×

zbMATH — the first resource for mathematics

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

MSC:
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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Fefferman, C.; Stein, E. M., H^p spaces of several variables, Acta Math., 189, 137-193 (1972) · Zbl 0257.46078
[2] Nikol’Skij, S. M., Approximation of functions of several variables and imbedding theorems (1969), Moscow: Nauka, Moscow
[3] Stein, E. M., Singular integrals and differentiability properties of functions (1970), Princeton: Princeton Univ. Press, Princeton · Zbl 0207.13501
[4] Storozenko, E. A., Approximation of functions of class H^p, 0<p≤1, Mat. Sb., 105, 147, 601-621 (1978)
[5] Storozenko, E. A., Theorems of Jackson type in H^p, 0<p<1, Izv. Akad. Nauk U.S.S.R. Ser. Mat., 44, 946-962 (1980) · Zbl 0455.42002
[6] Taibleson, M. H., On the theory of Lipschitz spaces of distributions in euclidean n-space, I, J. Math. Mec., 13, 407-480 (1964)
[7] Taibleson, M. H.; Weiss, G., The molecular characterization of certain Hardy spaces, Astérisque, 77, 67-149 (1980) · Zbl 0472.46041
[8] Zygmund, A., Trigonometric series (1968), Cambridge: Cambridge Univ. Press, Cambridge · JFM 58.0296.09
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.