Frequency domain Walsh functions and sequences: An introduction

In this letter, a new set of orthogonal band-limited basis functions is introduced. This set of basis functions is derived from the inverse Fourier transform of the frequency domain Walsh functions. The Fourier transforms of the Walsh functions were calculated by Siemens and Kitai in 1973 but they have been overlooked in the literature. Some of the properties of these functions are studied in this paper. Moreover, the orthogonal discrete version of these functions is obtained by truncation, sampling and orthogonalization utilizing the orthogonal Procrustes problem.     

Recent Developments In Harmonic Approximation,With Applications

A theorem of J.L. Walsh (1929) says that if E is a compact subset of Rⁿ with connected complement and if u is harmonic on a neighbourhood of E, then u can be uniformly approximated on E by functions harmonic on the whole of Rⁿ. In Part I of this article we survey some generalizations of Walsh’s theorem from the period 1980–94. In Part II we discuss applications of Walsh’s theorem and its generalizations to four diverse topics: universal harmonic functions, the Radon transform, the maximum principle, and the Dirichlet problem.     

Equiconvergence of some complex interpolatory polynomials

Summary Walsh showed the close relation between the Lagrange interpolant in then ^th roots of unity and the corresponding Taylor expansion for functions belonging to a certain class of analytic functions. Recent extensions of this phenomena to Hermite interpolation and other linear processes of interpolation have been surveyed in [3, 5]. Following a recent idea of L. Yuanren [7], we show how new relations between other linear operators can be derived which exhibit Walsh equiconvergence.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayThese authors were supported by NSERC A3094     

Walsh函数的一个迭代方程体系

Shanks曾用迭代方程产生离散佩利编号Walsh函数。作者在[4]中,给出了产生离散沃尔什编号Wa1sh函数的迭代方程. 本文在上述基础上,提出了一个产生离散哈德玛编号Walsh函数的迭代方程,推出了离散哈德玛编号Walsh函数的表示式及其变换(FWHT)的快速计算公式. 上述三个极为类似的迭代方程,已构成了离散Walsh函数的迭代方程体系.连续的Walsh函数,也能用迭代方程这种形式来描述.     

迭代方程与离散叙率Walsh函数

在J.L.Shanks的基础上,给出了产生离散叙率Walsh函数的迭代方程,由迭代 方程推出了离散Walsh 函数的表达式和Walsh变换的速算 法(FWWT).     

广义Walsh变式与一极值问题

<正> 设p为大于1的整数,t为非负实数,t的p进表示为     

A modulus of smoothness based on an algebraic addition

Summary The purpose of this paper is to present a new approach to smoothness of nonperiodic functions. We consider the space of continuous functions on [−1, 1] as well as the weighted L^p-space and introduce a modulus of smoothness that is based on an algebraic addition ⊕ defined on [−1, 1]. The present paper is mainly concerned with general properties and groundwork, whereas a second paper [4] is devoted to more complex properties, in particular to an equivalent K-functional and to the characterization of best algebraic approximation. Moreover the equivalence with the Butzer-Stens modulus will be shown there.     

Best Approximation with Walsh Atoms

We consider the approximation in L ² R of a given function using finite linear combinations of Walsh atoms, which are Walsh functions localized to dyadic intervals, also called Haar—Walsh wavelet packets. It is shown that up to a constant factor, a linear combination of K atoms can be represented to relative error ɛ by a linear combination of orthogonal atoms. In finite dimension N, best approximation with K orthogonal atoms can be realized with an algorithm of order . A faster algorithm of order solves the problem with indirect control over K. Therefore the above result connects algorithmic and theoretical best approximation. Date received: July 6, 1995. Date revised: January 8, 1996.     

Geometric construction of quintic parametric B-splines

The aim of this paper is to present a new class of B-spline-like functions with tension properties. The main feature of these basis functions consists in possessing C³$C^3$ or even C⁴$C^4$ continuity and, at the same time, being endowed by shape parameters that can be easily handled. Therefore they constitute a useful tool for the construction of curves satisfying some prescribed shape constraints. The construction is based on a geometric approach which uses parametric curves with piecewise quintic components.     

Fast integral wavelet transform on a dense set of the time-scale domain

Summary. The objective of this paper is to introduce a fast algorithm for computing the integral wavelet transform (IWT) on a dense set of points in the time-scale domain. By applying the duality principle and using a compactly supported spline-wavelet as the analyzing wavelet, this fast integral wavelet transform (FIWT) is realized by applying only FIR (moving average) operations, and can be implemented in parallel. Since this computational procedure is based on a local optimal-order spline interpolation scheme and the FIR filters are exact, the IWT values so obtained are guaranteed to have zero moments up to the order of the cardinal spline functions. The semi-orthogonal (s.o.) spline-wavelets used here cannot be replaced by any other biorthogonal wavelet (spline or otherwise) which is not s.o., since the duality principle must be applied to some subspace of the multiresolution analysis under consideration. In contrast with the existing procedures based on direct numerical integration or an FFT-based multi-voice per octave scheme, the computational complexity of our FIWT algorithm does not increase with the increasing number of values of the scale parameter. Received March 3, 1994     

Integral-type operators on continuous function spaces on the real line

In this paper we introduce some new sequences of positive linear operators, acting on a sufficiently large space of continuous functions on the real line, which generalize Gauss–Weierstrass operators.We study their approximation properties and prove an asymptotic formula that relates such operators to a second order elliptic differential operator of the form Lu?αu^′′+βu^′+γu.Shape-preserving and regularity properties are also investigated.     

