Walsh System Research Articles

Absolute convergence of multiple series of Fourier coefficients with respect to multiplicative systems

In this paper, we consider series of the Fourier coefficients of functions of several variables with respect to multiplicative systems generalizing the Walsh system. We present a sufficient condition for the convergence of these series in terms of the complete group modulus of continuity. We prove that this condition cannot be improved in the function classes H~ ~,'2 ..... ,~.. Let us present the relevant definitions and notation. Suppose that {p )}k~=t (J = 1, 2, ..., n) are

Asymptotic expansion for the Lebesgue constants of the Walsh system

Let L k denote the Lebesgue constants of the Walsh system. The following exact result is established by means of Mellin transforms: ∑ 1≤k<n Lk= n 4 log 2n+nF( log 2n)− L n 2 forn⩾1 where F( u) is a continuous periodic function with period 1 whose Fourier coefficients can be explicitly expressed in terms of Riemann's zeta function. This improves an old result of Fine.

An Inverse Sidon Type Inequality for the Walsh System

Upper estimations for the integral norm of the linear combinations of Dirichlet kernels expressed only in terms of the coefficients are called Sidon type inequalities. They can be applied in various fields of harmonic analysis. Among them are integrability and L1 convergence problems and summation methods. Several improvements of Sidon′s original inequality have been proved, most of them invariant with respect to the rearrangements of the coefficients, but no nontrivial lower estimation has been given so far. In this paper we give the best rearrangement invariant expression, which corresponds to an Orlicz norm, of coefficients for a Sidon type inequality. Examples for applications for convergence in L1 and in the dyadic Hardy space are also given. These questions are considered for the Walsh system.

Generalized localization for the multiple Walsh-Fourier series of functions in ,

In this paper the concept of generalized localization almost everywhere (GL) is analyzed for the multiple Fourier-Walsh series of functions in , , summable over rectangles. (For multiple trigonometric series and Fourier integrals GL was introduced and analyzed earlier by one of the authors.) If , then it is proved that GL holds for double Walsh-Fourier series on any open set. It breaks down on any set which is not dense in if and and also in the class if . All results on the Walsh system obtained in this paper are identical to the results on GL for Fourier series in the trigonometric system obtained earlier by one of the authors.

The Size of Trigonometric and Walsh Series and Uniform Distribution Mod 1

We characterize the class of non-decreasing functions f such that for any increasing sequence {nk} of integers lim N → ∞ Σ k ⩽ N cos 2 π n k ω N 1 / 2 f ( N ) = 0 a.e. Combined with an inequality of Koksma our results prove the existence of an increasing sequence {nk} of integers such that the discrepancy DN(ω) of the sequence {nkω} mod 1 satisfies lim sup N → ∞ ( N / log N ) 1 / 2 D N ( ω ) = ∞ a.e. This disproves conjectures of Erdős [9] and R. C. Baker [2]. We prove the analogue of the above result for the Walsh system, thereby solving a problem of Révész [18]. Finally we solve a problem raised in [16], by showing the existence of sequences {nk} of integers with n k + 1 n k ⩾ 1 + ρ k for k ⩾ 1 , where {ρk} is a given non-increasing sequence of real numbers, for which the discrepancy DN(ω) of {nkω} mod 1 fails the upper half of the law of the iterated logarithm by a factor of log log (1/ρN).

On the category of u-sets for series in the Walsh system

On the category of u-sets for series in the Walsh system

Mean Convergence of Vector‐valued Walsh Series

Given any Banach space $X$, let $L_2^X$ denote the Banach space of all measurable functions $f:[0,1]\to X$ for which ||f||_2:=(int_0^1 ||f(t)||^2 dt)^{1/2} is finite. We show that $X$ is a UMD--space (see \cite{BUR:1986}) if and only if \lim_n||f-S_n(f)||_2=0 for all $f\in L_2^X$, where S_n(f):=sum_{i=0}^{n-1} (f,w_i)w_i is the $n$--th partial sum associated with the Walsh system $(w_i)$.

