In this note noncommutative versions of Etemadi's SLLN and Petrov's SLLN are given. As a noncommutative counterpart of the classical almost sure convergence, the almost uniform convergence of measurable operators is used.
Generalized Wiener classes are considered. For these classes the exact order of Fourier coefficients with respect to the trigonometric
system is established and the estimation of ‖Sn(·, f)-f(·)‖C[0,2π] where Sn(·, f) are the Fourier partial sums, is given. In particular, a uniform convergence criterion for the Fourier trigonometric series
Weighted LP convergence of Hermite and Hermite-Fejér interpolations of higher order on the zeros of Jacobi polynomials is investigated. The results which cover the classical Hermite-Fejér case give necessary and in many cases sufficient conditions for such convergence for all continuous functions. Uniform convergence is considered, too.
A sufficient condition for the strict insertion of a continuous function between two comparable upper and lower semicontinuous
functions on a normal space is given. Among immediate corollaries are the classical insertion theorems of Michael and Dowker.
Our insertion lemma also provides purely topological proofs of some standard results on closed subsets of normal spaces which
normally depend upon uniform convergence of series of continuous functions. We also establish a Tietze-type extension theorem
characterizing closed Gδ-sets in a normal space.
Authors:A. Caterino, R. Ceppitelli, L’. Holá, and L. Zampogni
We study the completeness of three (metrizable) uniformities on the sets D(X, Y) and U(X, Y) of densely continuous forms and USCO maps from X to Y: the uniformity of uniform convergence on bounded sets, the Hausdorff metric uniformity and the uniformity UB. We also prove that if X is a nondiscrete space, then the Hausdorff metric on real-valued densely continuous forms D(X, ℝ) (identified with their graphs) is not complete. The key to guarantee completeness of closed subsets of D(X, Y) equipped with the Hausdorff metric is dense equicontinuity introduced by Hammer and McCoy in .
, S. P. , Zhou , P. and Yu , D. S. , Ultimate generalization for monotonicity for uniformconvergence of trigonometric series , arXiv: math.CA/0611805 v1 November 27, 2006 , preprint; Science China , 53 ( 2010 ), 1853 – 1862