be of the form (A.4.3) with each r_{j} ≥ 1. Without loss of generality we may assume that |f′(π)| = ∥f′∥_{K}, and it is sufficient to prove only that

|f′(π)| ≤ c_{3}N∥f∥_{K} .

By E.13 we may assume that each z_{j} is real in (A.4.3). Hence, by E.2, g := |f′| satisfies the assumption of Lemma A.4.9. Denote the endpoints of the interval I coming from Lemma A.4.9 by a < b. We can now deduce that

∥f′∥_{K} = |f′(π)| ≤ \frac{e}{b − a} ∫_{a}^{b} |f′(θ)| dθ

≤ \frac{eN}{c_{5}} ∫_{a}^{b} |f′(θ)| dθ = \frac{eN}{c_{5}} |f(b) − f(a)| ≤ c_{3}N∥f∥_{K}

with c_{3} := ec_{5}^{−1}, and the proof is finished. ⊓ ⊔

Proof of Theorem A.4.7. The theorem follows from Theorem A.4.6 by using the substitution x = cos θ and E.10 b]. ⊓ ⊔

Proof of Theorem A.4.8. Let α := 1−(1+N)^{−2}. Using Theorem A.4.7 and then E.14, we obtain

∥f′∥_{[−α,α]} ≤ c_{3}N(N + 1)∥f∥_{[−1,1]}
≤ c_{3}N(N + 1)c_{9}∥f∥_{[−α,α]} ≤ c_{4}N^{2}∥f∥_{[−α,α]} ,

and then the theorem follows by a linear transformation. ⊓ ⊔

Now we establish Remez-, Bernstein-, and Markov-type inequalities for generalized nonnegative polynomials in L_{p}. In the proofs we use the inequalities proved in this appendix so far, and the methods illustrate how one can combine the “basic” inequalities in the proofs of various other inequalities for generalized nonnegative polynomials. First we state the main results.

Theorem A.4.10 (L_{p} Remez-Type Inequality for GAP_{N}). Let χ be a nonnegative nondecreasing function defined on [0, ∞) such that χ(x)/x is nonincreasing on [0, ∞). There exists an absolute constant c ≤ 5\sqrt{2} such that

∫_{−1}^{1}(χ(f(x)))^{p} dx ≤ (1 + exp(cpN\sqrt{s})) ∫_{A}(χ(f(x)))^{p} dx

for every f ∈ GAP_{N}, A ⊂ [−1, 1] with m([−1, 1] \ A) ≤ s ≤ 1/2, and for every p ∈ (0, ∞).

Theorem A.4.11 (L_{p} Remez-Type Inequality for GTP_{N}). Let χ be a nonnegative nondecreasing function defined on [0, ∞) such that χ(x)/x is nonincreasing on [0, ∞). There exists an absolute constant c ≤ 8 such that

∫_{−π}^{π}(χ(f(θ)))^{p} dθ ≤ (1 + exp(cpNs)) ∫_{A} (χ(f(θ)))^{p} dθ

for every f ∈ GTP_{N}, A ⊂ [−π, π] with m([−π, π] \ A) ≤ s ≤ π/2, and for every p ∈ (0, ∞).

Theorem A.4.12 (L_{q} Bernstein-Type Inequality for GTP_{N}). Let χ be a nonnegative, nondecreasing, convex function defined on [0, ∞). There exists an absolute constant c such that

∫_{−π}^{π} χ(N^{−q}|f′(θ)|^{q}) dθ ≤ ∫_{−π}^{π} χ(cf(θ)^{q}) dθ

for every f ∈ GTP_{N} of the form (A.4.3) with each r_{j} ≥ 1, and for every q ∈ (0, 1].

Corollary A.4.13 (L_{p} Bernstein-Type Inequality for GTP_{N}). The inequality

∫_{−π}^{π} |f′(θ)|^{p} dθ ≤ c^{p+1}N^{p} ∫_{−π}^{π} |f(θ)|^{p} dθ

holds for every f ∈ GTP_{N} of the form (A.4.3) with each r_{j} ≥ 1, and for every p ∈ (0, ∞), where c is as in Theorem A.4.12.

Theorem A.4.14 (L_{p} Markov-Type Inequality for GAP_{N}). There exists an absolute constant c such that

∫_{−1}^{1} |f′(x)|^{p} dx ≤ c^{p+1}N^{2p} ∫_{−1}^{1} |f(x)|^{p} dx

for every f ∈ GAP_{N} of the form (A.4.1) with each r_{j} ≥ 1, and for every p ∈ (0, ∞).

Theorems A.4.1 and A.4.2 can be easily obtained from their L_{∞} analogs, Theorems A.4.1 and A.4.2, respectively; see E.15 and E.16, where hints are given.

Proof of Theorem A.4.12. For n := ⌊N⌋ let

D_{n}(θ) := | ∑_{j=−n}^{n} e^{ijθ}|

be the modulus of the nth Dirichlet kernel. Choose q ∈ (0, 1], and set m := 2q^{−1} ≥ 2. Let g ∈ GTP_{N} be of the form (A.4.3) with each r_{j} ≥ 1. On applying the Nikolskii-type inequality of Theorem A.4.3 to

G := gD_{n}^{m} ∈ GTP_{N+2nq^{−1}} ,

we obtain

(A.4.11)  ∥gD_{n}^{m}∥_{K}^{q} ≤ c_{1} (1 + q (N + 2nq^{−1})) ∥gD_{n}^{m}∥_{L_{q}(K)}^{q}
≤ c_{1}(1 + 3N) ∫_{−π}^{π}(g(θ)D_{n}^{m}(θ))^{q} dθ
= c_{1}(1 + 3N) ∫_{−π}^{π} g^{q}(θ)D_{n}^{2}(θ) dθ .

