Let 0 ≦ a1 < a2 < ⋯ be an infinite sequence of integers and let r1(A, n) = |(i;j): ai + aj = n, i ≦ j|. We show that if d > 0 is an integer, then there does not exist n0 such that d ≦ r1 (A, n) ≦ d + [√2d + ] for n > n0.
Authors:Jorge Bustamante, Abisaí Carrillo-Zentella, and José M. Quesada
We present direct and strong converse theorems for a general sequence of positive linear operators satisfying some functional equations. The results can be applied to some extensions of Baskakov and Szász–Mirakyan operators.
Let a1<a2<... be an infinite sequence of positive integers, let k≥2 be a fixed integer and denote by Rk(n) the number of solutions of n=ai1+ai2+...+aik. P. Erdős and A. Srkzy proved that if F(n) is a monotonic increasing arithmetic function with F(n)→+∞ and F(n)=o(n(log
n)-2) then |R2(n)-F(n)| =o((F(n))1/2) cannot hold. The aim of this paper is to extend this result to k>2.
Montgomery and Vaughan improved a theorem of Erdős and Fuchs for an arbitrary sequence. Srkzy extended this theorem of Erdős
and Fuchs for two arbitrary sequences which are "near" in a certain sense. Using the idea of Jurkat (differentiation of the
generating function), we will extend similarly the result of Montgomery and Vaughan for "sufficiently near" sequences.
In this paper we introduced the general sequence of linear positive operators via generating functions. Approximation properties of these operators are obtained with the help of the Korovkin Theorem. The order of convergence of these operators computed by means of modulus of continuity Peetre’s K-furictiorial and the elements of the usual Lipschitz class. Also we introduce the r-th order generalization of these operators and we evaluate this generalization by the operators defined in this paper. Finally we give some applications to differential equations.