Investigating the Orthogonality Conditions of Wavelets Based on Jacobi Polynomials
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INVESTIGATING THE ORTHOGONALITY CONDITIONS OF WAVELETS BASED ON JACOBI POLYNOMIALS V. Semenov1 and J. Prestin2
UDC 519.615
Abstract. The properties of wavelets based on Jacobi polynomials are analyzed. The conditions are considered under which these wavelets are mutually orthogonal and under which the wavelet basis is characterized by a minimum Riesz ratio. These problems lead to the solution of systems of nonlinear equations by a method proposed earlier by the authors. Keywords: wavelet, Jacobi polynomial, orthogonality condition, rootfinding. INTRODUCTION The last decades are characterized by the development of the theory of wavelets and its practical applications. The scope of wavelets is extended owing to their property to describe a local behavior of signals in different time scales, which makes it possible to use them for problems of telecommunications, geophysics, astrophysics, processing of audioand videosignals, biomedicine, etc. [1]. Each practical problem requires the use of a special class of wavelets. This article considers an important class of wavelets, namely, polynomial wavelets based on Jacobi polynomials. When carrying out the analysis, the theory of orthogonal polynomials presented in [2, 3] was used. The objective of this analysis is the obtaining of answers to the following questions: — what are the conditions under which the wavelets being considered are orthogonal? — what are the wavelet parameters ensuring a minimum Riesz ratio? The search for answers to these questions leads to the solution of systems of nonlinear equations. Some of the obtained systems of equations are solved by the method from [4] proposed by the authors. In Sec. 1, an advance information on polynomial wavelets based on Jacobi polynomials is given. The questions of orthogonality and minimization of the Riesz ratio are considered in Secs. 2 and 3, respectively. 1. ADVANCE INFORMATION Following [2], we will use the following definition of a polynomial wavelet of the nth order:
y n, r (x ) =
2n
å Pk (t r )Pk (x )
k = n+ 1
for some fixed set of parameters t1 , t 2 , K , t n . The Jacobi polynomial of the nth order that depends on the parameters a and b is defined as follows: 1
DELTA SPE Scientific and Production Enterprise, Kyiv, Ukraine, [email protected]. 2Institute of Mathematics of the University of L&& u beck, L&& u beck, Germany, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 4, July–August, 2018, pp. 182–190. Original article submitted September 19, 2017. 678
1060-0396/18/5404-0678 ©2018 Springer Science+Business Media, LLC
Pn( a , b) ( x )
m
n G ( a + n + 1) G ( a + b + n + m + 1) æ x - 1 ö C nm = ÷ , ç å n! G ( a + b + n + 1) m = 0 G ( a + m + 1) è 2 ø
(1)
where G(×) denotes a gamma function. Moreover, polynomial (1) is usually normalized by the multiplier
K=
2 a + b+ 1 G ( n + a + 1) G ( n + b + 1) (2n + a + b + 1) G ( n + a + b + 1) n!
so that the orthonormality condition á Pn( a , b) ( x ), Pm( a , b) ( x )ñ = d m n is satisfied. In what follows, a matrix B
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