If g ∈ GTP_{N} is of the form (A.4.3) with each r_{j} ≥ 1, then m ≥ 2 implies that G ∈ GTP_{N} is of the form (A.4.3) with each r_{j} ≥ 1 as well. If we apply the Bernstein-type inequality of Theorem A.4.6 to G and use (A.4.11), we can deduce that

|g′(θ)D_{n}^{m}(θ) + mD_{n}^{m−1}(θ)D′_{n}(θ)g(θ)|_{K}^{q}
≤ (c_{3} (N + 2nq^{−1}))^{q} ∥gD_{n}^{m}∥_{K}^{q}
≤ c_{3}^{q}N^{q} (1 + 2q^{−1})^{q} c_{1}(1 + 3N) ∫_{−π}^{π} g^{q}(θ)D_{n}^{2}(θ) dθ

for every θ ∈ K (we take one-sided derivatives everywhere). By putting θ = 0, and noticing that

D′_{n}(0) = 0 and D_{n}(0)^{m} = (2n + 1)^{2/q} ≥ N^{2/q} ,

we get

(A.4.12)  |g′(0)|^{q} ≤ cN^{q} ∫_{−π}^{π} g^{q}(θ)(2π)^{−1}(2n + 1)^{−1}D_{n}^{2}(θ) dθ

with an absolute constant c. Now let f ∈ GTP_{n} be of the form (A.4.3) with each r_{j} ≥ 1. Let τ ∈ K be fixed. On applying (A.4.12) to g(θ) := f(θ + τ), we conclude that

|f′(τ)|^{q} ≤ cN^{q} ∫_{−π}^{π} f^{q}(θ)(2π)^{−1}(2n + 1)^{−1}D_{n}^{2}(θ − τ) dθ .

Hence

(A.4.13)  N^{−q}|f′(τ)|^{q} ≤ ∫_{−π}^{π} cf^{q}(θ)(2π)^{−1}(2n + 1)^{−1}D_{n}^{2}(θ − τ) dθ .

Since

(A.4.14)  ∫_{−π}^{π} (2π)^{−1}(2n + 1)^{−1}D_{n}^{2}(θ − τ) dθ = 1 ,

Jensen’s inequality (see E.7) and (A.4.3) imply that

χ(N^{−q}|f′(τ)|^{q}) ≤ ∫_{−π}^{π} χ(cf^{q}(θ))(2π)^{−1}(2n + 1)^{−1}D_{n}^{2}(θ − τ) dθ .

If we integrate both sides with respect to τ, Fubini’s theorem and (A.4.14) (on interchanging the role of θ and τ) yield the inequality of the theorem.

⊓ ⊔

Proof of Corollary A.4.13. If 0 < p ≤ 1, then Theorem A.4.12 yields the corollary with q = p and χ(x) = x. If 1 ≤ p < ∞, then the corollary follows from Theorem A.4.12 again with q = 1 and χ(x) = x^{p}. ⊓ ⊔

Proof of Theorem A.4.14. We distinguish two cases.

Case 1: p ≥ 1. Let f ∈ GAP_{N} be of the form (A.4.1) with each r_{j} ≥ 1. Then by E.10 b], g(θ) := f(cos θ) ∈ GTP_{N} is of the form (A.4.3) with each r_{j} ≥ 1. With the substitution x = cos θ, Corollary A.4.13 and Theorem A.4.11 imply that

∫_{−1}^{1} |f′(x)|^{p}(1 − x^{2})^{(p−1)/2} dx (A.4.15)

≤ c_{1}^{p}N^{p} ∫_{−1}^{1} f^{p}(x)(1 − x^{2})^{−1/2} dx
≤ c_{1}^{p}N^{p} exp(c_{2}pNN^{−1}) ∫_{−δ}^{δ} f^{p}(x)(1 − x^{2})^{−1/2} dx ,

where δ := max{1−N^{−2}, cos(π/16)} and c_{1} and c_{2} are appropriate absolute constants. Since p − 1 ≥ 0, it follows from (A.4.15) that

∫_{−δ}^{δ} |f′(x)|^{p} dx (A.4.16)

≤ (1 − δ^{2})^{(1−p)/2} ∫_{−δ}^{δ} |f′(x)|^{p}(1 − x^{2})^{(p−1)/2} dx
≤ (1 − δ^{2})^{(1−p)/2}c_{1}^{p}N^{p} exp(c_{2}p)(1 − δ^{2})^{−1/2} ∫_{−δ}^{δ} f^{p}(x) dx
≤ c_{3}^{p}N^{p−1}N^{p}N ∫_{−δ}^{δ} f^{p}(x) dx
= c_{3}^{p}N^{2p} ∫_{−δ}^{δ} f^{p}(x) dx ,

where c_{3} is also an absolute constant. Since (A.4.16) is valid for every f ∈ GAP_{N} of the form (A.4.1) with each r_{j} ≥ 1, the theorem follows by a linear shift from [−δ, δ] to [−1, 1].

