On Approximation Properties of Multivariate Class of Nonlinear Singular Integral Operators

Gumrah Uysal; Vishnu Narayan Mishra; Ertan Ibikli

Gumrah Uysal Department of Computer Technologies, Karabuk University, Karabuk, Turkey
Vishnu Narayan Mishra Department of Mathematics, Indira Gandhi University, Lalpur, Madhya Pradesh, India
Ertan Ibikli Ankara University, Faculty of Science, Department of Mathematics, Ankara, Turkey

Keywords: generalized Lebesgue point; rate of convergence; Lipschitz condition; nonlinear analysis.

Abstract

In the present paper, we study the pointwise approximation of nonlinear multivariate singular integral operators having convolution type kernels of the form:
T (f; x) =
Z
D
K (t 􀀀 x; f(t)) dt; x 2 D; 2 ;
where D =
n
i=1 hai; bii is open, semi-open or closed multidimensional arbitrary
bounded box in Rn or D = Rn and is non-empty the set of non-negative
indices, at a -generalized Lebesgue point of f 2 Lp(D): Also, we investigate
the corresponding rates of convergences at this point.

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