A construction of multiresolution analysis by integral equations

Authors:
Dong-Myung Lee, Jung-Gon Lee and Sun-Ho Yoon

Journal:
Proc. Amer. Math. Soc. **130** (2002), 3555-3563

MSC (2000):
Primary 41A17, 42C15, 46A45, 46C99

DOI:
https://doi.org/10.1090/S0002-9939-02-06713-8

Published electronically:
July 2, 2002

MathSciNet review:
1920033

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we present a versatile construction of multiresolution analysis of two variables by means of eigenvalue problems of the integral equation, for . As a consequence we show that if is the solution of the equation with , then constructs a two-variable multiresolution analysis.

**[1]**J. Aguirre, M. Escobedo, J.C. Peral, and P.H. Tchamitchian,*Basis of wavelets and atomic decompositions of**and*, Proceedings AMS**111**(1991), 683-693. MR**91k:42037**.**[2]**P. G. Casazza and A.O. Christensen,*Frames containing a Riesz basis and preservation of this property under perturbations*, SIAM J. Math. Anal.**29**(1998), 266-278. MR**99i:42043****[3]**C. K. Chui,*An Introduction to Wavelets*, Academic Press, Inc (1992). MR**93f:42055****[4]**M. G.Cui, D. M. Lee, and J. G. Lee,*Fourier Transforms and Wavelet Analysis*, Kyung Moon Press (2001).**[5]**R. Coifman, and Y. Meyer,*Remarques sur L'analyse de Fourier a fenetre*, C. R. Acad. Sci. pairs**t.312**(1991), 259-261. MR**92k:42042****[6]**I. Daubechies and J. C. Lagarias,*Two-Scale Difference Equations I, Existence and global regularity of Solutions*, SIAM J. Math. Anal.**22**(1991), 1388-1410. MR**92d:39001****[7]**I. Daubechies,*Ten Lectures on Wavelets*, SIAM, Philadelphia, PA (1992). MR**93e:42045****[8]**K. Grochenig and W. R. Madych,*Multiresolution Analysis, Haar Bases, and Self-Similar Tilings of Rn*, IEEE Trans. on Information Theory**38**(1992), 556-568. MR**93i:42001****[9]**A. Grossmann and J. Morlet,*Transforms associated to square integrable group representations I*, J. Math. Phys.**26**(1986), 2473-2479. MR**86k:22013****[10]**B. Jawerth and W. Sweldens,*An overview of wavelet based multiresolution analysis*, SIAM Rev.**36(3)**(1994), 377-412. MR**95f:42002****[11]**W. Lawton,*Necessary and sufficient conditions for constructing orthonormal wavelet bases*, J. Math. Phys.**32**(1) (1991), 57-61. MR**91m:81100****[12]**J. G. Lee and D. M. Lee,*A Filtering Formula on Wavelets*, Korean Annales of Math.**15**(1998), 247-255.**[13]**S. Mallat,*A theory for multiresolution signal decomposition: the wavelet representation*, IEEE Trans. Pattern Anal. Machine Intell.**11**(1989).

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Additional Information

**Dong-Myung Lee**

Affiliation:
College of Mathematics Science, Won Kwang University, 344-2 Shinyongdong Ik-San, Chunbuk 570-749, Korea

Email:
dmlee@wonkwang.ac.kr

**Jung-Gon Lee**

Affiliation:
College of Mathematics Science, Won Kwang University, 344-2 Shinyongdong Ik-San, Chunbuk 570-749, Korea

**Sun-Ho Yoon**

Affiliation:
College of Mathematics Science, Won Kwang University, 344-2 Shinyongdong Ik-San, Chunbuk 570-749, Korea

DOI:
https://doi.org/10.1090/S0002-9939-02-06713-8

Keywords:
Fourier transform,
wavelet analysis,
integral equation,
multiresolution analysis,
Riesz basis

Received by editor(s):
August 23, 2000

Published electronically:
July 2, 2002

Additional Notes:
This paper was supported by Won Kwang University in 2002

Communicated by:
Christopher D. Sogge

Article copyright:
© Copyright 2002
American Mathematical Society