Case 2: 0 < p ≤ 1. Let f ∈ GAP_{N} be of the form (A.4.1) with each r_{j} ≥ 1. Using the inequality |a + b|^{p} ≤ |a|^{p} + |b|^{p} for p ∈ (0, 1], we can deduce that

∫_{A}(|f′(cos θ)|| sin θ|^{1/p+1})^{p} dθ (A.4.17)

≤ ∫_{A}|(f(cos θ)| sin θ|^{1/p})′|^{p} dθ
+ ∫_{A}(f(cos θ)p^{−1}| sin θ|^{1/p−1}| cos θ|)^{p} dθ

for every measurable subset A of [−π, π). Applying Theorem A.4.12 (with χ(x) = x) to

g(θ) := f(cos θ)| sin θ|^{1/p} ∈ GTP_{N+1/p} ,

then using (A.4.17) with A := [−δ, δ], δ := 1 − (N + 1)^{−2}, we conclude, by the substitution x = cos θ, that

∫_{−δ}^{δ} |f′(x)|^{p}(1 − x^{2})^{p/2} dx (A.4.18)

≤ c_{1}(N + 1/p)^{p} ∫_{−1}^{1} f^{p}(x) dx + p^{−p} ∫_{−δ}^{δ} f^{p}(x)(1 − x^{2})^{−p/2} dx ,

where c_{1} is an absolute constant. Note that Theorem A.4.10, 0 < p ≤ 1, and the choice of δ imply that

(A.4.19)  ∫_{−1}^{1} f^{p}(x) dx ≤ c_{2} ∫_{−δ}^{δ} f^{p}(x) dx

with an absolute constant c_{2}. A combination of (A.4.18) and (A.4.19) yields

∫_{−δ}^{δ} |f′(x)|^{p} dx (A.4.20)

≤ (1 − δ^{2})^{−p/2} ∫_{−δ}^{δ} |f′(x)|^{p}(1 − x^{2})^{p/2} dx
≤ (1 − δ^{2})^{−p/2}(c_{1}c_{2}(N + 1/p)^{p} + p^{−p}(1 − δ^{2})^{−p/2}) ∫_{−δ}^{δ} f^{p}(x) dx
≤ 2^{p/2}(N + 1)^{p}(c_{1}c_{2}(N + 1/p)^{p} + p^{−p}2^{p/2}(N + 1)^{p}) ∫_{−δ}^{δ} f^{p}(x) dx
≤ c_{3}N^{2p} ∫_{−δ}^{δ} f^{p}(x) dx ,

where c_{3} is an absolute constant. Since (A.4.20) is valid for every f ∈ GAP_{N} of the form (A.4.1) with each r_{j} ≥ 1, the theorem follows by a linear shift from [−δ, δ] to [−1, 1]. ⊓ ⊔ Comments, Exercises, and Examples.

Most of the results of this section can be found in Erdélyi [91a] and [92a]; Erdélyi, Máté, and Nevai [92]; and Erdélyi, Li, and Saff [94]; however, the proofs are somewhat simplified here. For polynomials f ∈ P_{n} and for arbitrary q ∈ (0, ∞), Theorem A.4.5 was also proved by Kemperman and Lorentz [79]. An early version of Markov’s inequality in L_{p} for ordinary polynomials is proven in Hille, Szegő, and Tamarkin [37]. Weighted Markov- and Bernstein-type analogs of Theorems A.4.6 to A.4.8 are obtained in Erdélyi [92b]. Applications of the inequalities of this section are given in Erdélyi, Magnus, and Nevai [94] and in Erdélyi and Nevai [92], where bounds are established for orthonormal polynomials and related functions associated with (generalized) Jacobi weight functions. Further applications in the theory of orthogonal polynomials may be found in Freud [71] and Erdélyi [91d].

L_{p} extensions of Theorem 5.1.4 (Bernstein’s inequality) and Theorem 5.1.8 (Markov’s inequality) have been studied by a number of authors. The sharp L_{p} version of Bernstein’s inequality for trigonometric polynomials was first established by Zygmund [77] for p ≥ 1. Using an interpolation formula of M. Riesz, he proved that

(A.4.21)  ∫_{−π}^{π} |t′(θ)|^{p} dθ ≤ n^{p} ∫_{−π}^{π} |t(θ)|^{p} dθ

for every t ∈ T_{n} (see E.5 h] of Appendix 4). For 0 < p < 1, first Klein [51] and later Osval’d [76] proved (A.4.21) with a multiplicative constant c(p). Nevai [79a] showed that c(p) ≤ 8p^{−1}. Subsequently, Máté and Nevai [80] showed the validity of (A.4.21) with a multiplicative absolute constant, and then Arestov [81] proved (A.4.21) (with the best possible constant 1) for every 0 < p < 1. Golitschek and Lorentz [89] gave a very elegant proof of Arestov’s theorem.

The L_{p} analog of Markov’s inequality states that

(A.4.22)  ∫_{−1}^{1} |Q′(x)|^{p} dx ≤ c^{p+1}n^{2p} ∫_{−1}^{1} |Q(x)|^{p} dx

for every Q ∈ P_{n} and 0 < p < ∞, where c is an absolute constant. This can be obtained from Arestov’s theorem similarly to the way that Theorem A.4.14 is proven from Corollary A.4.13. To find the best possible constant in (A.4.22) is still an open problem even for p = 2 or p = 1.

