On the convergence and summability of -dimensional Fourier series with respect to the Walsh-Paley systems in the spaces , .
On the denseness of Jacobi polynomials.
On the direct kernels with respect to the Walsh-Kaczmarz system.
On the Fejér kernel functions with respect to the Walsh-Kaczmarz system.
On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems
In one of the earliest monographs that involve the notion of a Schauder basis, Franklin showed that the Gram-Schmidt orthonormalization of a certain Schauder basis for the Banach space of functions continuous on [0,1] is again a Schauder basis for that space. Subsequently, Ciesielski observed that the Gram-Schmidt orthonormalization of any Schauder system is a Schauder basis not only for C[0,1], but also for each of the spaces , 1 ≤ p < ∞. Although perhaps not probable, the latter result would...
On the Hausdorff Dimension of Rademacher Riesz Products.
On the Hermite expansions of functions from the Hardy class
Considering functions f on ℝⁿ for which both f and f̂ are bounded by the Gaussian , 0 < a < 1, we show that their Fourier-Hermite coefficients have exponential decay. Optimal decay is obtained for O(n)-finite functions, thus extending a one-dimensional result of Vemuri.
On the hypercontractivity property.
On the integrability of Walsh-Fourier transform.
On the norm of the weighted maximal function of Fejér kernels with respect to the Walsh-Kaczmarz system.
On the maximal operator of Walsh-Kaczmarz-Fejér means
In this paper we prove that the maximal operator where is the -th Fejér mean of the Walsh-Kaczmarz-Fourier series, is bounded from the Hardy space to the space
On the non-equivalence of rearranged Walsh and trigonometric systems in
We consider the question of whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in for some p ≠ 2. We show that this question is closely related to a combinatorial problem. This enables us to prove non-equivalence for a number of rearrangements. Previously this was known for the Walsh-Paley order only.
On the Nörlund means of Vilenkin-Fourier series
We prove and discuss some new -type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients . These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of general Nörlund...
On the partial sums of Walsh-Fourier series
We investigate convergence and divergence of specific subsequences of partial sums with respect to the Walsh system on martingale Hardy spaces. By using these results we obtain a relationship of the ratio of convergence of the partial sums of the Walsh series and the modulus of continuity of the martingale. These conditions are in a sense necessary and sufficient.
On the pointwise estimation of Cesàro kernel of negative order with respect to Walsh-Paley system.
On the uniform convergence and -convergence of double Fourier series with respect to the Walsh-Kaczmarz system.
On the uniform convergence and L¹-convergence of double Walsh-Fourier series
In 1970 C. W. Onneweer formulated a sufficient condition for a periodic W-continuous function to have a Walsh-Fourier series which converges uniformly to the function. In this paper we extend his results from single to double Walsh-Fourier series in a more general setting. We study the convergence of rectangular partial sums in -norm for some 1 ≤ p ≤ ∞ over the unit square [0,1) × [0,1). In case p = ∞, by we mean , the collection of uniformly W-continuous functions f(x, y), endowed with the...
On unconditional convergence of Haar series
On weighted transplantation and multipliers for Laguerre expansions.
Once more on the double Fourier-Haar series