A series expansion with remainder for functions in a Sobolev space is derived in terms of the classical Bernoulli polynomials, the B-spline scale-space and the continuous wavelet transforms with the derivatives of the standardized B-splines as mother wavelets. In the limit as their orders tend to infinity, the B-splines and their derivatives converge to the Gaussian function and its derivatives...

We present a generalization of the commutation formula to irregular subdivision schemes and wavelets. We show how, in the noninterpolating case, the divided differences need to be adapted to the subdivision scheme. As an example we include the construction of an entire family of biorthogonal compactly supported irregular knot B-spline wavelets starting from Lagrangian interpolation.

Abstract: The idea of summing pairs of so-called semi-wavelets has been found to be very useful for constructing piecewise linear wavelets over refinements of arbitrary triangulations. In this paper we demonstrate the versatility of the semi-wavelet approach by using it to construct bases for the piecewise linear wavelet spaces induced by uniform refinements of four-directional box-spline grids.

This paper is to construct Riesz wavelets with short support. Riesz wavelets with short support are of interests in both theory and application. In theory, it is known that a B-spline of order m has the shortest support among all compactly supported refinable functions with the same regularity. However, it remained open whether a Riesz wavelet with the shortest support and m vanishing moments c...

The objective of this paper is to introduce a general scheme for the construction of interpolatory approximation formulas and compactly supported wavelets by using spline functions with arbitrary (nonuniform) knots. Both construction procedures are based on certain “optimally local” interpolatory fundamental spline functions which are not required to possess any approximation property.

This thesis is a theorical and numerical contribution to wavelet transform in image processing and surface computing. Group theory approach, multiresolution approach and lters banc approach of wavelets basis are reviewed. The wavelet transform applications concern image compression, discrete curve representation and surface approximation by radial functions. We begin by reviewing di erent image...

We present spline wavelets of class Cn(R) supported by sequences of aperiodic discretizations of R. The construction is based on multiresolution analysis recently elaborated by G. Bernuau. At a given scale, we consider discretizations that are sets of left-hand ends of tiles in a self-similar tiling of the real line with finite local complexity. Corresponding tilings are determined by two-lette...

The notion of tight (wavelet) frames could be viewed as a generalization of orthonormal wavelets. By allowing redundancy, we gain the necessary flexibility to achieve such properties as “symmetry” for compactly supported wavelets and, more importantly, to be able to extend the classical theory of spline functions with arbitrary knots to a new theory of spline-wavelets that possess such importan...

We explicitly construct compactly-supported wavelets associated with L-spline spaces. We then apply the theory to develop multiresolution methods based on L-splines. x

Following the approach of Chui and Quak, we investigate semi-orthogonal spline wavelets on the unit interval [0, 1]. We give a slightly different construction of boundary wavelets. As a result, we are able to prove that the inner wavelets and the newly constructed boundary wavelets together constitute a Riesz basis for the wavelet space at each level with the Riesz bounds being level-independen...