On uniqueness sets for multiple Walsh series

We study uniqueness sets for multiple Walsh series under ρ-regular (or bounded) convergence in rectangles. We prove that a countable set is a uniqueness set for such a series under this convergence. We construct a class of perfect uniqueness sets for multiple Walsh series under this convergence. We show that the notion of index of a perfect set does not solve the problem of whether this set belongs to the class of uniqueness sets. We note that the results of this paper remain valid for several rearranged multiple Walsh series.     

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.     

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.     

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.     

Universality properties of Walsh–Fourier series

It is shown that Walsh–Fourier series of \(W\) -continuous functions can have maximal sets of limit functions on small subsets of the unit interval.     

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...     

Fractals and distributions on the -torus

This paper establishes non-Cartesian product sets, called fractal carpets and fractal foam, as sets of uniqueness for a class of trigonometric series.

     

Walsh Series Analysis of the L 2-Discrepancyof Symmetrisized Point Sets

 We present a method to estimate the L ₂-discrepancy of symmetrisized point sets from above and from below with the help of Walsh series analysis. We apply the method to a class of two-dimensional net-type point sets, thereby generalizing results of Halton and Zaremba and of Proinov.     

Connection between sets of uniqueness for multiplicative systems of functions and sets of uniqueness for multiplicative transforms

A problem of uniqueness for series over multiplicative systems of functions and for multiplicative transforms is considered. It is shown that each set of uniqueness for a multiplicative transform is specified by a countable collection of sets of uniqueness for series over the corresponding multiplicative system of functions. Each set of uniqueness for a series over a multiplicative system of functions is a portion on [0, 1) of some set of uniqueness for the corresponding multiplicative transform.     

Walsh Series Analysis of the L 2-Discrepancyof Symmetrisized Point Sets

 We present a method to estimate the L ₂-discrepancy of symmetrisized point sets from above and from below with the help of Walsh series analysis. We apply the method to a class of two-dimensional net-type point sets, thereby generalizing results of Halton and Zaremba and of Proinov. (Received 14 September 2000)     

Semimartingales on rays,Walsh diffusions,and related problems of control and stopping

We introduce a class of continuous planar processes, called “semimartingales on rays”, and develop for them a change-of-variable formula involving quite general classes of test functions. Special cases of such processes are diffusions which choose, once at the origin, the rays for their subsequent voyage according to a fixed probability measure in the manner of Walsh (1978). We develop existence and uniqueness results for these “Walsh diffusions”, study their asymptotic behavior, and develop tests for explosions in finite time. We use these results to find an optimal strategy, in a problem of stochastic control with discretionary stopping involving Walsh diffusions.     

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.     

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.     

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.     

Strong approximation by Marcinkiewicz means of two-dimensional Walsh–Kaczmarz–Fourier series

In this paper we study the exponential uniform strong approximation of Marcinkiewicz type of two-dimensional Walsh–Kaczmarz–Fourier series. In particular, it is proved that the Marcinkiewicz type of two-dimensional Walsh–Kaczmarz–Fourier series of every continuous function f is uniformly strong summable to the function f exponentially in the power 1/2. Moreover, it is proved that this result is the best possible.     

Theorems on unions ofU-sets

The properties of families of subsets of topological spaces that make it possible to separate out the common core of the proofs of theorems on unions of sets of uniqueness for series in various systems of functions are considered. As a consequence of the results obtained, it is proved, in particular, that the countable union of closed sets of uniqueness for the trigonometric series is a set of uniqueness for the Abel-Poisson summation method. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 778–787, May, 2000.     

Uniqueness sets for multiple Haar series

Criteria are found for the membership of multidimensional sets in the class of M-sets for multiple Haar series with various conditions imposed on the coefficients. Several generalizations of the uniqueness theorem are established.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 789–798, December, 1973.     

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.