Then there exists a continuous 2π-periodic function f0 for which Rn[f0]≦βn for all n=1 and yet the Fourier series of f0 diverges at x=0.Rn[f]may be replaced in these theorems byMn[f], whereMn[f] is the minimal uniform deviation off(x) from piecewise monotonie functionsМn(х) of order ≦n.
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 ℝ2. Our theorem is a two-dimensional extension of a theorem of Móricz (see Theorem 3 in ) concerning the single Fourier integrals, which is more general than the two-dimensional extension given by Móricz himself (see Theorem 3 in ).
The aim of this paper is to continue our investigations started in , 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 (Cρ, ‖‖ρ) of continuous functions (ρ denotes (another) weight). In  we formulated several conditions with respect to w, ρ, (Cρ, ‖‖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  hold in this case. The order of convergence
will also be considered.
Chaundry and Jolliffe  proved that if ak is a nonnegative sequence tending monotonically to zero, then a necessary and sufficient condition for the uniform convergence
of the series Σk=1∞ak sin kx is limk→∞kak = 0. Lately, S. P. Zhou, P. Zhou and D. S. Yu  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 
on the L1-convergence of Fourier series.