Inequalities for entire functions of finite degree and their application to the theory of differentiable functions of several variables

“…Inequality (2.6) is a special case of an inequality between different metrics, or the Nikol'skii inequality. Such inequalities appeared for the first time in Nikol'skii's paper [15] and, shortly after that, in a paper by Szegő and Zygmund [18]. Similar inequalities and, more generally, inequalities between the uniform norm and weighted integral norms of algebraic and trigonometric polynomials and their derivatives have been studied over a period of more than 150 years, starting with the works of Chebyshev and his students-the Markov brothers.…”

Section: Pointwise Nikol'skii Inequality For Algebraic Polynomials Onmentioning

confidence: 90%

A Characterization of Extremal Elements in Some Linear Problems

Arestov¹

2017

UMJ

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Abstract:We give a characterization of elements of a subspace of a complex Banach space with the property that the norm of a bounded linear functional on the subspace is attained at those elements. In particular, we discuss properties of polynomials that are extremal in sharp pointwise Nikol'skii inequalities for algebraic polynomials in a weighted L q -space on a finite or infinite interval.Key words: Complex Banach space, Bounded linear functional on a subspace, Algebraic polynomial, Pointwise Nikol'skii inequality. Bounded linear functionals in complex Banach spaces Introduction. Statement of the problemLet X = X C be a complex Banach space (more precisely, a Banach space over the field C of complex numbers), let S(X) be its unit sphere, and let X * = X * C be the dual space of X, i.e., the space of complex-valued bounded linear (over the field C of complex numbers) functionals F on X with the norm F X * = sup{|F (x)| : x ∈ X, x X = 1}.Let P be a (closed) subspace of X, and let ψ be a bounded linear functional on P . We denote by D(ψ; P ) = sup{|ψ(p)| : p ∈ P, p X = 1} (1.1) the norm of the functional ψ on the subspace P . In what follows, we assume that ψ ≡ 0, so that D(ψ; P ) > 0. The value D(ψ; P ) is the smallest possible (the best) constant in the inequalityNonzero elements p of the subspace P with the property that inequality (1.2) turns into an equality for them (if such elements exist) will be called extremal elements in this inequality. Elements p of the unit sphere S(P ) = S(X) ∩ P of the subspace P that solve problem (1.1), i.e., those with the property that the supremum in (1.1) is attained at p, will be called extremal elements in problem (1.1). We will use the same terminology also in other similar situations. It is clear that an element ∈ P is extremal in inequality (1.2) if and only if the element / X is extremal in problem (1.1). In this sense, extremal elements in problem (1.1) and inequality (1.2) coincide. The aim of this papers is exactly to characterize extremal elements in inequality (1.2) or in problem (1.1), which is the same.

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“…In particular, the sufficient conditions for a sequence of complex numbers {λ k } k∈Z n in order to make it a multiplier of multiple trigonometric Fourier series from Lp[0;1] n to…”

Section: Introductionmentioning

confidence: 99%

“…It is said that the sequence of complex numbers λ = {λ k } k∈Z is a trigonometrical Fourier series multiplier from Lp [0,1] Analogously the problem about Fourier transform multipliers can be formulated as follows:…”

Section: Introductionmentioning

confidence: 99%