Walsh System Research Articles

Functions Almost Universal in the Sense of Signs with Respect to the Trigonometric System and the Walsh System

Functions Almost Universal in the Sense of Signs with Respect to the Trigonometric System and the Walsh System

Convergence of matrix transform means with respect to the Walsh–Kaczmarz system

Abstract In this paper the Walsh system will be considered in the Kaczmarz rearrangement. We estimate the difference between matrix transform means of Walsh–Kaczmarz–Fourier series and the corresponding function in norm, and the upper estimation is given by the modulus of continuity of the function. We also prove norm convergence with similar conditions.

Uniform Boundedness of Sequence of Operators Associated with the Walsh System and Their Pointwise Convergence

Uniform Boundedness of Sequence of Operators Associated with the Walsh System and Their Pointwise Convergence

Approximation by Nörlund means with respect to Walsh system in Lebesgue spaces

Approximation by Nörlund means with respect to Walsh system in Lebesgue spaces

On the universal pair with respect to the generalized Walsh system

On the universal pair with respect to the generalized Walsh system

On the Uniform Convergence of Spherical Partial Sums of Fourier Series by the Double Walsh System

On the Uniform Convergence of Spherical Partial Sums of Fourier Series by the Double Walsh System

Application of Walsh System in Data Encryption

This article explores the application of the Walsh system in modern data encryption methods. The advantages and limitations of this system are reviewed and its effectiveness is analyzed in comparison to other encryption schemes. The article describes the mathematical basis of the Walsh transform, its use for encryption and decryption, and its practical applications. In conclusion, prospects for further research in this area are considered.

Estimates for entropy numbers of multiplier operators of multiple series

The asymptotic behavior for entropy numbers of general Fourier multiplier operators of multiple series with respect to an abstract complete orthonormal system {ϕm}m∈N0d on a probability space and bounded in L∞, is studied. The orthonormal system can be of the type ϕm(x)=ϕm1(1)(x1)⋯ϕmd(d)(xd), where each {ϕl(j)}l∈N0 is an orthonormal system, that can be different for each j, for example, it can be a Vilenkin system, a Walsh system on a real sphere or the trigonometric system on the unit circle. General upper and lower bounds for the entropy numbers are established by using Levy means of norms constructed using the orthonormal system. These results are applied to get upper and lower bounds for entropy numbers of specific multiplier operators, which generate, in particular cases, sets of finitely and infinitely differentiable functions, in the usual sense and in the dyadic sense. It is shown that these estimates have order sharp in various important cases.

(H_{p}-L_{p})-Type inequalities for subsequences of Nörlund means of Walsh–Fourier series

We investigate the subsequence {t_{2^{n}}f } of Nörlund means with respect to the Walsh system generated by nonincreasing and convex sequences. In particular, we prove that a large class of such summability methods are not bounded from the martingale Hardy spaces H_{p} to the space mathit{weak-}L_{p} for 0< p<1/(1+alpha ) , where 0<alpha <1. Moreover, some new related inequalities are derived. As applications, some well-known and new results are pointed out for well-known summability methods, especially for Nörlund logarithmic means and Cesàro means.

Estimates of entropy for multiplier operators of systems of orthonormal functions

We obtain upper and lower estimates for ɛ -entropy and entropy numbers of multiplier operators of systems of orthonormal functions bounded from L p to L q . Upper estimates in our study require that a Marcinkiewicz-type multiplier theorem is available for the system. As application we obtain estimates for ɛ -entropy and entropy numbers of the multiplier operators associated with the sequences ( k − γ ln k − ξ ) k = 2 ∞ and ( e − γ k r ) k = 0 ∞ where γ > 0 , ξ ≥ 0 and 0 < r ≤ 1 . Some of these estimates are order sharp. We verify that the trigonometric system on the circle, the Vilenkin system and the Walsh system satisfy the conditions of our study. We also study analogous results for the Haar system and the Walsh systems on spheres.

On the $$L^{p}$$-Greedy Universal Functions with Respect to the Generalized Walsh System

On the $$L^{p}$$-Greedy Universal Functions with Respect to the Generalized Walsh System

Burkholder meets Gundy: Bellman function method for general operators on martingales

It is shown that the Bellman function method can be applied to study the Lp-norms of general operators on martingales, i.e., of operators that are not necessarily martingale transforms. Informally, we provide a single Bellman-type function that “encodes” the Lp-boundedness of “almost all” operators from Gundy's extrapolation theorem. As examples of such operators, we consider the Haar transforms and the operator whose Lp-boundedness underlies Rubio de Francia's inequality for the Walsh system.