The magnitude of

(A.4.23)  \frac{∥f′w∥_{[−1,1]}}{∥fw∥_{[−1,1]}} ,

(A.4.24)  \frac{|f′(y)w(y)|}{∥fw∥_{[−1,1]}} ,  −1 ≤ y ≤ 1 ,

and their corresponding L_{p} analogs for f ∈ P_{n} and generalized Jacobi weight functions

(A.4.25)  w(z) = ∏_{j=1}^{k} |z − z_{j}|^{r_{j}} ,  z_{j} ∈ ℂ , r_{j} ∈ (−1, ∞)

have been examined by several people. See, for example, Ditzian and Totik [88], Khalilova [74], Konjagin [78], Lubinsky and Nevai [87], Nevai [79a] and [79b], and Protapov [60], but a multiplicative constant depending on the weight function appears in these estimates. The magnitude of (A.4.23), (A.4.24), and their L_{p} analogs are examined in Erdélyi [92b] and [93], when both f and w are generalized nonnegative polynomials. In these inequalities only the degree of f, the degree of w, and a multiplicative absolute constant appear. The most general results are the following:

Theorem A.4.15. There exists an absolute constant c such that

∫_{−π}^{π} |f′(θ)|^{p}w(θ) dθ ≤ c^{p+1}(N + M)^{p}(Mp^{−1} + 1)^{p} ∫_{−π}^{π} |f(θ)|^{p}w(θ) dθ

and

∥f′w∥_{[−π,π]} ≤ c(N + M)(M + 1)∥fw∥_{[−π,π]}

for every f ∈ GTP_{N} of the form (A.4.3) with each r_{j} ≥ 1, for every w ∈ GTP_{M}, and for every p ∈ (0, ∞).

Theorem A.4.16. There exists an absolute constant c such that

∫_{−1}^{1} |f′(x)|^{p}w(x) dx ≤ c^{p+1}(N + M)^{2p}(Mp^{−1} + 1)^{2p} ∫_{−1}^{1} |f(x)|^{p}w(x) dx

and

∥f′w∥_{[−1,1]} ≤ c(N + M)^{2}∥fw∥_{[−1,1]}

for every f ∈ GAP_{N} of the form (A.4.1) with each r_{j} ≥ 1, for every w ∈ GAP_{M}, and for every p ∈ (0, ∞).

E.1 Another Representation of Generalized Nonnegative Polynomials.

a] Show that if f = ∏_{j=1}^{k} |Q_{j}|^{r_{j}} with each Q_{j} ∈ P_{n_{j}} and r_{j} > 0, then f ∈ GAP_{N} with N ≤ ∑_{j=1}^{k} r_{j}n_{j}. Similarly, if f = ∏_{j=1}^{k} |Q_{j}|^{r_{j}} with each Q_{j} ∈ T_{n_{j}} and r_{j} > 0, then f ∈ GTP_{N} with N ≤ ∑_{j=1}^{k} r_{j}n_{j}.

b] Show that if f ∈ GAP_{N} is of the form (A.4.1), then f = ∏_{j=1}^{m} |Q_{j}|^{r_{j}/2} with each Q_{j} ∈ P_{2} and 0 ≤ Q_{j} on ℝ. Similarly, if f ∈ GTP_{N} is of the form (A.4.3), then f = ∏_{j=1}^{m} |Q_{j}|^{r_{j}/2} with each Q_{j} ∈ T_{1} and 0 ≤ Q_{j} on ℝ.

E.2 The Derivative of an f ∈ GTP_{N} with Real Zeros. Show that if f ∈ GTP_{N} is of the form (A.4.3) with each r_{j} ≥ 1 and z_{j} ∈ ℝ, then |f′| ∈ GTP_{N} is of the form (A.4.3) with each r_{j} > 0 and z_{j} ∈ ℝ, and at least one of any two adjacent (in K) zeros of |f′| has multiplicity exactly 1.

E.3 Generalized Nonnegative Polynomials with Rational Exponents. Show that if f ∈ GAP_{N} is of the form (A.4.1) with rational exponents r_{j} = q_{j}/q, where q_{j}, q ∈ ℕ, then f^{2q} ∈ P_{2qN}, while if f ∈ GTP_{N} is of the form (A.4.3) with rational exponents r_{j} of the above form then f^{2q} ∈ T_{2qN}.

E.4 Proof of (A.4.6). Prove (A.4.6) from the explicit formula (2.1.1) for the Chebyshev polynomial T_{n}.

E.5 Sharpness of the Remez-Type Inequality for GAP_{N}. Let

f_{n}(x) := |T_{n} (\frac{2x + s}{2 − s})|^{N/n} ∈ GAP_{N} , n = 1, 2, ….

Show that

m({x ∈ [−1, 1] : f_{n}(x) ≤ 1}) = m([−1, 1 − s]) = 2 − s

and

lim_{n→∞} f_{n}(1) = (\frac{\sqrt{2} + \sqrt{s}}{\sqrt{2} − \sqrt{s}})^{N} .

The upper bound in Theorem A.4.1 is actually not achieved by an element of GAP_{N}; see Erdélyi, Li, and Saff [94].

