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Abstract  

We prove that some results on uniform convergence of sequences of unconditionally convergent series, in Banach spaces, can be generalized to sequences of weakly unconditionally Cauchy series.

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Abstract  

We introduce a new class of sequences called NBVS to generalize GBVS, essentially extending monotonicity from “one sided” to “two sided”, while some important classical results keep true.

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series: coefficients criteria Studia Math. 193 285 – 306 10.4064/sm193-3-5 . [3] Le , R. J. , Zhou , S. P. 2005 A new condition for the uniform convergence of certain trigonometric series

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Abstract  

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 ‖S n(·, f)-f(·)‖C [0,2π] where S n(·, f) are the Fourier partial sums, is given. In particular, a uniform convergence criterion for the Fourier trigonometric series is obtained.

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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.

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, J. P. , Uniform convergence of Fourier series , Rice Inst. Pamphlet , ( 1953 ), 31 – 57 . [8] Nikol’skii , S. M. 1946 The Fourier series with a given modulus

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, S. P. , Zhou , P. and Yu , D. S. , Ultimate generalization for monotonicity for uniform convergence of trigonometric series , arXiv: math.CA/0611805 v1 November 27, 2006 , preprint; Science China , 53 ( 2010 ), 1853 – 1862

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Abstract  

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 G δ-sets in a normal space.

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. 350 568 – 589 10.1016/j.jmaa.2008.03.058 . [5] Beer , G. , Levi , S. 2010 Uniform continuity, uniform convergence, and shields Set-Valued Var. Anal. 18 251 – 275 10.1007/s11228

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Abstract  

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 U B. 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 [7].

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