Abstract
We study the rate of uniform approximation by Nörlund means of the rectangular partial sums of double Fourier series of continuous functionsf(x, y), 2π-periodic in each variable. The results are given in terms of the modulus of symmetric smoothness defined by
As a special case we obtain the rate of uniform approximation to functionsf(x,y) in Lip({α, β}), the Lipschitz class, and inZ({α, β}), the Zygmund class of ordersα andβ, 0<α,β ≤ l, as well as the rate of uniform approximation to the conjugate functions\(\tilde f^{(1,0)} (x,y), \tilde f^{(0,1)} (x,y)\) and\(\tilde f^{(1,1)} (x,y)\).
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References
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Communicated by Tom. H. Koornwinder.
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Móricz, F., Rhoades, B.E. Approximation by Nörlund means of double fourier series to continuous functions in two variables. Constr. Approx 3, 281–296 (1987). https://doi.org/10.1007/BF01890571
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DOI: https://doi.org/10.1007/BF01890571
Key words and phrases
- Double Fourier series of continuous functions
- Modulus of continuity
- Modulus of smoothness
- Lipschitz classes and Zygmund classes of functions in two variables
- Doubleconjugate series
- Conjugate functions in two variables
- Nörlund means
- (C, γ, δ)-means
- Kernelestimates
- The rate of uniform approximation