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Wavelet approximation and Fourier widths of classes of periodic functions of several variables. I

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Abstract

We obtain characterizations (and prove the corresponding equivalence of norms) of function spaces B sm pq (\( \mathbb{I} \) k) and L sm pq (\( \mathbb{I} \) k) of Nikol’skii-Besov and Lizorkin-Triebel types, respectively, in terms of representations of functions in these spaces by Fourier series with respect to a multiple system \( \mathcal{W}_m^\mathbb{I} \) of Meyer wavelets and in terms of sequences of the Fourier coefficients with respect to this system. We establish order-sharp estimates for the approximation of functions in B sm pq (\( \mathbb{I} \)) and L sm pq (\( \mathbb{I} \) k) by special partial sums of these series in the metric of L r (\( \mathbb{I} \) k) for a number of relations between the parameters s, p, q, r, and m (s = (s 1, ..., s n ) ∈ ℝ n+ , 1 ≤ p, q, r ≤ ∞, m = (m 1, ..., m n ) ∈ ℕn, k = m 1 +... + m n , and \( \mathbb{I} \) = ℝ or \( \mathbb{T} \)). In the periodic case, we study the Fourier widths of these function classes.

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  1. Institute of Mathematics, ul. Pushkina 125, Almaty, 050010, Kazakhstan

    D. B. Bazarkhanov

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Original Russian Text © D.B. Bazarkhanov, 2010, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2010, Vol. 269, pp. 8–30.

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Bazarkhanov, D.B. Wavelet approximation and Fourier widths of classes of periodic functions of several variables. I. Proc. Steklov Inst. Math. 269, 2–24 (2010). https://doi.org/10.1134/S0081543810020021

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