A Non-linear Approach to Signal Processing by Means of Vector Measure Orthogonal Functions (original) (raw)

Vector Measure Orthonormal Systems and Self-weighted Functions Approximation

If λ is a positive vector measure on l 2 , the notion of λ-orthonormal system of functions leads to a natural generalization of the relation between orthogonality and best approximation in Hilbert spaces for spaces L 2 (λ) of square integrable functions with respect to λ. We provide a vector orthogonality criterion that induces the definition of a particular projection on a subspace of L 2 (λ) that we call the self-weighted approximation. As an application, we show a new extrapolation technique.

Orthogonal systems in L2L^2L2 spaces of a vector measure

TURKISH JOURNAL OF MATHEMATICS, 2013

Let m : Σ → X be a Banach space valued countably additive vector measure. In this paper we present a procedure to construct an m-orthogonal system in the space L 2 (m) of square integrable functions with respect to m. If the vector measure is constructed from a family of indeterminate scalar measures, it is possible to obtain a family of polynomials that is orthogonal with respect to this vector measure. On the other hand, if the vector measure is fixed, then we can obtain sequences of orthogonal functions using the Kadec − P elczyński disjointification method.

Orthogonal series representations for generalized functions

Journal of Mathematical Analysis and Applications, 1988

A test function space AL,, of smooth weighted LP-functions is constructed whose elements possess orthonormal series expansion converging in the space. Conditions for convergence in ALWP -space of any orthogonal series are obtained. The space of continuous linear functionals on ALW, is investigated and orthogonal series representations for its elements are given. Various special cases of the expansion are discussed, and their applications are pointed out.

Some Orthogonalities in Approximation Theory

Journal of Mathematical Sciences, 2006

In approximation theory, a number of quantities are interrelated via orthogonal transformations. The revelation of these orthogonalities allows one to obtain useful relations and, in particular, to use known theorems on the Fourier coefficients. The paper considers this approach in application to the Fourier transforms of finite functions, to the best L2-approximations, and to trigonometric polynomials with coefficients in a vector space. Bibliography: 11 titles.

The reproducing kernel structure arising from a combination of continuous and discrete orthogonal polynomials into Fourier systems

2007

We study mapping properties of operators with kernels defined via a combination of continuous and discrete orthogonal polynomials, which provide an abstract formulation of quantum (q-) Fourier type systems. We prove Ismail´s conjecture regarding the existence of a reproducing kernel structure behind these kernels, by establishing a link with Saitoh´s theory of linear transformations in Hilbert space. The results are illustrated with Fourier kernels with ultraspherical weights, their continuous q-extensions and generalizations. As a byproduct of this approach, a new class of sampling theorems is obtained, as well as Neumann type expansions in Bessel and q-Bessel functions.

The approximation of almost time and band limited functions by their expansion in some orthogonal polynomials bases

The aim of this paper is to investigate the quality of approximation of almost time and almost band-limited functions by its expansion in three classical orthogonal polynomials bases: the Hermite, Legendre and Chebyshev bases. As a corollary, this allows us to obtain the quality of approximation in the L 2 −Sobolev space by these orthogonal polynomials bases. Also, we obtain the rate of the Legendre series expansion of the prolate spheroidal wave functions. Some numerical examples are given to illustrate the different results of this work.

A Comparison between Different Concepts of Almost Orthogonal Polynomials

In this paper, we will discuss the notion of almost orthogonality in a functional sequence.Especially, we will define a few sequences of almost orthogonal polynomials which can be used successfully for modeling of electronic systems which generate orthonormal basis. We will include quasi-orthogonality and examine its influence on the behavior of these sequences.

Inner Product Space in Fourier Approximation.

International Journal of Engineering Sciences & Research Technology, 2014

et f be an element and S a subset of a normed linear space x. A basic Problem of approximation theory is to find an element of S, which is as close as possible to f, i.e. seek an element S* of S such that ‖ f - S* ‖ ≤ ‖f - s ‖ for all S* in S. This work seeks to use the fourier approximation method using the Inner product space to obtain a best approximation of functions. The fourier approximation in calculus is shown to be a special least square approximation. Define an inner product and norm on c [ -π, π] by the equation. f.g =∫ �(𝑥) 𝜋 −𝜋 �(𝑥) �𝑥 𝑎�� ‖ f ‖ = [∫ � 𝜋 2 −𝜋 (𝑥)�𝑥] 1 2 𝐿 Let Jn be the set of all trigonometric polynomial of degree at most n. We seek S* of Jn such that the integral ∫ 𝜋 −𝜋 [f (x) - S* (x)] 2dx is minimum. The inner product space, or orthogonal projections, least square approximations and Fourier approximations are defined. The inner product space and orthogonal projection are applied to the least square approximation and subsequently to Fourier approximation in calculus.

Generalizations of orthogonal polynomials

Journal of Computational and Applied Mathematics, 2005

We give a survey of recent generalizations for orthogonal polynomials that were recently obtained. It concerns not only multidimensional (matrix and vector orthogonal polynomials) and multivariate versions, or multipole (orthogonal rational functions) variants of the classical polynomials but also extensions of the orthogonality conditions (multiple orthogonality). Most of these generalizations are inspired by the applications in which they are applied. We also give a glimpse of the applications, but they are usually also generalizations of applications where classical orthogonal polynomials play a fundamental role: moment problems, numerical quadrature, rational approximation, linear algebra, recurrence relations, random matrices.

Approximation of signals (functions) belonging to the weighted W(Lp,ξ(t))-class by linear operators

International Journal of Mathematics and Mathematical Sciences, 2006

Mittal and Rhoades (1999–2001) and Mittal et al. (2006) have initiated the studies of error estimatesEn(f)through trigonometric Fourier approximations (TFA) for the situations in which the summability matrixTdoes not have monotone rows. In this paper, we determine the degree of approximation of a functionf˜, conjugate to a periodic functionfbelonging to the weightedW(Lp,ξ(t))-class(p≥1), whereξ(t)is nonnegative and increasing function oftby matrix operatorsT(without monotone rows) on a conjugate series of Fourier series associated withf. Our theorem extends a recent result of Mittal et al. (2005) and a theorem of Lal and Nigam (2001) on general matrix summability. Our theorem also generalizes the results of Mittal, Singh, and Mishra (2005) and Qureshi (1981-1982) for Nörlund(Np)-matrices.