On Universal Fourier Series in the Walsh System

On Universal Fourier Series in the Walsh System

The Walsh-Fourier transform on the real line

The element of the Walsh system, that is the Walsh functions map from the unit interval to the set {−1, 1}. They can be extended to the set of nonnegative reals, but not to the whole real line. The aim of this article is to give an Walsh-like orthonormal and complete function system which can be extended on the real line.

Some new results and inequalities for subsequences of N\xf6rlund logarithmic means of Walsh\u2013Fourier series

We prove that there exists a martingale fin H_{p} such that the subsequence {L_{2^{n}}f } of Nörlund logarithmic means with respect to the Walsh system are not bounded from the martingale Hardy spaces H_{p} to the space weak-L_{p} for 0< p<1 . We also prove that for any fin L_{p}, pgeq 1 , L_{2^{n}}f converge to f at any Lebesgue point x. Moreover, some new related inequalities are derived.

Matrix and tensor rigidity and Lp-approximation

We apply methods originated in Complexity theory to some problems of Approximation. We notice that the construction of Alman and Williams that disproves the rigidity of Walsh-Hadamard matrices, provides good ℓp-approximation for p<2. Hence the first n functions of Walsh system can be approximated by a low-dimensional linear space:dn1−δ({w1,…,wn},Lp[0,1])⩽n−δ,1≤p<2, where δ=δ(p)>0. We do not know if this is possible for the trigonometric system. We show that the algebraic method of Alon–Frankl–Rödl for bounding the number of low-signum-rank matrices, works for tensors: almost all signum-tensors have large signum-rank and can't be ℓ1-approximated by low-rank tensors. This implies lower bounds for Θm — the error of m-term approximation of multivariate functions by sums of tensor products u1(x1)⋯ud(xd). For the set of trigonometric polynomials with spectrum in ∏j=1d[−nj,nj] and of norm ‖t‖∞⩽1, and m⩽c(d)∏nj/max⁡{nj}, we haveΘm(T(n1,…,nd)∞,L1[−π,π]d)⩾c1(d)>0. Sharp bounds follow for classes of dominated mixed smoothness in the case 2⩽p⩽∞,1⩽q⩽2:Θm(Wp(r,r,…,r),Lq[0,1]d)≍m−rdd−1.

Applications of Affine Systems of Walsh Type to Generate Smooth Basis

The question on affine Riesz basis of Walsh affine systems is considered. An affine Riesz basis is constructed, generated by a continuous periodic function that belongs to the space on the real line, which has a derivative almost everywhere; in connection with the construction of this example, we note that the functions of the classical Walsh system suffer a discontinuity and their derivatives almost vanish everywhere. A method of regularization (improvement of differential properties) of the generating function of Walsh affine system is proposed, and a criterion for an affine Riesz basis for a regularized generating function that can be represented as a sum of a series in the Rademacher system is obtained.

An efficient weak Euler–Maruyama type approximation scheme of very high dimensional SDEs by orthogonal random variables

We will introduce Euler–Maruyama approximations given by an orthogonal system in L2[0,1] for high dimensional SDEs, which could be finite dimensional approximations of SPDEs. In general, the higher the dimension is, the more one needs to generate uniform random numbers at every time step in numerical simulation. The schemes proposed in this paper, in contrast, can deal with this problem by generating very few uniform random numbers at every time step. The schemes save time in the simulation of very high dimensional SDEs. In particular, we conclude that an Euler–Maruyama approximation based on the Walsh system is efficient in high dimensions.

Estimates for the rate of convergence of Fourier series in the Sobolev orthogonal functional system generated by the Walsh system

Estimates for the rate of convergence of Fourier series in the Sobolev orthogonal functional system generated by the Walsh system

A Generalized Walsh System and Its Fast Algorithm

In this paper, we propose a new class of discontinuous orthogonal system called Walsh-U system, which is composed of piecewise polynomials of degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> in Hilbert space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}[0, 1]$</tex-math></inline-formula> . The Walsh-U system generalizes the Walsh system from piecewise constant to piecewise polynomials, which is capable of representing discontinuous signals. Besides, the basis functions of the Walsh-U system not only possess a more concise construction, but also have stronger sparse representation capabilities than the U-system. In the construction of the Walsh-U system, the Legendre polynomials are adopted as the generators at first, and then a series of basis functions are produced through the generators and basic operations of scaling and replication. These basis functions are formed into different groups based on the generation order. Furthermore, a fast algorithm for the Walsh-U system is designed in this paper. The experimental results demonstrate that when compared with Fourier series, Walsh functions, wavelet functions, and U-system, the Walsh-U system has fast convergence and high approximation precision in analog signal reconstruction and denoising. It is verified that the fast algorithm for the Walsh-U system can effectively improve the computation speed for discontinuous signals.