An orthonormal system on the construction of the generalized cantor set

This paper presents a new complete orthonormal system of functions defined on the interval[0,1]and whose supports shrink to nothing. This system related to the construction of the Cantor ternary set. We defined the canonicalmap ξand proved the equivalence between this system and the Walsh system. The generalized Cantor set with any dissection ratio is established and the constructed system is defined in the general case.

A note on some expansions of p-adic functions

Introduction. Recently J. Rutkowski (see [3]) has defined the p-adic analogue of the Walsh system, which we shall denote by $(ϕₘ)_{m∈ ℕ₀}$. The system $(ϕₘ)_{m∈ ℕ₀}$ is defined in the space C(ℤₚ,ℂₚ) of ℂₚ-valued continuous functions on ℤₚ. J. Rutkowski ha

Law of iterated logarithm for lacunary series with respect to the Walsh system

Law of iterated logarithm for lacunary series with respect to the Walsh system

On the integrability and L1-convergence of Walsh series with coefficients of bounded variation

We study the integrability and the convergence in the L 1(0, 1)-metric of Walsh series ∑ k = 0 ∞ a k w k ( x), where { a k } is a null sequence of bounded variation. Our basic tool is the following Sidon type inequality: for every 1 < p ⩽ 2, sequence { a k } of real numbers, and integer n ⩾ 1, we have ∫ 0 1 ∑ k=1 n a kD k(x) dx⩽C pn 1− 1 p ∑ k=1 n |a k| p 1 p where D k ( x) = ∑ j = 0 k − 1 W j ( x) is the Dirichlet kernel for the Walsh system and C p = 2 p (p −) .

Lacunary series with respect to periodic multiplicative systems

co A system X = {X k (X)}k= 0 of complex-valued functions that are defined on the segment [0, i] is said to be multiplicative if for any two functions X k and Xs in it the functions (Xk)-I and XkXE also belong to it. The classical examples of multiplicative systems are the trigonometric system {exp " (2~inX)}n=_ ~ and the Walsh system. In [I-3] the properties of periodic multiplicative systems and, in particular, of certain lacunary series with respect to these systems have been investigated. For the trigonometric system, the most widely known and important lacunarity of a series is the Hadamard--lacunarity, i.e., the condition

Tree population changes in treated rain forest at Omo Forest Reserve, south-western Nigeria

ABSTRACTA review of previous evaluations of long-term changes in treated Nigerian rain forest is presented, to show that these studies have concentrated on economic species with little consideration of other species, or of forest biomass and structure.We examine regeneration and successional patterns in Omo Forest Reserve by comparing enumeration data taken before (1952) and after (1981) treatment of the forest by selective canopy opening and climber-cutting (Plot A), or clear-felling and burning (Walsh system) (Plot C).Before treatment, about 50 species in 25 families were found of stems ≥10 cm dbh, in 4.05 ha of the forest, the Euphorbiaceae contributing the greatest number of species while the medium-sized trees Diospyros alboflavescens (Ebenaceae) and Strombosia pustulata (Olacaceae) contributed more than 40% of the stems.Twenty-eight years after treatment (1981) the number of species and families remained similar to those in 1952, although smaller areas (0.75 ha) were enumerated; the treated plots were, however, dominated by early succession species such as Macaranga barteri, Musanga cecropioides, Cleistopholis patens, Funtumia elastica and Fagara macrophylla, and lacked an abundance of the ‘economic’ species that treatment had been expected to induce; the medium-sized trees that were dominant in 1952 were still abundant in Plot A but not in Plot C.For stems ≥ 30 cm dbh tree diversity (reciprocal of Simpson's index) was highest (15.7) in a 1952 plot and least (4.8) in the clear-felled plot enumerated in 1981; diversity of the 1952 plots, however, fell markedly to values lower than those for the 1981 plots when computation was based on all stems ≥10 cm dbh, presumably because of increase in abundance of small-stemmed species like Diospyros spp., Strombosia sp. and Rinorea sp., each represented by a large number of stems.Basal area was greatest (29.6 m2 ha−1) in the 1952 plots and least (12.7 m2 ha−1) in the clear-felled plots enumerated in 1981, but the relative distribution of basal area and number of stems in size-classes was similar in all the plots.Mean annual increment, computed by dividing the mean diameter (7.50 cm) of the stems in the clear-felled plots by the number of years (28) over which they had grown, was 0.27 cm.Nauclea diderrichii dominated the seedling regeneration from the first year after clear-felling and burning (1954) till the sixth year (1960), when seedlings of the Meliaceae entered the regeneration list and overall seedling density was 395 per hectare.Among the plots assessed in 1981, the standing crop was greatest in an untreated Control plot (229.6 t ha−1) followed by Plot A (159.7 t ha−1) and Plot C (91.1 t ha7−1), but the relative allocation of biomass to stem, branch, leaf, root and fruit fractions was comparable for all plots.The data are discussed in relation to other Nigerian forest studies and it is suggested that the main qualitative features of structural organization and the species composition of the top canopy synusia of mature secondary rain forest may be determined quite early in the development of the stand.

