Walsh Series Research Articles

Weighted Inequalities for Vector Operators on Martingales

Sufficient conditions are given to get weighted inequalities between two maximal operators on Banach valued regular martingales. As an application we obtain generalizations with weights of the inequalities in the definitions of UMD- and MT-Banach spaces and weighted estimates for the vector valued square function operator. We also obtain results for Haar and Walsh series. In particular it is shown that LpX-convergence of the Walsh-Fourier series characterizes UMD spaces.

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).

Rate of convergence and dyadic differentiability of Walsh series

Rate of convergence and dyadic differentiability of Walsh series

Representation of Functions as Walsh Series to Different Bases and an Application to the Numerical Integration of High-Dimensional Walsh Series

We will prove the following theorem on Walsh series, and we will derive from this theorem an effective and constructive method for the numerical integration of Walsh series by number-theoretic methods. Further, concrete computer calculations are given. Theorem. For base b > 2, dimension s > 1, and o > 1, c > 0 {b, s € N; c, a e R), let i,E~°(c) be the class of all functions f: (0, l)s -» C which are representable by absolutely convergent Walsh series to base b with Walsh coefficients W(h\,... ,hs) with the following property: \W(h\,... , hs) 2, provided that a > 1 + s/,, where

On the Numerical Integration of Walsh Series by Number-Theoretic Methods

In analogy to the theory of good lattice points for the numerical integration of rapidly converging Fourier series, a theory for the fast numerical integration of Walsh series is developed. The basis for this theory is provided by a class of very well-distributed point sets in the s-dimensional unit cube, the so-called (t, m, s)-nets.

THE GENERALIZED BARI THEOREM FOR THE WALSH SYSTEM

For Walsh series in the Paley arrangement the author proves a generalized Bari theorem on the union of sets of uniqueness, from which it follows in particular that the union of two -sets, one of which is simultaneously an -set and a -set, is a -set, and the union of two disjoint -sets of type is again a -set. It is shown that the last two assertions hold for sets of uniqueness of those classes of series for which the principle of localization of the kernel holds. Bibliography: 11 titles.

Representation of functions as Walsh series to different bases and an application to the numerical integration of high-dimensional Walsh series

Representation of functions as Walsh series to different bases and an application to the numerical integration of high-dimensional Walsh series

On the numerical integration of Walsh series by number-theoretic methods

In analogy to the theory of good lattice points for the numerical integration of rapidly converging Fourier series, a theory for the fast numerical integration of Walsh series is developed. The basis for this theory is provided by a class of very well-distributed point sets in the s-dimensional unit cube, the so-called (t, m, s)-nets.

Multilayer associative neural networks (MANN's): storage capacity versus perfect recall

The objective of this paper is to to resolve important issues in artificial neural nets-exact recall and capacity in multilayer associative memories. These problems have imposed restrictions on coding strategies. We propose the following triple-layered hybrid neural network: the first synapse is a one-shot associative memory using the modified Kohonen's adaptive learning algorithm with arbitrary input patterns; the second one is Kosko's bidirectional associative memory consisting of orthogonal input/output basis vectors such as Walsh series satisfying the strict continuity condition; and finally, the third one is a simple one-shot associative memory with arbitrary output images. A mathematical framework based on the relationship between energy local minima (capacity of the neural net) and noise-free recall is established. The robust capacity conditions of this multilayer associative neural network that lead to forming the local minima of the energy function at the exact training pairs are derived. The chosen strategy not only maximizes the total number of stored images but also completely relaxes any code-dependent conditions of the learning pairs.

Local BMO d Estimates for Lacunary Series

Convergent lacunary Walsh series are in BMO d , and so their restrictions of arbitrary measurable sets, E, belong to BMO d ( E) in the sense of Wolff. We give precise BMO d ( E) norm estimates for these restrictions.

Waveform synthesis using Walsh functions

A software and a microprocessor based hardware for waveform synthesis using Walsh functions are described. The software is based on Walsh function generation using Hadamard matrices and on the truncated Walsh series expansion for the waveform to be synthesized. The hardware employs six microprocessor controlled programmable Walsh function generators (PWFGs) for generating the first six non-vanishing terms of the truncated Walsh series. Improved approximation to a given waveform may be achieved by employing additional PWFGs.

Uniqueness theorems for Walsh series under a strong condition

It= ~ ~(k)wk(x) k--0 be a Walsh series and `4 a certain class of Walsh series. When E is a subset of the dyadic group, it is called a set of uniqueness for ,4, if tt E A and 2n--I lim E ft(k)wk(x) = 0 everywhere except on E k=O imply tha t ft(k) = 0 for k = 0, 1,2, . . . . When E is not a set of uniqueness for `4, it is called a set of multiplicity for ,4. It is easy to see tha t a subset of the dyadic group is a set of uniqueness for the class of all Walsh series It such tha t

Paley sets and term by term dyadic differentiation of Walsh series

Paley sets and term by term dyadic differentiation of Walsh series

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 $(w_i)$.

On aU-set for multiple Walsh series

В этой статье доказыв ается существование множества единственностиm-мерн ого (m≧2) ряда Уолша сходимост и по прямоугольникам, которое не является множеством единствен-н ости сходимости по п рямоугольникам со ст оронами, отношения которых ог раничены.

Book Review: Walsh series and transforms

Book Review: Walsh series and transforms

Lacunary Walsh series

Lacunary Walsh series

Note on single-term Walsh series method for singular systems

By considering linear time-varying singular systems in discrete time it is shown that the single-term Walsh series technique is not applicable to singular systems of index three.

Optimal control of singular systems via single-term walsh series

In this paper the single-term Walsh series (STWS) technique is used to study the optimal control of linear singular systems with quadratic cost. This method can be easily implemented on a digital computer. The method is verified by an illustrative example.

On the integrability andL 1-convergence of double Walsh series

On the integrability andL 1-convergence of double Walsh series