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Published in:
Introduction to Fractals Read chapter Read first chapter

2014 | OriginalPaper | Chapter

Inverse Representation Theorem for Matrix Polynomials and Multiscaling Functions

Authors : M. Mubeen, V. Narayanan

Published in: Fractals, Wavelets, and their Applications

Publisher: Springer International Publishing

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Abstract

Wavelet analysis provides suitable bases for the class of L 2 functions. The function to be represented is approximated at different resolutions. The desirable properties of a basis are orthogonality, compact supportedness and symmetricity. In the scalar case, the only wavelet with these properties is Haar wavelet. Theory of multiwavelets assumes significance since it offers symmetric, compactly supported, orthogonal bases for L 2(R). The properties of a multiwavelet are determined by the corresponding Multiscaling Function. A multiscaling function is characterized by its symbol function which is a matrix polynomial in complex exponential. The inverse representation theorem of matrix polynomials provides a method to construct a matrix polynomial from its Jordan pair. Our objective is to find the properties that characterize a Jordan pair of a symbol function of a multiscaling function with desirable properties.

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previous chapter Interval Wavelet Sets Determined by Points on the Circle
next chapter A Remark on Reconstruction of Splines from Their Local Weighted Average Samples
Literature
1.
2.
3.
Daubechies, I.: Ten Lectures on Wavelets, vol. 61. Society for Industrial and Applied Mathematics, Philadelphia (1992)CrossRefMATH
4.
Geronimo, J.S., Hardin, D.P., Massopust, P.R.: Fractal functions and wavelet expansions based on several scaling functions. J. Approx. Theory 78(3), 373–401 (1994)MathSciNetCrossRefMATH
5.
Gohberg, I., Rodman, L., Lancaster, P.: Matrix Polynomials. Academic, London, (1982)MATH
6.
7.
Heil, C., Colella, D.: Matrix refinement equations: existence and uniqueness. J. Fourier Anal. Appl. 2, 363–378 (1996)MathSciNetMATH
8.
Heil, C., Strang, G.: Continuity of the joint spectral radius: application to wavelets. In: Linear Algebra for Signal Processing, pp. 51–61. Springer, New York (1995)
9.
10.
Kaiser, G.: A Friendly Guide to Wavelets. Birkhauser, Basel (1994)MATH
11.
Keinert, F.: Wavelets and Multiwavelets. Chapman and Hall/CRC, London (2004)MATH
12.
13.
Markus, A.S.: Introduction to the Spectral Theory of Polynomial Operator Pencils, vol. 71. AMS Bookstore, Providence (1988)MATH
14.
Plonka, G., Strela, V.: From Wavelets to Multiwavelets. Mathematical Methods for Curves and Surfaces II (eds- M. Dahlen, T. Lyche, L.L. Schumaker), Vanderbilt University Press 1998, Nashville, TN, pp. 1–25 (1997)
15.
Plonka, G., Strela, V.: Construction of multiscaling functions with approximation and symmetry. SIAM J. Math. Anal. 29(2), 481–510 (1998)MathSciNetCrossRefMATH
16.
Sjoestrand, J.: Spectral properties of non-self-adjoint operators. arXiv preprint arXiv:1002.4844 (2010)
17.
Stanimirovic, P., Stankovic, M.: Determinants of rectangular matrices and Moore-Penrose inverse. Novi Sad J. Math 27(1), 53–69 (1997)MathSciNetMATH
18.
19.
Strang, G., Strela, V.: Orthogonal multiwavelets with vanishing moments. Opt. Eng. 33(7), 2104–2107 (1994)CrossRef
20.
Strela, V.: Multiwavelets: regularity, orthogonality and symmetry via two scale similarity transform. Stud. Appl. Math. 98(4), 335–354 (1997)MathSciNetCrossRefMATH
21.
Theys, J.: Joint spectral radius: theory and approximations. Diss. Ph.D. thesis, Universite catholique de Louvain (2005)
22.
Thielemann, H.: Polynomial functions are refinable. arXiv preprint arXiv:1012.2453 (2010)
23.
Turcajova, R.: An algorithm for the construction of symmetric orthogonal multiwavelets. SIAM J. Matrix Anal. Appl. 25(2), 532–550 (2003)MathSciNetCrossRefMATH
24.
Turcajova, R., Kautsky, J.: Block Toeplitz-like operators and multiwavelets. In: SPIE’s 1995 Symposium on OE/Aerospace Sensing and Dual Use Photonics. International Society for Optics and Photonics (1995)
Metadata
Title
Inverse Representation Theorem for Matrix Polynomials and Multiscaling Functions
Authors
M. Mubeen
V. Narayanan
Copyright Year
2014
Book
Fractals, Wavelets, and their Applications

Print ISBN: 978-3-319-08104-5

Electronic ISBN: 978-3-319-08105-2

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-319-08105-2_20

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