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In this paper, we present a new computational method for solving linear Fredholm integral equations of the second kind, which is based on the use of B-spline quasi-affine tight framelet systems generated by the unitary and oblique extension principles. We convert the integral equation to a system of linear equations. We provide an example of the construction of quasi-affine tight framelet systems. We also give some numerical evidence to illustrate our method. The numerical results confirm that the method is efficient, very effective and accurate.

The line between the displayed formulas (16) and (17) was copied incorrectly from [41, Theorem 1].

An important tool for the construction of tight wavelet frames is the Unitary Extension Principle first formulated in the Fourier-domain by Ron and Shen. We show that the time-domain analogue of this principle provides a unified approach to the construction of tight frames based on many variations of multiresolution analyses, e.g., regular refinements of bounded L-shaped domains, refinements of subdivision surfaces around irregular vertices, and nonstationary subdivision. We consider the case of nonnegative refinement coefficients and develop a fully local construction method for tight frames. Especially, in the shift-invariant setting, our construction produces the same tight frame generators as the Unitary Extension Principle.

We construct several families of tight wavelet frames in L2(R2)L2(R2) associated to the quincunx matrix. A couple of those families has five generators. Moreover, we construct a family of tight wavelet frames with three generators. Finally, we show families with only two generators. The generators have compact support, any given degree of regularity, and any fixed number of vanishing moments. Our construction is made in Fourier space and involves some refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. In addition, we will use well known results on construction of tight wavelet frames with two generators on RR with the dyadic dilation. The refinable functions we use are constructed from the Daubechies low pass filters and are compactly supported. The main difference between these families is that while the refinable functions associated to the five generators have many symmetries, the refinable functions used in the construction of the others families are merely even. The first author was partially supported by MEC/MICINN grant #MTM2011-27998 (Spain).

Wavelets with matrix dilation are studied. An explicit formula for masks providing vanishing moments is found. The class of interpolatory masks providing vanishing moments is also described. For an interpolatory mask, formulas for a dual mask which also provides vanishing moments of the same order and for wavelet masks are given explicitly. An example of construction of symmetric and antisymmetric wavelets for a concrete matrix dilation is presented.

In this paper, we introduce matrix-valued multi-resolution structure and matrix-valued bivariate wavelet wraps. A constructive method of semi-orthogonal matrix-valued bivari-ate wavelet wraps is presented. Their properties have been characterized by using time-frequency analysis method, unitary extension principle and operator theory. The direct decom-position relation is obtained.

We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudo-spline tight frames, and symmetric bi-frames with short supports and high approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we discuss briefly as well.  2002 Elsevier Science (USA). All rights reserved.

We analyze the internal structure of the multiresolution analyses of L^2(R^d) defined by the unitary extension principle (UEP) of Ron and Shen. Suppose we have a wavelet tight frame defined by the UEP. Define V"0 to be the closed linear span of the shifts of the scaling function and W"0 that of the shifts of the wavelets. Finally, define V"1 to be the dyadic dilation of V"0. We characterize the conditions that V"1=W"0, that V"1=V"0@?W"0 and V"1=V"0@?W"0. In particular, we show that if we construct a wavelet frame of L^2(R) from the UEP by using two trigonometric filters, then V"1=V"0@?W"0; and show that V"1=W"0 for the B-spline example of Ron and Shen. A more detailed analysis of the various 'wavelet spaces' defined by the B-spline example of Ron and Shen is also included.

Wavelet frames with matrix dilation are studied. We found a necessary condition and a sufficient condition under which a given pair of refinable functions generates dual wavelet systems with a given number of vanishing moments. Explicit methods for construction of compactly supported dual and tight frames with vanishing moments are suggested. Examples of tight frames with symmetric/antisymmetric wavelet functions found by means of this method are presented.