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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

##### 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)
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##### 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
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