On the Nörlund Method of Signal Processing Involving Coifman Wavelets

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.433-440.3378 ◽

2012 ◽

Vol 433-440 ◽

pp. 3378-3387

Author(s):

Sarjoo Prasad Yadav ◽

Rakesh Kumar Yadav ◽

Dinesh Kumar Yadav

Keyword(s):

Signal Processing ◽

Real Line ◽

Jacobi Polynomials ◽

Jacobi Expansions ◽

Coifman Wavelets

We consider on real line R a space of signals which are p-power (1 ≤ p ≤∞ ) Lebesgue integrable with weight w(x) = (1 - x)α (1 + x)β , ( α, β > -1) on [-1, 1] R. A subspace χabNvof Xabvis recognized by restricting the types of signals, so that the signals are represented by Jacobi Polynomials. Then by the derivability of Jacobi polynomials, we reach to the conclusion that the signals of the subspace XαβNv can be represented by the Coifman wavelets. The method involves the N rlund summation of Fourier-Jacobi expansions and the properties of Jacobi polynomials in [--1, 1] R

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Time-and-band limiting for matrix orthogonal polynomials of Jacobi type

Random Matrices Theory and Application ◽

10.1142/s2010326317400019 ◽

2017 ◽

Vol 06 (04) ◽

pp. 1740001 ◽

Cited By ~ 2

Author(s):

M. Castro ◽

F. A. Grünbaum

Keyword(s):

Signal Processing ◽

Differential Operator ◽

Orthogonal Polynomials ◽

Random Matrix Theory ◽

Random Matrix ◽

Jacobi Polynomials ◽

Matrix Theory ◽

Matrix Orthogonal Polynomials

We extend to a situation involving matrix-valued orthogonal polynomials a scalar result that plays an important role in Random Matrix Theory and a few other areas of mathe-matics and signal processing. We consider a case of matrix-valued Jacobi polynomials which arises from the study of representations of [Formula: see text], a group that plays an important role in Random Matrix Theory. We show that in this case an algebraic miracle, namely the existence of a differential operator that commutes with a naturally arising integral one, extends to this matrix-valued situation.

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Construction of wavelets and framelets on a bounded interval

Analysis and Applications ◽

10.1142/s0219530518500045 ◽

2018 ◽

Vol 16 (06) ◽

pp. 807-849 ◽

Cited By ~ 6

Author(s):

Bin Han ◽

Michelle Michelle

Keyword(s):

Signal Processing ◽

Boundary Conditions ◽

Real Line ◽

Construction Method ◽

The Real ◽

Bounded Interval ◽

Orthogonal Wavelets ◽

Simple Boundary

Many problems in applications are defined on a bounded interval. Therefore, wavelets and framelets on a bounded interval are of importance in both theory and application. There is a great deal of effort in the literature on constructing various wavelets on a bounded interval and exploring their applications in areas such as numerical mathematics and signal processing. However, many papers on this topic mainly deal with individual examples which often have many boundary wavelets with complicated structures. In this paper, we shall propose a method for constructing wavelets and framelets in [Formula: see text] from symmetric wavelets and framelets on the real line. The constructed wavelets and framelets in [Formula: see text] often have a few simple boundary wavelets/framelets with the additional flexibility to satisfy various desired boundary conditions. To illustrate our construction method, from several spline refinable vector functions, we present several examples of (bi)orthogonal wavelets and spline tight framelets in [Formula: see text] with very simple boundary wavelets/framelets.

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Fast algorithms for Jacobi expansions via nonoscillatory phase functions

IMA Journal of Numerical Analysis ◽

10.1093/imanum/drz016 ◽

2019 ◽

Vol 40 (3) ◽

pp. 2019-2051

Author(s):

James Bremer ◽

Haizhao Yang

Keyword(s):

Differential Equation ◽

Phase Function ◽

Numerical Experiments ◽

Jacobi Polynomials ◽

Source Code ◽

Fast Algorithms ◽

Affine Function ◽

Jacobi Expansions ◽

Sturm Liouville ◽

Phase Functions

Abstract We describe a suite of fast algorithms for evaluating Jacobi polynomials, applying the corresponding discrete Sturm–Liouville eigentransforms and calculating Gauss–Jacobi quadrature rules. Our approach, which applies in the case in which both of the parameters $\alpha $ and $\beta $ in Jacobi’s differential equation are of magnitude less than $1/2$, is based on the well-known fact that in this regime Jacobi’s differential equation admits a nonoscillatory phase function that can be loosely approximated via an affine function over much of its domain. We illustrate this with several numerical experiments, the source code for which is publicly available.

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