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We investigate the pointwise and uniform convergence of the symmetric rectangular partial (also called Dirichlet) integrals of the double Fourier integral of a function that is Lebesgue integrable and of bounded variation over ℝ^{2}. Our theorem is a two-dimensional extension of a theorem of Móricz (see Theorem 3 in [10]) concerning the single Fourier integrals, which is more general than the two-dimensional extension given by Móricz himself (see Theorem 3 in [11]).

## Abstract

Let

*P*and

*Q*denote polynomials (algebraic or trigonometric) of degree ≦

*n*. Theorem 2a. If for a continuous 2π-periodic function f the condition

*β*

_{n}} be a non-increasing sequence of positive numbers such that

_{0}for which R

_{n}[f

_{0}]≦β

_{n}for all n=1 and yet the Fourier series of f

_{0}diverges at x=0.

*R*

_{n}

*[f]*may be replaced in these theorems by

*M*

_{n}

*[f]*, where

*M*

_{n}

*[f]*is the minimal uniform deviation of

*f(x)*from piecewise monotonie functions

*М*

_{n}

*(х)*of order ≦

*n*.

## Abstract

The aim of this paper is to continue our investigations started in [15], where we studied the summability of weighted Lagrange
interpolation on the roots of orthogonal polynomials with respect to a weight function *w*. Starting from the Lagrange interpolation polynomials we constructed a wide class of discrete processes which are uniformly
convergent in a suitable Banach space (*C*
_{ρ}, ‖‖_{ρ}) of continuous functions (ρ denotes (another) weight). In [15] we formulated several conditions with respect to *w*, ρ, (*C*
_{ρ}, ‖‖_{ρ}) and to summation methods for which the uniform convergence holds. The goal of this part is to study the special case when
*w* and ρ are Freud-type weights. We shall show that the conditions of results of [15] hold in this case. The order of convergence
will also be considered.

## Abstract

Chaundry and Jolliffe [1] proved that if *a*
_{k} is a nonnegative sequence tending monotonically to zero, then a necessary and sufficient condition for the uniform convergence
of the series Σ_{k=1}
^{∞}
*a*
_{k} sin *kx* is lim_{k→∞}
*ka*
_{k} = 0. Lately, S. P. Zhou, P. Zhou and D. S. Yu [4] generalized this classical result. In this paper we propose new classes
of sequences for which we get the extended version of their results. Moreover, we generalize the results of S. Tikhonov [2]
on the *L*
^{1}-convergence of Fourier series.