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A construction of multiresolution analysis by integral equations

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

Institution:	College of Mathematics Science, Won Kwang University, 344-2 Shinyongdong Ik-San, Chunbuk 570-749, Korea ; College of Mathematics Science, Won Kwang University, 344-2 Shinyongdong Ik-San, Chunbuk 570-749, Korea Sun-Ho Yoon ; College of Mathematics Science, Won Kwang University, 344-2 Shinyongdong Ik-San, Chunbuk 570-749, Korea

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.

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

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