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Ukrainian Mathematical Journal

, Volume 52, Issue 2, pp 249–259 | Cite as

A generalization of the rogosinski-rogosinski theorem

  • S. Ya. Dekanov
  • H. O. Mykhalin
Article
  • 22 Downloads

Abstract

We establish necessary and sufficient conditions for numerical functions αj(x), jN, xX, under which the conditions K(f j K(f 1) ∀j≥2 and \(\mathop {\lim }\limits_{U_r } \sum\nolimits_{j = 1}^\infty {\alpha _j (x)f_j (x) = a} \) yield \(\mathop {\lim }\limits_{U_r } f_1 (x) = a.\) The functions fj(x) are uniformly bounded on the set X and take values in a boundedly compact space L, and K(fj) is the kernel of the function fj. The well-known Rogosinski-Rogosinski theorem follows from the proved statements in the case where X = N, α j (x) ≡ αj, and the space L is the m-dimensional Euclidean space.

Keywords

Compact Space Numerical Function Partial Limit Functional Sequence Closed Convex Hull 
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References

  1. 1.
    R. G. Cooke, Infinite Matrices and Sequence Spaces [Russian translation] Fizmatgiz, Moscow 1960.zbMATHGoogle Scholar
  2. 2.
    G. H. Hardy, Divergent Series [Russian translation] Inostrannaya Literatura, Moscow 1951.Google Scholar
  3. 3.
    A. V. Revenko, “Imbedding of kernels by regular transformations,” Ukr. Mat. Zh., 36, No. 5, 662–666 (1984).MathSciNetGoogle Scholar
  4. 4.
    W. W. Rogosinski and H. P. Rogosinski, Jr., “An elementary companion to a theorem of J. Mercer,” Anal. Math., 14, 311–322 (1965).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 2000

Authors and Affiliations

  • S. Ya. Dekanov
  • H. O. Mykhalin

There are no affiliations available

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