E.6 Proof of Theorem A.4.2.

Hint: First assume that f ∈ GTP_{N} is of the form (A.4.3) with rational exponents r_{j} = q_{j}/q, where q_{j}, q ∈ ℕ. Then p := f^{2q} ∈ T_{2qN}, and the desired inequality can be obtained from Theorem 5.1.2 as in the proof of Theorem A.4.1. ⊓ ⊔

E.7 Corollaries of Theorems A.4.1 and A.4.2.

a] The inequality

m (\{x ∈ [−1, 1] : f(x) ≥ exp(−N\sqrt{s})∥f∥_{[−1,1]}\}) ≥ \frac{s}{8}

holds for every f ∈ GAP_{N} and 0 < s < 2. In particular,

m ({x ∈ [−1, 1] : f(x) ≥ e^{−1}∥f∥_{[−1,1]}}) ≥ (8N^{2} + 4)^{−1}

holds for every f ∈ GAP_{N}.

Inequalities for Generalized Polynomials in L_{p} 409

b] The inequality

m (\{θ ∈ [−π, π) : f(θ) ≥ exp(−Ns)∥f∥_{K}\}) ≥ \frac{s}{4}

holds for every f ∈ GTP_{N} and 0 < s < 2π. In particular,

m ({θ ∈ [−π, π) : f(θ) ≥ e^{−1}∥f∥_{K}}) ≥ (4(N + 1))^{−1}

holds for every f ∈ GTP_{N}.

E.8 Nonnegative Trigonometric Polynomials. Part a] restates E.3 c] of Section 2.4. Parts b] and c] discuss simple related observations. a] Let t ∈ T_{n} be nonnegative on ℝ. Show that there is a p ∈ P_{n}^{c} such that t(θ) = |p(e^{iθ})|^{2} for every θ ∈ ℝ. b] Let p ∈ P_{n}^{c} and t(θ) := |p(e^{iθ})|^{2} for every θ ∈ ℝ. Then t ∈ T_{n} and t is nonnegative on ℝ. c] Show that if t ∈ T_{n}^{c}, then there is a p ∈ P_{2n}^{c} such that |t(θ)| = |p(e^{iθ})| for every θ ∈ ℝ.

E.9 A Crucial Inequality in the Proof of Theorem A.4.5. Show that

|P (z)| ≤ (\frac{2}{1 + r} )^{n} |P (rz)| , z ∈ ∂D , r ∈ [0, 1]

for every P ∈ P_{n}^{c} having all its zeros outside the open unit disk D. Hint: Let P (z) = c ∏_{j=1}^{m} (z − z_{j}), where c ∈ ℂ, z_{j} ∈ ℂ \ D, and m ≤ n. Show that

|z − z_{j}| ≤ \frac{2}{1 + r} |rz − z_{j}| , j = 1, 2, …, m , r ∈ [0, 1] .

⊓ ⊔

E.10 f ∈ GAP_{N} Implies f(cos θ) ∈ GTP_{N}. a] Show that if f ∈ GAP_{N}, then g(θ) = f(cos θ) ∈ GTP_{N}. b] Show that if f ∈ GAP_{N} is of the form (8.1.1) with each r_{j} ≥ 1, then g(θ) = f(cos θ) ∈ GTP_{N} is of the form (A.4.3) with each r_{j} ≥ 1.

E.11 A Property of Q_{n,ω}. Let Q_{n,ω} be defined by (A.4.9) and (A.4.10). Show that there is an absolute constant c_{6} > 1 such that Q_{n,ω}(π) ≥ c_{6}. Hint: Use the explicit formula (2.1.1) for the Chebyshev polynomial T_{n}. ⊓ ⊔ E.12 A Property of the Zeros of a g ∈ GTP_{N}. Assume that g ∈ GTP_{N} is of the form (A.4.3) and let

M := ∑_{{j=1 \atop z_{j}∈[π−ρ,π+ρ]}}^{m} r_{j} .

Show that M ≤ 3Nρ∥g∥_{[−1,1]}^{−1} .

Hint: First assume that each r_{j} is rational with common denominator q ∈ ℕ, and apply E.12 of Section 5.1 to p := g^{2q} ∈ T_{2qN}. ⊓ ⊔

E.13 Extremal Functions for the Bernstein-Type Inequality.

a] Let r_{j} ≥ 1, j = 1, 2, …, m, be fixed real numbers. Show that there exists an f̃ ∈ GTP_{N} of the form

(A.4.26)  f̃(z) = ∏_{j=1}^{m} | sin((z − ζ̃_{j})/2)|^{r_{j}} ,  ζ̃_{j} ∈ ℂ

such that

\frac{|f̃′(π)|}{∥f̃∥_{K}} = sup_{f} \frac{|f′(π)|}{∥f∥_{K}} ,

where the supremum in the right-hand side is taken for all f ∈ GTP_{N} of the form

(A.4.27) f(z) = |ω| ∏_{j=1}^{m} | sin((z − z_{j})/2)|^{r_{j}} ,  z_{j} ∈ ℂ , 0 ≠ ω ∈ ℂ .

