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Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series

Earl Berkson (2014)

Studia Mathematica

Let f ∈ V r ( ) ∪ r ( ) , where, for 1 ≤ r < ∞, V r ( ) (resp., r ( ) ) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values, the condition...

Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences

Ferenc Móricz (2013)

Studia Mathematica

Let s: [1,∞) → ℂ be a locally Lebesgue integrable function. We say that s is summable (L,1) if there exists some A ∈ ℂ such that l i m t → ∞ τ ( t ) = A , where τ ( t ) : = 1 / ( l o g t ) ∫ 1 t s ( u ) / u d u . (*) It is clear that if the ordinary limit s(t) → A exists, then also τ(t) → A as t → ∞. We present sufficient conditions, which are also necessary, in order that the converse implication hold true. As corollaries, we obtain so-called Tauberian theorems which are analogous to those known in the case of summability (C,1). For example, if the function s is slowly...

Ordinary convergence follows from statistical summability (C,1) in the case of slowly decreasing or oscillating sequences

Ferenc Móricz (2004)

Colloquium Mathematicae

Schmidt’s Tauberian theorem says that if a sequence (xk) of real numbers is slowly decreasing and l i m n → ∞ ( 1 / n ) ∑ k = 1 n x k = L , then l i m k → ∞ x k = L . The notion of slow decrease includes Hardy’s two-sided as well as Landau’s one-sided Tauberian conditions as special cases. We show that ordinary summability (C,1) can be replaced by the weaker assumption of statistical summability (C,1) in Schmidt’s theorem. Two recent theorems of Fridy and Khan are also corollaries of our Theorems 1 and 2. In the Appendix, we present a new proof of Vijayaraghavan’s...

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