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Approximation of bandlimited functions by finite exponential sums

Authors:

S Norvidas

Institution:

(1) Institute of Mathematics and Informatics, Akademijos 4, LT-08663 Vilnius, Lithuania

Abstract:

For a compact set K in ℝⁿ, let B ² _K be the set of all functions f ∈ L ²(ℝ²) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B ² _K by finite exponential sums

$\sum\limits_{k \in {\mathbb{Z}^n} \cap \left( {{\tau \mathord{\left/{\vphantom {\tau \pi }} \right.} \pi }} \right)K} {{c_k}{{\text{e}}^{i\frac{\pi }{\tau }\left( {x,k} \right)}}}$

in the space $L^{\rm 2}(\tau\mathbb{T}^n),\ \mathbb{T}^n=-1,1]^n$ , as τ → ∞.

Keywords:

Fourier transforms bandlimited functions entire functions of exponential type exponential sums trigonometric polynomials Bernstein spaces Paley– Wiener spaces

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