Hint: Write each f of the form (A.4.27) for which the supremum is taken as

f(z) = |ω_{0}| ∏_{j=1}^{m} |ω_{j} sin((z − z_{j})/2) sin((z − \overline{z}_{j})/2)|^{r_{j}/2} ,

where the numbers ω_{j} > 0 are defined by

∥ω_{j} sin((z − z_{j})/2) sin((z − \overline{z}_{j})/2)∥_{K} = 1 , j = 1, 2, …, m ,

and then use a compactness argument for each factor separately. ⊓ ⊔

b] Let f̃ be as in part a]. Show that each zero of f̃ is real, that is, ζ̃_{j} ∈ ℝ for each j in (A.4.26).

Hint: Assume that there is an index 1 ≤ j ≤ m such that ζ_{j} ∈ ℂ \ ℝ. Then

f̃_{ϵ}(z) := f̃(z) (1 − \frac{ϵ sin^{2}((z − π)/2)}{sin((z − ζ_{j})/2) sin((z − \overline{ζ}_{j})/2)})^{r_{j}} ∈ GTP_{N}

with sufficiently small ϵ > 0 contradicts the maximality of f̃. ⊓ ⊔ E.14 A Corollary of the Remez-Type Inequality for GAP_{N}. Suppose that α := 1 − (1 + N)^{−2}. Show that there is an absolute constant c_{9} > 0 such that

∥f∥_{[−1,1]} ≤ c_{9}∥f∥_{[−α,α]}

for every f ∈ GAP_{N}.

Hint: This is a corollary of Theorem A.4.1. ⊓ ⊔

E.15 Proof of Theorem A.4.10.

Outline. For f ∈ GAP_{N}, let

I(f) := \{x ∈ [−1, 1] : (χ(f(x)))^{p} ≥ exp(−5pN\sqrt{2s})∥χ(f)∥_{[−1,1]}^{p}\} .

From Theorem 5.1.1, 0 < s ≤ 1/2, and the assumptions prescribed for χ it follows that m(I(f)) ≥ 2s. Let I := A ∩ I(f). Since m([−1, 1] \ A) ≤ s, m(I) ≥ s. Therefore

∫_{[−1,1]\A}(χ(f(x)))^{p}dx ≤ ∫_{[−1,1]\A} ∥χ(f)∥_{[−1,1]}^{p} dx
≤ exp(5pN\sqrt{2s}) ∫_{I}(χ(f(x)))^{p} dx
≤ exp(5pN\sqrt{2s}) ∫_{A} (χ(f(x)))^{p} dx .
⊓ ⊔

E.16 Proof of Theorem A.4.11.

Hint: For f ∈ GTP_{N}, let

I(f) := {θ ∈ [−π, π] : (χ(f(θ)))^{p} ≥ exp(−8pNs)∥χ(f)∥_{K}^{p}} .

From Theorem 5.2.2, 0 < s ≤ π/4, and the assumptions for χ, it follows that m(I(f)) ≥ 2s. Now finish the proof as in the hint for E.1. ⊓ ⊔

E.17 Sharpness of Theorem A.4.10.

a] Show that there exists a sequence of polynomials Q_{n} ∈ P_{n} and an absolute constant c > 0 such that

∫_{−1}^{1} |Q_{n}(x)|^{p} dx ≥ s(1 + exp(cpn\sqrt{s})) ∫_{−1}^{1−s} |Q_{n}(x)|^{p} dx

for every n ∈ ℕ, s ∈ (0, 1], and p ∈ (0, ∞).

Hint: Study the Chebyshev polynomials T_{n} transformed linearly from [−1, 1] to [−1, 1 − s]. ⊓ ⊔ b] Show that there exist a sequence of trigonometric polynomials t_{n} ∈ T_{n} and an absolute constant c > 0 such that

∫_{−π}^{π} |t_{n}(θ)|^{p} dθ ≥ s(1 + exp(cpns)) ∫_{−π+s/2}^{π−s/2} |t_{n}(θ)|^{p} dθ

for every p ∈ ℕ, s ∈ (0, π], and p ∈ (0, ∞).

Hint: Study Q_{n,ω} defined by (A.4.9) with ω := π − s/2. ⊓ ⊔

E.18 Sharpness of Corollary A.4.13 and Theorem A.4.14.

a] Find a sequence of trigonometric polynomials t_{n} ∈ T_{n} that shows the sharpness of Corollary A.4.13 up to the constant c > 0 for every p ∈ (0, ∞) simultaneously.

Hint: Take t_{n}(θ) := cos nθ. ⊓ ⊔

b] For every p ∈ (0, ∞), find a sequence of polynomials Q_{n,p} ∈ P_{n} which shows the sharpness of Theorem A.4.14 up to the constant c > 0.

Outline. Let L_{k} ∈ P_{k} be the kth orthonormal Legendre polynomial on [−1, 1] (see E.5 of Section 2.3), and let

G_{m}(x) := c ∑_{k=0}^{m} L′_{k}(1)L_{k}(x) ,

where c is chosen so that

(A.4.28)  ∫_{−1}^{1} G_{m}^{2}(x) dx = 1 .

Show that there exist absolute constants c_{1} > 0 and c_{2} > 0 (independent of m) such that

(A.4.29)  |G_{m}(1)| ≥ c_{1}m  and  |G′_{m}(1)| ≥ c_{2}m^{3} .

