A Characterization of Extremal Elements in Some Linear Problems

Abstract

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.

On One Inequality of Different Metrics for Trigonometric Polynomials

“…As mentioned above, statements similar to Theorem 2 can be found in the authors' papers, personal and with co-authors: [5][6][7]9]. The ideas contained in these statements have been summarized in [3]. The fact that we need [3] holds for inequalities similar to inequality (1.16) for algebraic polynomials in spaces L p with weight.…”

“…We find it difficult in some cases to make precise references to this paper. Therefore, we have to repeat some arguments from [3] with clarifications and explanations.…”

“…The ideas contained in these statements have been summarized in [3]. The fact that we need [3] holds for inequalities similar to inequality (1.16) for algebraic polynomials in spaces L p with weight. Because of this, we first rewrite inequality (1.16) in terms of algebraic polynomials, use Theorem 4 from [3], and then make a conclusion related to inequality (1.16).…”

“…P r o o f. Most of the results of this lemma are contained in [3]. We find it difficult in some cases to make precise references to this paper.…”

“…In Theorem 4 of [3], in particular, the following properties of a polynomial ̺ n extremal in the inequality (3.12) were proved.…”