Spline wavelets on the interval with homogeneous boundary conditions
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Spline wavelets on the interval with homogeneous boundary conditions Rong-Qing Jia
Received: 13 April 2007 / Accepted: 1 December 2007 / Published online: 9 April 2008 © Springer Science + Business Media, LLC 2008
Abstract In this paper we investigate spline wavelets on the interval with homogeneous boundary conditions. Starting with a pair of families of B-splines on the unit interval, we give a general method to explicitly construct wavelets satisfying the desired homogeneous boundary conditions. On the basis of a new development of multiresolution analysis, we show that these wavelets form Riesz bases of certain Sobolev spaces. The wavelet bases investigated in this paper are suitable for numerical solutions of ordinary and partial differential equations. Keywords Spline wavelets · Wavelets on the interval · Slant matrices · Multiresolution analysis · Riesz bases · Sobolev spaces Mathematics Subject Classifications (2000) 42C40 · 41A15 · 46B15
1 Introduction In this paper we investigate spline wavelets on the interval [0, 1] with homogeneous boundary conditions. In [3] Chui and Wang initiated the study of semi-orthogonal wavelets generated from cardinal splines. Following their work, Chui and Quak [4] constructed semi-orthogonal spline wavelets on the interval [0, 1]. In [10] Jia modified the construction of boundary wavelets in [4] and established the stability of
Communicated by Tim Goodman. Supported in part by NSERC Canada under Grant OGP 121336. R.-Q. Jia (B) Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada T6G 2G1 e-mail: [email protected]
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wavelet bases in Sobolev spaces. Concerning applications of wavelets to numerical solutions of ordinary and partial differential equations, we are interested in wavelets on the interval [0, 1] with homogeneous boundary conditions. Using Hermite cubic splines, Jia and Liu in [11] constructed wavelet bases on the interval [0, 1] and applied those wavelets to numerical solutions of the Sturm-Liouville equations with the Dirichlet boundary condition. In this paper, starting with cardinal B-splines, we will construct a family of wavelets on the interval [0, 1] which satisfy homogeneous boundary conditions of arbitrary order. Let us introduce some notation. We use IN, ZZ, IR, and C to denote the set of positive integers, integers, real numbers, and complex numbers, respectively. Let IN0 := IN ∪ {0}. For a complex number c, we use c to denote its complex conjugate. For a complex-valued (Lebesgue) measurable function f on IR, let
1/ p
f p :=
| f (x)| p dx
for 1 ≤ p < ∞,
IR
and let f ∞ denote the essential supremum of | f | on IR. For 1 ≤ p ≤ ∞, by L p (IR) we denote the Banach space of all measurable functions f on IR such that f p < ∞. In particular, L2 (IR) is a Hilbert space with the inner product given by f, g :=
f (x)g(x) dx,
f, g ∈ L2 (IR).
IR
The Fourier transform of a function f ∈ L1 (IR) is defined by fˆ(ξ ) :=
f (x)e−ixξ dx,
ξ ∈ IR.
IR
The Fourier transform can be natura
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