(untitled) (Analysis in Theory and Applications, 2017)
In this paper some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type ( T_λ f) ( x,y)=_a^b_a^bf(t,s) K_λ(t-x,s-y) dsdt, x,y(a,b), λ Λ [ 0,∞ ), are given. Here f belongs to the function space L_1( a,b^2), where a,b is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator ( T_λ f)( x,y) and the others for its Fatou-type pointwise convergence to f(x_0,y_0), as (x,y,λ ) tends to (x_0,y_0,λ_0). In contrast to previous works, the kernel functions K_λ(u,v) don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1,6,8,10,11,13] in three dimensional frame and especially the very recent paper [15].