Wavelet approximation and Fourier widths of classes of periodic functions of several variables. I |
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Authors: | D. B. Bazarkhanov |
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Affiliation: | 1.Institute of Mathematics,Almaty,Kazakhstan |
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Abstract: | We obtain characterizations (and prove the corresponding equivalence of norms) of function spaces B pq sm ($
mathbb{I}
$
mathbb{I}
k ) and L pq sm ($
mathbb{I}
$
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}
$
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 pq sm ($
mathbb{I}
$
mathbb{I}
) and L pq sm ($
mathbb{I}
$
mathbb{I}
k ) by special partial sums of these series in the metric of L r ($
mathbb{I}
$
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 ) ∈ ℕ n , k = m 1 +... + m n , and $
mathbb{I}
$
mathbb{I}
= ℝ or $
mathbb{T}
$
mathbb{T}
). In the periodic case, we study the Fourier widths of these function classes. |
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Keywords: | |
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