Construction of Biorthogonal B-spline Type Wavelet Sequences with Certain Regularities Tian-Xiao He∗ Department of Mathematics and Computer Science Illinois Wesleyan University, Bloomington, IL 61702-2900, USA
[email protected] Abstract In this paper we study a refinement equation of the form φ(x) = P n∈Z hnφ(2x − n), where {hn} is a finitely supported sequence. 1 P n Let the symbol of the sequnce m(z) := 2 n hnz take the form 1+z N k0 Pk j P P j m(z) = ( 2 ) z j=0 ajz where j aj = 1 and j(−1) aj 6= P 2 P 2 0. Under the assumption 0≤odd j≤k |aj| + 0≤even j≤k |aj| 2N−1 < 2 we show that the corresponding φ is in L2. Then the B-spline type wavelet sequences that possess the largest possible regularities and required vanishing moments are characterized. AMS Subject Classification: 39A70, 41A80, 65B10. Key Words and Phrases: symbolic summation operator, power series, generating function, Euler’s series transform. 1 Introduction We start by setting some notations. We define a low-pass filter as ∗The author would like to thank the Illinois Wesleyan University for a sabbatical leave during which the research in this paper was carried out. 1 2 Tian-Xiao He −1 X −inξ m0(ξ) = 2 cne . (1.1) n Here, we assume that only finitely many cn are nonzero. However, some of our results can be extended to infinite sequences that have sufficient decay for |n| → ∞. Next, we define φ by ˆ ∞ −j φ(ξ) = Πj=1m0(2 ξ).