Almost everywhere convergence of lacunary partial sums of Vilenkin-Fourier series

We prove that if , and is any lacunary sequence of positive integers, then the sequence of th partial sums of Vilenkin-Fourier series of converges almost everywhere to .

     

Almost sure convergence of weighted partial sums

Sufficient conditions of covariance type are presented for weighted averages of random variables with arbitrary dependence structure to converge to 0, both for logarithmic and general weighting. As an application, an a.s. local limit theorem of Csáki, Földes and Révész is revisited and slightly improved.     

Almost everywhere divergence of lacunary subsequences of partial sums of fourier series

If an increasing sequence {n _m } of positive integers and a modulus of continuity ω satisfy the condition Σ _m=1 ^∞ ω(1/n _m )/m < ∞, then it is known that the subsequence of partial sums \(S_{n_m } \left( {f,x} \right)\) converges almost everywhere to f(x) for any function f ∈ H ₁ ^ω . We show that this sufficient convergence condition is close to a necessary condition for a lacunary sequence {n _m }.     

Almost everywhere convergence of the inverse spherical transform onSL(2,R)

We prove the almost everywhere convergence of the inverse spherical transform ofL _p bi-K-invariant functions on the groupSL(2,R), 4/3<p≤2. The result appears to be sharp. Partially supported by the MPI Partially supported by the CMA     

Almost everywhere convergence of Fourier integrals revisited

A condition of proved worth guarantees almost everywhere convergence of Fourier integrals of functions from an essentially wider class than known earlier.     

Almost everywhere convergence of inverse Fourier transforms

We show that if , then the inverse Fourier transform of converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to 0. Our proof is based on a simple application of the Rademacher-Menshov Theorem. In the special case of spherical partial integrals, the theorem was proved by Carbery and Soria. We obtain some partial results when and . We also consider sequential convergence for general elements of .

     

The uniform convergence of spherical sums of Fourier differentiable functions

Asymptotic equalities are found for exact upper bounds of the deviations in the uniform metric of the spherical sums of a multiple trigonometric Fourier series on classes of functions with a mean-bounded Liouville derivative.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 47–53, January, 1991.     

Almost everywhere convergence of Banach space-valued Vilenkin-Fourier series

The duality between martingale Hardy and BMO spaces is generalized for Banach space valued martingales. It is proved that if X is a UMD Banach space and f ∈ L _p(X) for some 1 < p < ∞ then the Vilenkin-Fourier series of f converges to f almost everywhere in X norm, which is the extension of Carleson’s result. This paper was written while the author was researching at University of Vienna (NuHAG) supported by Lise Meitner fellowship No. M733-N04. This research was also supported by the Hungarian Scientific Research Funds (OTKA) No. T043769, T047128, T047132.     

Divergence almost everywhere of rectangular partial sums of multiple fourier series of bounded functions

In this paper we establish the following results, which are the multidimensional generalizations of well-known theorems:

Almost sure central limit theory for products of sums of partial sums

Consider a sequence of i.i.d. positive random variables. An universal result in almost sure limit theorem for products of sums of partial sums is established.We will show that the almost sure limit the...     

Almost everywhere convergence of series with respect to a price system

Moscow. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 47–52, September–October, 1992.     

Almost everywhere convergence of sequences of multiplier operators on local fields

LetK ⁿ be then-dimensional vector space over a local fieldK. Two maximal multiplier theorems onL ^p (K ⁿ ) are proved for certain multiplier operator sequences associated with regularization and dilation respectively. Consequently the a. e. convergence of such multiplier operator sequences is obtained. This sharpens Taibleson’s main result and applies to several important singular integral operators onK ⁿ .     

On almost everywhere convergence of Marcinkiewicz sums of double Fourier series

Let f be a function summable on the two-dimensional torus with Fourier series . The Marcinkiewicz means . where is a function defined on [0, 1], are considered. The following theorem is proved. Let > 0 and assume that the function , concave on [0, 1], is such that (0)=1, (1)=0 and its modulus of continuity satisfies the relation (,h)=0 (log^–2–(1+1/k)). Then for almost all x.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 148–156, 1991.