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Multisummability for some classes of difference equations

Boele L. J. Braaksma, Bernard F. Faber (1996)

Annales de l'institut Fourier

This paper concerns difference equations y ( x + 1 ) = G ( x , y ) where G takes values in C n and G is meromorphic in x in a neighborhood of ∞ in C and holomorphic in a neighborhood of 0 in C n . It is shown that under certain conditions on the linear part of G , formal power series solutions in x - 1 / p , p ∈ N , are multisummable. Moreover, it is shown that formal solutions may always be lifted to holomorphic solutions in upper and lower halfplanes, but in general these solutions are not uniquely determined by the formal solutions.

On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system

G. Gát (1998)

Studia Mathematica

Let G be the Walsh group. For f ∈ L 1 ( G ) we prove the a. e. convergence σf → f(n → ∞), where σ n is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator σ * f ≔ s u p n | σ n f | . We prove that σ* is of type (p,p) for all 1 < p ≤ ∞ and of weak type (1,1). Moreover, ∥ σ * f ∥ 1 ≤ c ∥ | f | ∥ H , where H is the Hardy space on the Walsh group.

Currently displaying 101 – 120 of 238