Martingale transforms and uniformly convergent orthogonal series

S. A. Vinogradov's method is adapted to prove for certain orthogonal product systems the analogue of his inequality concerning the trigonometric system. For example, for the Walsh system W={wn} the following holds. Let U(W) be the space of functions with a uniformly convergent Walsh-Fourier series. Then, for every functional F on U(W) we have the inequality\(mes\mathop {\left\{ {\left. {\sup } \right|} \right.}\limits_N \sum\limits_{n \leqslant 2^N } {F(w_n )w_n } \left. {\left| { > \lambda } \right.} \right\} \leqslant const\lambda ^{ - 1} \left\| F \right\|_{U(W)^* }\).

The approximate identity kernels of product type for the Walsh system

At present there are only a few approximate identity kernels for the Walsh system, for example, the p N -truncated Dirichlet kernel D p N − 1 ( t) = ∑ j = 0 p N − 1 w j ( t) [6]; the Abel-Poisson kernel λ γ ( t) = ∑ k = 0 ∞ γ k w k ( t) [3], and so on. In [6], Zheng has introduced a new kind of approximate identity kernels for the Walsh system—the kernels of product type. In the present paper we discuss the approximation properties of such product type kernels. Estimates of their moments as well as a direct approximation theorem are obtained. Then, to establish an inverse approximation theorem, we need the p-adic derivative of product type kernels and we estimate this derivative in L 1-norm.

Central limit theorem for weakly lacunary Walsh series

Moreover, let Wo(X)-=1. Then {w,,(x)} is a Walsh system in Paley's ordering. We call the set of functions in the system {wn(x)} , with indices from 2 k to 2 ~'I --! (k O, i .... ) inclusive, a "packet" system, and denote it by [2 ~, 2hq). The following fact was established in [I] for the system {~6, (x)} -the so-called central limit theorem (CLT). THEOREM A. Let us be given a sequence of natural numbers (n k} and a sequence of nonnegative numbers {ak}, satisfying the conditions

Central limit theorem for the Walsh system

Central limit theorem for the Walsh system

A Sequence of Piecewise Orthogonal Polynomials

In this paper we study the construction of an orthogonal sequence $(U_{k,n}^{(i)} )$ of piecewise polynomials of degree k which is complete in $L_2 $. Explicit expressions of $(U_{k,n}^{(i)} )$ for $k = 1,2,3$ are given. We also study the sign-change properties of this sequence and consider the convergence of the corresponding Fourier series. The results generalize those obtained earlier for Walsh system.

Bounded divergence of series with respect to a Walsh system

Bounded divergence of series with respect to a Walsh system

On systems of functions resembling the Walsh system.

On systems of functions resembling the Walsh system.