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Approximation in Hardy spaces. (English) Zbl 0558.41017
The author studies the extension of the classical Jackson-Bernstein theorems to Hardy spaces \(H^ p({\mathbb{R}}^ N)\), \(0<p<\infty\), over the n-dimensional Euclidean space \({\mathbb{R}}^ N\). Direct and converse theorems are proved on approximation for distributions in a space \(\Lambda (p,{\mathbb{R}}^ N)\), the space of all distributions in \(H^ p({\mathbb{R}}^ N)\) equipped with a suitable norm, by means of entire- functions of exponential type (the analogue in \({\mathbb{R}}^ N\) of trigonometric polynomials). As consequences of the main theorems, certain results on embedding theorems for spaces \(\Lambda (p,{\mathbb{R}}^ N)\) are also deduced.
Reviewer: G.D.Dikshit

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
42B30 \(H^p\)-spaces
41A30 Approximation by other special function classes
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Full Text: DOI
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