Nonstationary tight wavelet frames, II: unbounded intervals

From the definition of tight frames, normalized with frame bound constant equal to one, a tight frame of wavelets can be considered as a natural generalization of an orthonormal wavelet basis, with the only exception that the wavelets are not required to have norm equal to one. However, without the orthogonality property, the tight-frame wavelets do not necessarily have vanishing moments of order higher than one, although the associated multiresolution spaces may contain higher order polynomials locally. This observation motivated a relatively recent parallel development of the general theory of affine (i.e. stationary) tight frames by Daubechies-Han-Ron-Shen and the authors, with both papers published in this journal in 2002-2003. In the second issue of this volume of Special Issues, we introduced a general theory of nonstationary wavelet frames on a bounded interval, and emphasized, with illustrative examples, that in general such tight frames cannot be easily constructed by adopting the above-mentioned stationary wavelets as “interior” frame elements, even for the “uniform” setting. Hence, the results on nonstationary tight frames on a bounded interval obtained in our previous paper are definitely not follow-up of the present paper, in which we will introduce a general mathematical theory of nonstationary tight frames on unbounded intervals. While the “Fourier” and “matrix calculus” approaches were used in the above-mentioned works on stationary and nonstationary frames, respectively, we will engage a “kernel operator” approach to the development of the theory of nonstationary tight frames on unbounded intervals, and observe that this somewhat new approach could be considered as a unification of the previous considerations. The nonstationary notion discussed in this paper is very general, with (polynomial) splines of any (fixed) order on arbitrary but dense nested knot vectors as canonical examples, and in particular, eliminates the rigid assumptions of invariance in translations and scalings among different levels. In addition to the development of approximate duals and construction of compactly supported tight-frame wavelets with desirable order of vanishing moments, a unified formulation of the degree of approximation in Sobolev spaces of negative exponent, of order up to twice of that of the corresponding approximate dual, is established in this paper. A thorough development for the general spline setting is a major focus of our study, with examples of tight frames of splines with multiple knots included to illustrate our constructive approach.