Derivative-orthogonal Riesz wavelets in Sobolev spaces with applications to differential equations

Abstract

Riesz wavelets in the Sobolev space H m ( R ) with m ∈ N ∪ { 0 } , whose m th-order derivatives are orthogonal among different levels, are of particular interest and importance in computational mathematics, due to their many desirable properties such as small condition numbers and sparse stiffness matrices. We call such Riesz wavelets in the Sobolev space H m ( R ) as m th-order derivative-orthogonal Riesz wavelets. In this paper we shall comprehensively study and completely characterize all compactly supported m th-order derivative-orthogonal Riesz wavelets in the Sobolev space H m ( R ) . More precisely, from any given compactly supported refinable vector function ϕ = ( ϕ 1 , … , ϕ r ) T in H m ( R ) satisfying the refinement equation ϕ ˆ ( 2 ξ ) = a ˆ ( ξ ) ϕ ˆ ( ξ ) for some r × r matrix a ˆ of 2 π -periodic trigonometric polynomials, we prove that there exists a compactly supported m th-order derivative-orthogonal Riesz wavelet in H m ( R ) , which is derived from ϕ through the refinable structure, if and only if the refinable vector function ϕ has stable integer shifts and the filter a has at least order 2 m sum rules. This double order of sum rules over the smoothness order m is surprising but is necessary for constructing m th-order derivative-orthogonal Riesz wavelets in H m ( R ) . Then we shall present several examples of such derivative-orthogonal spline Riesz wavelets with short support derived from B-splines and Hermite splines. To illustrate the developed theory and its potential usefulness, we shall apply our constructed such m th-order derivative-orthogonal Riesz wavelets for the numerical solutions of differential equations such as Sturm–Liouville equations and biharmonic equations. Our constructed derivative-orthogonal spline Riesz wavelets on the interval [ 0 , 1 ] have a simple structure with only one boundary wavelet at each endpoint and can easily handle different types of boundary conditions. The resulting coefficient matrices are sparse and have very small condition numbers with some examples even having the optimal condition number one.