On divergence of the general terms of the double Fourier-Haar series

In the present article we prove that the sequence of the general terms corresponding to the rectangular and spherical partial sums of the double Fourier-Haar series of some integrable functions do not converge almost everywhere. Received: 7 May 2005; revised: 28 June 2005     

On the divergence of multiple Fourier-Haar series

It is proved that for any dimension n ?? 2, L(ln⁺ L) ^n?1 is the widest integral class in which the almost everywhere convergence of spherical partial sums of multiple Fourier-Haar series is provided. Moreover,it is shown that the divergence effects of rectangular and spherical general terms of multiple Fourier-Haar series can be achieved simultaneously on a set of full measure by an appropriate rearrangement of values of arbitrary summable function f not belonging to L(ln⁺ L) ^n?1.     

On absolute divergence of Fourier-Haar series of characteristic functions

The paper studies some questions related to almost everywhere, absolute divergence of the series in Haar system. It is constructed a measurable set E ⊂ [0, 1] such that the Fourier-Haar series of the characteristic function of the set E absolutely diverges almost everywhere.     

Waterman classes and spherical partial sums of double Fourier series

f(x,y) ₀BV(T²), ={1/n} _n=1 .

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Dedicated to Professor Károly Tandori, the outstanding mathematician and academician on his seventieth birthday

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This work was done under the financial support of the Russian Foundation for Fundamental Scientific Research, Grant 93-01-00240.     

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.     

On the order of growth of rectangular partial sums of double orthogonal series

We obtain estimates of the order of growth of rectangular partial sums of double orthogonal series and establish their unimprovability on the set of all double orthogonal systems. South-Ukrainian Pedagogical University, Odessa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10, pp. 1299–1310, October. 1999.     

On the evaluation of Tornheim sums and allied double sums

The object of study in this paper is some Tornheim type sums \(\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \frac{1}{n^{r}m^{s}(n+m)^{t}}\) which are close relatives of the so-called Euler sums \(\sum_{n=1}^{\infty}\frac{1}{n^{s}}\sum_{m=1}^{\infty}\frac{1}{m^{t}}\). Closed form evaluations of several such double sums are obtained using elementary summation techniques earlier developed by the same author.     

Absolute convergence of double series of Fourier-Haar coefficients for functions of bounded p-variation

We consider functions of two variables of bounded p-variation of the Hardy type on the unit square. For these functions we obtain a sufficient condition for the absolute convergence of series of positive powers of Fourier coefficients with power-type weights with respect to the double Haar system. This condition implies those for the absolute convergence of series of Fourier-Haar coefficients of one-variable functions which have a bounded Wiener p-variation or belong to the class Lip ??. We show that the obtained results are unimprovable. We also formulate N-dimensional analogs of the main result and its corollaries.     

On the divergence of Vilenkin-Fourier series

Acta Mathematica Hungarica -     

Waterman classes and triangular sums of double Fourier series

Let ^a ={nln^a (n+1)}, where a R. The following results are established: For every &fnof ^a BV ((- ]²), the triangular partial sums of its Fourier series are uniformly bounded if a = -1, and converge everywhere if a < -1.For every a>0, there exists &fnof ^a BV ((- ]²) such that the triangular partial sums of its Fourier series are unbounded at the point (0;0).     

On the partial sums of Vilenkin-Fourier series

The aim of this paper is to investigate weighted maximal operators of partial sums of Vilenkin-Fourier series. Also, the obtained results we use to prove approximation and strong convergence theorems on the martingale Hardy spaces H _p , when 0 < p ≤ 1.     

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 }.     

On the divergence of a certain series

We investigate for which real numbers α the series (4) converges, and prove that, even though it converges almost everywhere in the sense of Lebesgue to a periodic, with a period 1, odd function in L₂([0,1]), it is divergent at uncountably many points, the set of which is dense in [0,1]. Finally, we find the Fourier expansion of the function defined by the series (4).     

Stable summation of Fourier-Haar series with approximate coefficients

Mathematical Notes -