Summary: It is known that the integrability of a function does not guarantee the convergence of the corresponding Walsh-Fourier series. An additional condition that implies the convergence can be made by the \(L^1\) modulus of continuity-Dini-Lipschitz condition, or by requiring that the function belongs to a narrower space, say \(L^p[0,1)\) \((1<p\leq\infty)\). Another possibility is to give a convergence condition with respect to the Walsh-Fourier coefficients.
In this paper, we formalize such a condition by means of a shifted Sidon type inequality for the Walsh-Dirichlet kernels and by using the concepts of dyadic Hardy space and generalized de la Vallée Poussin means.