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Weighted approximation of entire functions of exponential type of several variables by trigonometric polynomials

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Abstract

Letf be an entire function (in Cn) of exponential type for whichf(x)=0(ϕ(x)) on the real subspace\(\mathbb{R}^w (\phi \geqslant 1,{\mathbf{ }}\mathop {\lim }\limits_{\left| x \right| \to \infty } \phi (x) = \infty )\) and ∀δ>0∃Cδ>0

$$\left| {f(z)} \right| \leqslant C_\delta \exp \left\{ {h_s (y) + S\left| z \right|} \right\},z = x + iy$$

where h, (x)=sup〈3, x〉, S being a convex set in ℝn. Then for any ɛ, ɛ>0, the functionf can be approximated with any degree of accuracy in the form p→\(\mathop {\sup }\limits_{x \in \mathbb{R}^w } \frac{{\left| {P(x)} \right|}}{{\varphi (x)}}\) by linear combinations of functions x→expi〈λx〉 with frequenciesX belonging to an ɛ-neighborhood of the set S.

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Authors
  1. V. E. Katznel'son
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Metematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 69–79, 1976.

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Katznel'son, V.E. Weighted approximation of entire functions of exponential type of several variables by trigonometric polynomials. J Math Sci 16, 1095–1101 (1981). https://doi.org/10.1007/BF02427718

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