ISSN:
1432-0940
Keywords:
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
;
Primary 41A50
;
Secondary 42B05
;
40G05
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 $$\begin{gathered} \omega _2 \left( {f,\delta _1 ,\delta _2 } \right) = \mathop {\sup }\limits_{x,y} \mathop {\sup }\limits_{\left| u \right| \leqslant \delta _1 ,\left| v \right| \leqslant \delta _2 } \left| {f\left( {x + u,y + v} \right)} \right. + f\left( {x + u,y - v} \right) + f\left( {x - u,y + v} \right) \hfill \\ + \left. {f\left( {x - u,y - v} \right) + 4f\left( {x,y} \right)} \right| for \delta _1 ,\delta _2 \geqslant 0. \hfill \\ \end{gathered} $$ 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 $$\tilde f^{(1,1)} (x,y)$$ .
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01890571
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