Almost Everywhere Convergence of Fejér Means of Two-dimensional Triangular Walsh–Fourier Series

Abstract

In 1987 Harris proved [11] -among others-that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ L p such that its triangular Walsh-Fourier series diverges almost everywhere. In this paper we investigate the Fejér (or (C, 1)) means of the triangle two variable Walsh-Fourier series of L 1 functions. Namely, we prove the a.e. convergence

On almost everywhere convergence of the generalized Marcienkiwicz means with respect to two dimensional Vilenkin-like systems

“…convergence relation σ 2 n f → f . This result for the whole sequence of the triangular mean operators in the Walsh case is given by the first author [9]. In the Vilenkin situation there is nothing proved yet.…”

“…To demonstrate the proof of this, see some calculations below [9] between the triangle and the one dimensional Dirichlet kernels.…”

Mixed Martingale Hardy Spaces

“…It is easy to see that for all (s, − → p )-atoms a, (S, − → p )-atoms a or (M, − → p )-atoms a and A ∈ F τ , s(aχ A ) = s(a)χ A , S(aχ A ) = S(a)χ A and M(aχ A ) = M(a)χ A . This means that the operators s, S and M satisfy condition (13).…”

“…If all (S, − → p )-atoms (resp. (M, − → p )-atoms) a satisfy (13), then for all f ∈ Q− → p (resp. f ∈ P− → p ),…”

“…Details for the martingale Hardy spaces can be found in Burkholder [4,5], Burkholder and Gundy [6], Garsia [12], Long [33], or Weisz [39,40]. For the application of martingale Hardy spaces in Fourier analysis, see Gát [13,14], Goginava [15,16] or Weisz [40,41].…”