On the divergence of spherical sums of double Fourier-Haar series

. : [0, +) [0, +) - , u+ (u) (u)=o(u lnu). [0, 1]² f , ¦f¦ L([0, 1]²), - [0, 1]².     

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.     

Trigonometric series with positive partial sums

Translated from Matematicheskie Zametki, Vol. 53, No. 3, pp. 149–152, March, 1993.     

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.     

Non-Fourier-Lebesgue trigonometric series with nonnegative partial sums

It is proved that a trigonometric cosine series of the form _n ^Emphasis>=0/ a _n cos(nx) with nonnegative coefficients can be constructed in such a way that all of its partial sums are positive on the real axis. It converges to zero almost everywhere and is not a Fourier-Lebesgue series. Some other properties of trigonometric series with nonnegative partial sums are also studied.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 24–41, January, 1996.     

On the order of partial sums of general orthogonal series

It is shown that for convergence of every orthonormal system _n(x) given on [0, l],it is necessary and sufficient that, under the condition on tlie increasing function W(x) and for there hold almost everywhere on [0, 1].Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 451–462, October, 1969.     

On the growth order of partial sums of non-orthogonal series

a _k f _k , f _k L ₂, w-, (2), w(n) — . a _k f _k N {a _k }l ₂, {a _k }l ₂ ( 1, 2, 1a, 2a). ( 2) [8]. , {a _k } w-.     

多尺度紧支撑向量值正交小波的构造

引入了多尺度向量值多分辨分析．利用矩阵理论,给出多尺度紧支撑向量值正交小波存在的必要条件及其构造方法．     

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.     

Stable summation of Fourier-Haar series with approximate coefficients

Mathematical Notes -     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

On uniform approximation by partial sums of Jacobi series on elliptic region

In this paper, we discuss the relation between the partial sums of Jacobi series on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the partial sums of Jacobi series on an elliptic region.