d0300thttpwww.comonIDE≥L LETTER TO THE EDITOR Explicit Construction of Framelets Alexander petukhov 2 Communicated by Charles Chui Received July 11, 2000; revised October 27, 2000; published online July 10, 2001 Abstract-We study tight wavelet frames associated with given refinable func tions which are obtained with the unitary extension principles. All possible solu- tions of the corresponding matrix equations are found. It is proved that the problem of the extension may be always solved with two framelets. In particular, if symbols of the refinable functions are polynomials(rational functions), then the correspond- with polynomial (rational)symbols can be for 2001 Academic Press Key Words: tight frames; multiresolution analysis; wavelets 1 INTRODUCTION The main goal of our paper' is to present an explicit construction of an arbitrary wavelet frames generated by a refinable function. After submission this paper to Applied and Computational Harmonic Analysis we received information that the editorial portfolio already contains the paper by C. Chui and w. He [3] that contains similar results. In this paper, we shall consider only functions of one variable in the space L (R)with the inner product (f,g) f(x)g(x)dx As usual, f(o)denotes the Fourier transform of f(x)ELZ(R), defined by f(o)= f(r)e-d Suppose a real-valued function E LZ(R)satisfies the following conditions (a)(2o)=mo(o)(a), where mo is essentially bounded 2T-periodic function; (b)lima→0(a)=(2x)-1 then the function is called refinable or scaling mo is called a symbol of and the relation in item(a) is called a refinement equation I This work was partially supported under Grants NSF KDI 578A045, DoD-N00014-97-1-0806, ONR/ARO- DEPSCoR-DAAG55-98-1-0002, and by Russian Foundation for Basic Research under Grant #OO-01-00467 2 Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, E-mail petukhov@math. sc. edu, petukhov@ pdmi.ras.ru 3 This paper is a reduced version of preprint [71 063-520313500 Copyright G 2001 by Academic All rights of reproductiApplied and Computational Harmonic Analysis 11, 313–327 (2001) doi:10.1006/acha.2000.0337, available online at http://www.idealibrary.com on LETTER TO THE EDITOR Explicit Construction of Framelets 1 Alexander Petukhov 2 Communicated by Charles Chui Received July 11, 2000; revised October 27, 2000; published online July 10, 2001 Abstract—We study tight wavelet frames associated with given refinable func￾tions which are obtained with the unitary extension principles. All possible solu￾tions of the corresponding matrix equations are found. It is proved that the problem of the extension may be always solved with two framelets. In particular, if symbols of the refinable functions are polynomials (rational functions), then the correspond￾ing framelets with polynomial (rational) symbols can be found.  2001 Academic Press Key Words: tight frames; multiresolution analysis; wavelets. 1. INTRODUCTION The main goal of our paper 3 is to present an explicit construction of an arbitrary wavelet frames generated by a refinable function. After submission this paper to Applied and Computational Harmonic Analysis we received information that the editorial portfolio already contains the paper by C. Chui and W. He [3] that contains similar results. In this paper, we shall consider only functions of one variable in the space L 2 (R) with the inner product
f, g =
∞ −∞ f (x)g(x) dx. As usual, f (ˆ ω) denotes the Fourier transform of f (x) ∈ L 2 (R), defined by f (ˆ ω) =
∞ −∞ f (x)e −ixω dx. Suppose a real-valued function ϕ ∈ L 2 (R) satisfies the following conditions: (a) ϕ(ˆ 2ω) = m0(ω)ϕ(ˆ ω), where m0 is essentially bounded 2π-periodic function; (b) limω→0 ϕ(ˆ ω) = (2π )−1/2 ; then the function ϕ is called refinable orscaling, m0 is called a symbol of ϕ, and the relation in item (a) is called a refinement equation. 1 This work was partially supported under Grants NSF KDI 578AO45, DoD-N00014-97-1-0806, ONR/ARO￾DEPSCoR-DAAG55-98-1-0002, and by Russian Foundation for Basic Research under Grant #00-01-00467. 2 Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, E-mail: petukhov@math.sc.edu, petukhov@pdmi.ras.ru. 3 This paper is a reduced version of preprint [7]. 313 1063-5203/01 $35.00 Copyright  2001 by Academic Press All rights of reproduction in any form reserved.