Uniqueness for Cesàro summable double Walsh series

We prove a Tauberian theorem for Walsh series of two variables, and use it to obtain several results about uniqueness of Cesàro summable double Walsh series. Namely, we show that up to sets of measure zero, Cesàro summability of double Walsh series is the same as convergence of the square dyadic partial sums and, under a suitable growth condition, that uniqueness holds for Cesàro summable double Walsh series.     

A class of uniqueness sets for double Walsh series

A problem of uniqueness for multiple Walsh series is considered. A class of uniqueness sets for double Walsh series is obtained, this class contains known classes of uniqueness sets.     

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.     

Uniqueness of multiple Walsh series for the convergence on binary cubes

We consider uniqueness problems for multiple Walsh series convergent on binary cubes on a multidimensional binary group. We find conditions under which a given finite or countable set is a set of uniqueness.     

Uniqueness Questions for Some Classes of Haar Series

Sets of relative uniqueness for Haar series are studied. Whole classes of conditions on the behavior of a Haar series, including the Arutyunyan--Talalyan condition, are considered. New numerical characteristic of perfect sets are introduced. They are used to obtain necessary conditions and sufficient conditions for a given set to be a set of relative uniqueness under certain assumptions. Thereby, the 1967 results of G. M. Mushegyan are generalized. Moreover, for 0<p<2, the existence of perfect U-sets with the G(p)-conditions introduced by W. Wade in 1981 is proved and a method for constructing such sets is given.     

A class of sets of uniqueness for multiple Walsh series

The uniqueness problem for multiple series over orthogonal systems of functions is considered. Classes of sets of uniqueness for multiple Walsh series and for multiple series over a mixed system of functions are obtained, which extends known classes of sets of uniqueness. A many-dimensional analogue of Privalov’s theorem is established.     

Multiple walsh series and Zygmund sets

The classical Zygmund theorem claims that, for any sequence of positive numbers {? _n } monotonically tending to zero and any δ > 0, there exists a set of uniqueness for the class of trigonometric serieswhose coefficients aremajorized by the sequence {? _n } whosemeasure is greater than 2π ?δ. In this paper, we prove the analog of Zygmund’s theorem for multiple series in the Walsh system on whose coefficients rather weak constraints are imposed.     

Some properties of the reduced modulus in space

We obtain upper bounds for the conformal modulus of a condenser with uniformly perfect plates and for the reduced modulus of a uniformly perfect set E at . For the reduced moduli of α-uniformly perfect sets we prove the continuity property with respect to the kernel convergence of the complements to these sets in the sense of Carathéodory.     

On the Uniqueness Sets of Multiple Walsh Series for Convergence in Cubes

Mathematical Notes - Let $$G^d$$ be a power of the Cantor binary group $$G$$ . The uniqueness problem for a multiple Walsh series on a power of the binary group in the case of convergence in cubes...     

Generalized Walsh Bases and Applications

We investigate convergence properties of generalized Walsh series associated with signals f∈L ¹[0,1]. We also show how the dependence of the generalized Walsh bases on N×N unitary matrices allows for applications in signal encoding and encryption, provided the signals are piece-wise constant on N-adic subintervals of [0,1].     

On convergence of greedy algorithm by Walsh system in the space <Emphasis Type="Italic">C</Emphasis>(0, 1)

The paper studies convergence of the greedy algorithm by the Walsh system in the space C(0, 1). Some sufficient conditions for uniform convergence are given. It is proved that there exists a function satisfying more restrictive conditions, for which the sequence of the partial sums of the Fourier-Walsh series diverges at the point 0.     

On the uniqueness condition for the representation of functions by walsh series

We prove that if we consider a set of positive integers of a specific arithmetic nature, then a uniqueness theorem holds for the series with respect to the Walsh system with partial sums convergent with respect to a subsequence of numbers from this set. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 187–197, February, 1977.     

Uniqueness theorem for locally antipodal Delaunay sets

We prove theorems on locally antipodal Delaunay sets. The main result is the proof of a uniqueness theorem for locally antipodal Delaunay sets with a given 2R-cluster. This theorem implies, in particular, a new proof of a theorem stating that a locally antipodal Delaunay set all of whose 2R-clusters are equivalent is a regular system, i.e., a Delaunay set on which a crystallographic group acts transitively.     

On a class of sets of uniqueness for double trigonometric series

We obtain a new class of sets of uniqueness for double trigonometric series in the case of rectangular convergence as well as prove the two-dimensional analog of Privalov’s theorem.     

An example of a zero series expansion in the Walsh system

We construct an example of a zero series expansion in the Walsh system which converges to zero outside some closed M set of zero measure and converges to + at each point of this set. This shows, in particular, that in the theorem which says that a Walsh series which converges everywhere to a finite symmetric function is a Fourier series it is impossible to omit the requirement of finiteness and allow convergence of the series on a set of zero measure to an infinity of specified sign.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 179–186, February, 1976.     

Nonlocal Dezin’s problem for Lavrent’ev–Bitsadze equation

Using the method of spectral analysis, for the mixed type equation u_xx + (sgny)u_yy = 0 in a rectangular domain we establish a criterion of uniqueness of its solution satisfying periodicity conditions by the variable x, a nonlocal condition, and a boundary condition. The solution is constructed as the sum of a series in eigenfunctions for the corresponding one-dimensional spectral problem. At the investigation of convergence of the series, the problem of small denominators occurs. Under certain restrictions on the parameters of the problem and the functions, included in the boundary conditions, we prove uniform convergence of the constructed series and stability of the solution under perturbations of these functions.     

Convergence of Walsh Series in Near- $$L^\infty $$ Spaces

We consider near- $L^\infty $ spaces that are the union of Orlicz classes and prove the convergence of Walsh series in such spaces.     

On the uniform convergence of the greedy algorithm in a generalized Walsh system

We consider the problems of the uniform convergence of the greedy algorithm in the generalized Walsh system Ψ _a , a ≥ 2, after correcting a function on a set of small measure.     

On solutions of a boundary value problem for the biharmonic equation

We study the uniqueness of the solution of a boundary value problem for the biharmonic equation in unbounded domains under the assumption that the generalized solution of this problem has a bounded Dirichlet integral with weight |x|^a. Depending on the value of the parameter a, we prove uniqueness theorems or present exact formulas for the dimension of the solution space of this problem in the exterior of a compact set and in a half-space.     

Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders

We study a mixed problem for the wave equation with integrable potential and with two-point boundary conditions of distinct orders for the case in which the corresponding spectral problem may have multiple spectrum. Based on the resolvent approach in the Fourier method and the Krylov convergence acceleration trick for Fourier series, we obtain a classical solution u(x, t) of this problem under minimal constraints on the initial condition u(x, 0) = ?(x). We use the Carleson–Hunt theorem to prove the convergence almost everywhere of the formal solution series in the limit case of ?(x) ∈ L _p[0, 1], p > 1, and show that the formal solution is a generalized solution of the problem.