For a fixed n ∈ ℕ, let u := ⌊2/p⌋ + 1 and m := ⌊n/u⌋. If m ≥ 1, then let Q_{n,p} := G_{m}^{u} ∈ P_{n}, otherwise let Q_{n,p}(x) := x ∈ P_{n}. If m = 0, then the calculation is trivial, so let m ≥ 1. Using (A.4.29) and the Nikolskii-type inequality of Theorem A.4.4, show that there exists an absolute constant c_{3} > 0 such that

(A.4.30)  ∫_{−1}^{1} |Q′_{n,p}(x)|^{p} dx ≥ c_{3}^{p+1}m^{(u+2)p}(1 + pn)^{−2} .

Use the inequality

∫_{−1}^{1} |Q_{n,p}(x)|^{p} dx = ∫_{−1}^{1} |G_{m}(x)|^{up} dx ≤ ∥G_{m}∥_{[−1,1]}^{up−2} ∫_{−1}^{1} G_{m}^{2}(x) dx ,

the Nikolskii-type inequality of Theorem A.4.4, and (A.4.28) to show that there is an absolute constant c_{4} > 0 such that

(A.4.31)  ∫_{−1}^{1} |Q_{n,p}(x)|^{p} dx ≤ c_{4}^{p+1}m^{up−2} .

Finally, combine (A.4.30) and (A.4.31). Note that if p > 2, then m = ⌊n⌋, while if p ≤ 2, then \frac{1}{4}p(n − 1) ≤ m ≤ 2pn. Note also that p^{p} is between two positive bounds for p ∈ (0, 2]. ⊓ ⊔

E.19 Sharpness of Theorems A.4.3 and A.4.4.

a] Let q ∈ (0, ∞) be fixed. Show that there exists a sequence of polynomials Q_{n,q} ∈ P_{n} and an absolute constant c > 0 such that

∥Q_{n,q}∥_{[−1,1]} ≥ c^{1+1/q}(1 + qn)^{2/q}∥Q_{n,q}∥_{L_{q}[−1,1]}

for every n ∈ ℕ.

Hint: Study Q_{n,p} with p = q in the hint to the previous exercise. ⊓ ⊔

b] Let Q_{n,q} ∈ P_{n} be the same as in part a]. Show that there exist absolute constants c_{1} > 0 and c_{2} > 0 such that

∥Q_{n,q}∥_{L_{p}[−1,1]} ≥ c_{1}^{1+1/q}c_{2}^{1/p}(1 + qn)^{2/q}(1 + pn)^{−2/p}∥Q_{n,q}∥_{L_{p}[−1,1]}

for every n ∈ ℕ and 0 < q < p ≤ ∞.

Hint: Combine part a] and the Nikolskii-type inequality of Theorem A.4.4. ⊓ ⊔

c] Let q ∈ (0, ∞) be fixed. Show that there exists a sequence of trigonometric polynomials t_{n,q} ∈ T_{n} and an absolute constant c > 0 such that

∥t_{n,q}∥_{K} ≥ c^{1+1/q}(1 + qn)^{1/q}∥t_{n,q}∥_{L_{q}(K)}

for every n ∈ ℕ.

Hint: Let t_{n,q}(θ) := Q_{n,q}(cos θ), where Q_{n,q} are the same as in part a]. Use part a] and the Schur-type inequality of Theorem A.4.5. ⊓ ⊔

d] Let t_{n,q} ∈ T_{n} be the same as in part c]. Show that there exist absolute constants c_{1} > 0 and c_{2} > 0 such that

∥t_{n,q}∥_{L_{p}(K)} ≥ c_{1}^{1+1/q}c_{2}^{1/p}(1 + qn)^{1/q}(1 + pn)^{1/p}∥t_{n,q}∥_{L_{q}(K)}

for every n ∈ ℕ and 0 < q < p ≤ ∞.

Hint: Combine part a] and the Nikolskii-type inequality of Theorem A.4.3. ⊓ ⊔ E.20 Jensen’s Inequality. Let χ be a real-valued convex function defined on [0, ∞), and let f and w be nonnegative Riemann integrable functions on the interval [a, b], where ∫_{a}^{b} w = 1. Show that

χ (∫_{a}^{b} f(x)w(x) dx) ≤ ∫_{a}^{b} χ(f(x))w(x) dx .

Hint: First assume that w is a step function. Use the fact that the convexity of χ implies its continuity, and hence the functions χ(f)w and fw are Riemann integrable. ⊓ ⊔

E.21 A Pointwise Remez-Type Inequality for GAP_{N}.

a] Show that there exists an absolute constant c > 0 such that

|p(y)| ≤ exp (cn min \{\frac{s}{\sqrt{1 − y^{2}}} , \sqrt{s}\}) , y ∈ [−1, 1]

for every p ∈ P_{n} and s ∈ (0, 1] satisfying

m({x ∈ [−1, 1] : |p(x)| ≤ 1}) ≥ 2 − s ,

that is, with the notation of Theorem 5.1.1, for every p ∈ P_{n}(s) with s ∈ (0, 1].

Proof. Assume, without loss of generality, that y ∈ [0, 1]. Let

a := y + \frac{1}{2}(1 − y), α := arccos a ,
β := 2 arccos y − arccos a , ω := π − \frac{1}{2}(β − α) ,

Q_{n,ω}(θ) := T_{2n} (\frac{sin(θ/2)}{sin(ω/2)}) ,

where T_{2n} is the Chebyshev polynomial of degree 2n defined by (2.1.1), and let

Q_{n,α,β}(θ) := Q_{n,ω} (\frac{1}{2}(θ − (2π − (α + β)))) .

