The aim of this paper is to prove that the Cesàro means of order α (0 < α < 1) of the Fourier series with respect to representative product systems converge to the function in L1-norm, only for certain values of α which depend on some parameter of the representative product system.
Simon  proved that the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space Hp to the space Lp for p > 1/(1 + α). In this paper we prove that this boundedness result does not hold if p ≦ 1/(1 + α). However, in the endpoint case p = 1/(1 + α) the maximal operator σ*α,k is bounded from the martingale Hardy space H1/(1+α) to the space weak-L1/(1+α).
Properties of Fourier–Haar coefficients of continuous functions are studied. It is established that Fourier–Haar coefficients of continuous functions are monotonic in a certain sense for convex functions. Questions of quasivariation of Fourier–Haar coefficients of continuous functions are also considered.
The main aim of this paper is to prove that the maximal operator σ0k*:= supn ∣σn,nk∣ of the Fej�r means of double Fourier series with respect to the Kaczmarz system is not bounded from the Hardy space H1/2 to the space weak-L1/2.
The main aim of this paper is to prove that there exists a martingale f ∈ H12/▭ such that the restricted maximal operators of Fejér means of twodimensional Walsh-Fourier series and conjugate Walsh-Fourier
series does not belong to the space weak-L1/2.
) of continuous functions. Some special cases are also considered, such as the Fejér, de la Vallée Poussin, Cesàro, Riesz and Rogosinski summations. Our aim is to give such conditions with respect to Jacobi weights
and to summation matrix Θ for which the uniform convergence holds for all
The paper is related to the lower and upper estimates of the norm for Mercer kernel matrices. We first give a presentation
of the Lagrange interpolating operators from the view of reproducing kernel space. Then, we modify the Lagrange interpolating
operators to make them bounded in the space of continuous function and be of the de la Vallée Poussin type. The order of approximation
by the reproducing kernel spaces for the continuous functions is thus obtained, from which the lower and upper bounds of the
Rayleigh entropy and the l2-norm for some general Mercer kernel matrices are provided. As an example, we give the l2-norm estimate for the Mercer kernel matrix presented by the Jacobi algebraic polynomials. The discussions indicate that the
l2-norm of the Mercer kernel matrices may be estimated with discrete orthogonal transforms.
The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the Walsh–Fourier series of bivariate integrable functions. That is, let such that aj(n+1)≧δsupk≦naj(n) (j=1,2, n∊ℕ) for some δ>0 and a1(+∞)=a2(+∞)=+∞. Then for each integrable function f∊L1(I2) we have the a.e. relation . It will be a straightforward and easy consequence of this result the cone restricted a.e. convergence of the two-dimensional Walsh–Fejér means of integrable functions which was proved earlier by the author and Weisz [3,8].