Generalized cardinal B-splines: Stability, linear independence, and appropriate scaling matrices

Generalized cardinal B-splines are defined as convolution products of characteristic functions of self-affine lattice tiles with respect to a given integer scaling matrix. By construction, these generalized splines are refinable functions with respect to the scaling matrix and therefore they can be used to define a multiresolution analysis and to construct a wavelet basis. In this paper, we study the stability and linear independence properties of the integer translates of these generalized spline functions. Moreover, we give a characterization of the scaling matrices to which the construction of the generalized spline functions can be applied.     

Continuous Gabor Transform for Strong Hypergroups

We define a continuous Gabor transform for strong hypergroups and prove a Plancherel formula, an L ² inversion formula and an uncertainty principle for it. As an example, we show how these techniques apply to the Bessel–Kingman hypergroups and to the dual Jacobi polynomial hypergroups. These examples have an interpretation in the setting of radial functions on R ^d and zonal functions on compact two-point homogeneous spaces, where they provide a new transform which possesses many properties of the classical Gabor transform.     

A shape-preserving quasi-interpolation operator satisfying quadratic polynomial reproduction property to scattered data

In this paper, we construct a univariate quasi-interpolation operator to non-uniformly distributed data by cubic multiquadric functions. This operator is practical, as it does not require derivatives of the being approximated function at endpoints. Furthermore, it possesses univariate quadratic polynomial reproduction property, strict convexity-preserving and shape-preserving of order 3 properties, and a higher convergence rate. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operator with that of Wu and Schaback’s quasi-interpolation scheme.     

On a Sequence of Positive Linear Operators Associated with a Continuous Selection of Borel Measures

In this paper we introduce and study a new sequence of positive linear operators acting on the space of Lebesgue-integrable functions on the unit interval. These operators are defined by means of continuous selections of Borel measures and generalize the Kantorovich operators. We investigate their approximation properties by presenting several estimates of the rate of convergence by means of suitable moduli of smoothness. Some shape preserving properties are also shown. Dedicated to the memory of Professor Aldo Cossu     

Mellin-type differential equations and associated sampling expansions

This paper is devoted to a self-contained approach to Mellin-type differential equations and associated ssampling expansions. Here the first order differential operator is not the normal d/dx but D_M,c=xd/dx+c,c E R being connected with the definition of the Mellin transform. Existence and uniqueness theorems are established for a system of first order Mellin equations and the properties of nth order linear equations are investigated. Then self adjoint Mellin-type second order Sturm-Liouville eigenvalue problems are considered and properties of the eigenvalues, eigenfunctions and Green's functions are derived. As applications. sampling representations for two classes of integral transforms arising from the eigenvalue problem are introduced. In the first class the kernesl are solutions of the problem and in the second they are expressed in terms of green's function.     

A quadrature formula for the Hankel transform

We present in this paper a quadrature formula for a certain Fourier-Bessel transform and, closely related to this, for the Hankel transform of order >–1. Such formulas originate in the context of a Galerkin-type projection of the weightedL ₂(–, ; ) space ( is the weight function mentioned below) used to get a discrete representation of a certain physical problem in Quantum Mechanics. The generalized Hermitee polynomialsH ₀ (x),H ₁ (x),..., with weight function (x), are used as the basis on which such a projection takes place. It is shown that theN-dimensional vectors representing certain projected functions as well as the entries of theN×N matrix representing the kernel of that Fourier-Bessel transform, approach the exact functional values at the zeros of theNth generalized Hermitee polynomial whenN.These properties lead to propose this matrix as a finite representation of the kernel of the Fourier-Bessel transform involved in this problem and theN zeros of the generalized Hermitee polynomialH _N (x) as abscissas to yield certain quadrature formulae for this integral and for the related Hankel transform. The error function produced by this algorithm is estimated at theN nodes and its is shown to be of a smaller order than 1/N. This error estimate is valid for piecewise continuous functions satisfying certain integral conditions involving their absolute values. The algorithm is presented with some numerical examples.     

Approximation inL 2 Sobolev spaces on the 2-sphere by quasi-interpolation

In this article we consider a simple method of radial quasi-interpolation by polynomials on the unit sphere in ℝ³, and present rates of covergence for this method in Sobolev spaces of square integrable functions. We write the discrete Fourier series as a quasi-interpolant and hence obtain convergence rates, in the aforementioned Sobolev spaces, for the discrete Fourier projection. We also discuss some typical practical examples used in the context of spherical wavelets.     

Constrained multi-degree reduction of triangular Bézier surfaces using dual Bernstein polynomials

We propose a novel approach to the problem of multi-degree reduction of Bézier triangular patches with prescribed boundary control points. We observe that the solution can be given in terms of bivariate dual discrete Bernstein polynomials. The algorithm is very efficient thanks to using the recursive properties of these polynomials. The complexity of the method is O(n²m²), n and m being the degrees of the input and output Bézier surfaces, respectively. If the approximation—with appropriate boundary constraints—is performed for each patch of several smoothly joined triangular Bézier surfaces, the result is a composite surface of global C^r continuity with a prescribed order r. Some illustrative examples are given.     

Uncertainty Principles for the Segal-Bargmann Transform

We recall some properties of the Segal-Bargmann transform; and we establish for this transform qualitative uncertainty principles: local uncertainty principle, Heisenberg uncertainty principle, Donoho-Stark''s uncertainty principle and Matolcsi-Sz\"ucs uncertainty principle.