Associated with p ∈ P_{n}(s), we introduce

g(θ) := p(cos θ)Q_{n,α,β}(θ) ∈ T_{2n} .

Obviously

|Q_{n,α,β}(θ)| ≤ 1 , θ ∈ [0, 2π) \ (α, β)

and

∥Q_{n,α,β}∥_{[0,2π]} = Q_{n,α,β} (\frac{1}{2}(α + β)) = Q_{n,ω}(π) .

The definition of ω implies that there exist absolute constants c_{1} > 0 and c_{2} > 0 such that

c_{1}\sqrt{1 − y^{2}} ≤ π − ω ≤ c_{2}\sqrt{1 − y^{2}} .

This, together with E.3 c], yields that there are absolute constants c_{3} > 0 and c_{4} > 0 such that

Q_{n,ω}(π) ≥ exp(c_{3}n\sqrt{1 − y^{2}}) ≥ exp(5n\sqrt{s})

whenever y ∈ [0, 1 − c_{4}s]. It follows now from Theorem 5.1.1 that

|g(θ)| ≤ exp(5n\sqrt{s}) ≤ Q_{n,ω}(π)

for every θ ∈ [0, 2π) \ (α, β) and y ∈ [0, 1 − c_{4}s]. Furthermore

|g(θ)| ≤ Q_{n,ω}(π)

for every θ ∈ (α, β) for which |p(cos θ)| ≤ 1. Note that

| cos θ| ≤ 1 − \frac{1}{2}(1 − y) , θ ∈ (α, β)

and hence p ∈ P_{n}(s) with s ∈ (0, 1] implies that there exists an absolute constant c_{5} > 0 such that

m(\{θ ∈ (α, β) : |p(cos θ)| ≥ 1\}) ≤ \frac{c_{5}s}{\sqrt{1 − y^{2}}} .

Therefore

f := (Q_{n,ω}(π))^{−1}g ∈ T_{2n}

satisfies

m({θ ∈ [0, 2π) : |f(θ)| ≤ 1}) ≥ 2π − s̃

with

s̃:= \frac{c_{5}s}{\sqrt{1 − y^{2}}} , y ∈ [0, 1 − c_{4}s] .

Applying Theorem 5.1.2 with f and s̃, we conclude that

|p(y)| = |p (cos (\frac{1}{2}(α + β)))| = (Q_{n,ω}(π))^{−1}g (\frac{1}{2}(α + β))
=f (\frac{1}{2}(α + β)) ≤ exp(4ns̃) ≤ exp (\frac{c_{5}s}{\sqrt{1 − y^{2}}} )

whenever s̃ ∈ (0, π/2] and y ∈ [0, 1 − c_{4}s]. If s ∈ (0, 1], but s̃ > π/2 or y ∈ [1 − c_{4}s, 1], then Theorem 5.1.1 yields the required inequality. ⊓ ⊔ b] Show that the inequality of part a] is sharp up to the absolute constant c > 0.

Hint: Assume, without loss, that y ∈ [0, 1]. Show that there exists an absolute constant c_{1} > 0 such that the polynomials

W_{n,y,s}(x) := T_{n} (\frac{2x}{2 − x} + \frac{s}{2 − s}) ∈ P_{n}(s) ,

where T_{n} is the Chebyshev polynomial of degree n defined by (2.1.1), satisfy

|W_{n,y,s}(y)| ≥ exp(c_{1}n\sqrt{s})

for every n ∈ ℕ, y ∈ [1 − s/2, 1], and s ∈ (0, 1].

If y ∈ [0, 1 − s/2], then let

a := y + \frac{s}{4} , α := arccos a ,

β := 2 arccos y − arccos a , ω := π − \frac{β − α}{2} ,

Q_{n,α,β}(θ) := T_{2n} (\frac{sin((θ + π − (α + β)/2)/2)}{sin(ω/2)}) ,
R_{n,α,β}(θ) := \frac{1}{2}(Q_{n,α,β}(θ) + Q_{n,α,β}(−θ)) ,

and we define W_{n,y,s} for every n ∈ ℕ, y ∈ [0, 1 − s/2), and s ∈ (0, 1] by

R_{n,α,β}(θ) = W_{n,y,s}(cos θ) , W_{n,y,s} ∈ P_{n} .

Show that W_{n,y,s} ∈ P_{n}(s) and that there exists an absolute constant c > 0 such that

|W_{n,y,s}(y)| ≥ exp (cn \frac{s}{\sqrt{1 − y^{2}}} )

for every n ∈ ℕ, y ∈ [0, 1 − s/2), and s ∈ (0, 1]. ⊓ ⊔

c] Extend the validity of part a] to the class GAP_{N}; that is, prove that there exists an absolute constant c > 0 such that

|f(y)| ≤ exp (cN min \{\frac{s}{\sqrt{1 − y^{2}}} , \sqrt{s}\}) , y ∈ [−1, 1]

for every f ∈ GAP_{N} and s ∈ (0, 1] satisfying

m({x ∈ [−1, 1] : f(x) ≤ 1}) ≥ 2 − s .