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.     

On absolute convergence of series by a general Franklin system

The paper considers general Franklin systems generated by strong regular partitions of the segment [0; 1]. For such systems we prove the following assertions: 1) the absolute convergence of a Fourier-Franklin series at a point is a local property; 2) the Fourier-Franklin series of a function of bounded variation absolute converges almost everywhere; 3) any almost everywhere finite measurable function can be represented by an almost everywhere absolutely convergent series by a general Franklin system.     

An example of the sequence of coefficients of a double trigonometric series

We construct an example of a double sequence a of nonnegative numbers that are monotone decreasing to zero in the first index for any fixed value of the second index and two Hadamard lacunary sequences of natural numbers such that the double trigonometric lacunary monotone series with the coefficients a constructed from the first lacunary sequence is squaredivergent almost everywhere and the one constructed from the second lacunary sequence is squareconvergent almost everywhere.     

On the order of magnitude of Haar-Fourier coefficients

In the paper, we investigate the connection between the properties of a function and its Haar-Fourier coefficients. We show that not only the Haar-Fourier coefficients have bounded order of magnitude, but also their differences do have.     

Comparison of Two Generalized Trigonometric Integrals

The P ²-integral of James is compared with the T ²-integral. A trigonometric series convergent almost everywhere to a function which is T ²-integrable but not P ²-integrable is constructed.     

Uniform convexity and smoothness,and their applications in Finsler geometry

We generalize the Alexandrov–Toponogov comparison theorems to Finsler manifolds. Under suitable upper (lower, resp.) bounds on the flag and tangent curvatures together with the 2-uniform convexity (smoothness, resp.) of tangent spaces, we show the 2-uniform convexity (smoothness, resp.) of Finsler manifolds. As applications, we prove the almost everywhere existence of the second order differentials of semi-convex functions and of c-concave functions with the quadratic cost function.     

Estimates of the Dirichlet kernel and divergent fourier series in the Walsh-Kaczmarz system

A new lower bound for the growth of the Dirichlet kernel for the Walsh-Kaczmarz system is obtained and an example of an almost everywhere divergent Fourier series with respect to this system from a class narrower than that examined earlier is constructed.     

Examples of divergent Fourier series for a wide class of rearranged Walsh-Paley system

A rearranged Walsh system is considered in the paper. An example of a Fourier series from the class $Lo\left( {\sqrt {\ln ^ + } } \right)L$ divergent almost everywhere is constructed.     

Certain properties of Cesàro derivatives of higher orders

It is shown that a function with a Cesàro C ₂-derivative greater than ?∞ everywhere on a segment is not necessarily VBG. We also construct a function having a finite approximate derivative almost everywhere on a segment, but its C ₂-derivative is equal to +∞ almost everywhere.     

On the Haar and Walsh Systems on a Triangle

In this paper we establish the Haar and Walsh systems on a triangle. These systems are complete in $L_2(\Delta)$. The uniform convergence of the Haar-Fourier series and the uniform convergence by group of the Walsh-Fourier series for any continuous function are proved.     

窗口Fourier变换反演公式的级数表示——献给陆善镇教授75华诞

本文研究连续窗口Fourier变换的反演公式.与经典的积分重构公式不同,本文证明当窗函数满足合适的条件时,窗口Fourier变换的反演公式可以表示为一个离散级数.此外,本文还研究这一重构级数的逐点收敛及其在Lebesgue空间的收敛性.对于L^2空间,本文给出重构级数收敛的充分必要条件.     

Integration of Banach-valued functions and Haar series with Banach-valued coefficients

It is proved that for any Banach space each everywhere convergent Haar series with coefficients from this space is the Fourier–Haar series in the sense of Henstock type integral with respect to a dyadic differential basis. At the same time, the almost everywhere convergence of a Fourier–Henstock–Haar series of a Banach-space-valued function essentially depends on properties of the space.     

Coefficients of convergent multiple Haar series

It is well-known that if a multiple trigonometric series almost everywhere converges in the square or restricted rectangular sense to a finite function, then its coefficients grow slower than any exponential function. In this paper we prove the existence of a multiple Haar series that converges in the square or restricted rectangular sense to a finite function and contains a subsequence of coefficients that grows faster than any sequence defined in advance. Moreover, we show that for such series conditions of the Arutyunyan-Talalyan type can be violated at some points.     

分片线性谱序列生成的广义Fourier级数的Lp收敛性

In the present paper, we discuss some properties of piecewise linear spectral sequences introduced by Liu and Xu. We have a study on the pointwise and almost everywhere convergence of its corresponding series. Also, it is shown that the set (G) constructed from piecewise linear spectral sequences are bases, but not unconditional bases, for LP(0,1) where 1 ＜ p ＜∞, p ≠ 2.     

Example of divergent Fourier series over rearranged Walsh-Paley system

A specially chosen class of Shipp’s rearrangements of the Walsh system is considered in the paper. An example of a Fourier series from the class L(ln⁺ ln⁺)^1−ε L divergent almost everywhere is constructed for the systems obtained here.     

The summation of orthogonal series by linear methods

A class of linear methods is distinguished which possesses the property: each method sums almost everywhere any orthogonal series in L₂ if and only if a subsequence of partial sums whose indices satisfy a certain condition and do not depend on the series converges almost everywhere. Questions are considered on the exact Weyl multiplier and strong summability.Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 697–705, December, 1968.The author would like to express his sincere gratitude to A. V. Efimov for interest in the investigation.     

Model oblique derivative problem for the heat equation with a discontinuous boundary function

The oblique derivative problem for the heat equation is considered in a model formulation with a boundary function that can be discontinuous and with the boundary condition understood as the limit in the normal direction almost everywhere on the lateral boundary of the domain. An example is given showing that the solution is not unique in this formulation. A solution is sought in the parabolic Zygmund space H ₁, which is an analogue of the parabolic Hölder space for an integer smoothness exponent. A subspace of H ₁ is introduced in which the existence and uniqueness of the solution is proved under suitable assumptions about the data of the problem.     

Uniqueness of double Walsh series

Summary We prove that under mild growth conditions, uniqueness holds for a multiple Walsh series whose square dyadic partial sums converge almost everywhere to an integrable function. We apply this result to obtain a new uniqueness result for Cesáro summable multiple Walsh series.     

Representation of measurable functions in Franklin general system

The paper proves that every almost everywhere finite measurable function is representable by an absolute convergent series in the Franklin systems generated by quasi-dyadic, weakly regular partitions.     

Maximal smoothness of the anti-analytic part of a trigonometric null series

We proved recently (C. R. Acad. Sci. Paris, Ser. I 336 (2003) 475–478) that the anti-analytic part of a trigonometric series, converging to zero almost everywhere, may belong to L² on the circle. Here we prove that it can even be C^∞, and we characterize precisely the possible degree of smoothness in terms of the rate of decrease of the Fourier coefficients. This sharp condition might be viewed as a ‘new quasi-analyticity’. To cite this article: G. Kozma, A. Olevski??, C. R. Acad. Sci. Paris, Ser. I 338 (2004).