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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: July 2, 2002
MathSciNet review: 1920033
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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 $\lambda =2$. As a consequence we show that if $\phi (x)$ is the solution of the equation $\phi (x) = \lambda \int _{\mathbb{R}} h (2x-y) \phi (y)dy$ with $supp \hat h(\omega ) = [-\pi , \pi ]$, then $V_{j} =span \{ \phi (2^{j} x_{1} -k_{1} )$ $ \phi (2^{j} x_{2} -k_{2} ) \vert k_{1} , k_{2} \in \mathbb{Z} \}$ constructs a two-variable multiresolution analysis.


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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: http://dx.doi.org/10.1090/S0002-9939-02-06713-8
PII: S 0002-9939(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