Smooth multiwavelet duals of Alpert bases by moment-interpolating refinement
The authors construct smooth multiwavelet bases of \(L^{2}[0,1]\) and \(L^{2}(R) \) which are duals of a generalized class of Alpert multiwavelets [cf. \textit{B. K. Alpert}, SIAM J. Math. Anal. 24, No. 1, 246-262 (1993; Zbl 0764.42017); \textit{B. Alpert, G. Beylkin, R. Coifman} and \textit{V. Rokhlin}, SIAM J. Sci. Comput. 14, No. 1, 159-184 (1993; Zbl 0771.65088)]. One begins by observing that the Alpert bases can be thought of as the first few Legendre polynomials cut off at the endpoints of \([0,1]\). By shifting to other unit intervals one then gets a basis for \(L^{2}(R)\) that gives rise to higher order multiwavelets such that inner products of \(f\) with these wavelets represent local moments of \(f\). The moment interpolation problem is that of starting with a sequence of moments at given scales, and determining local polynomials having a number of wavelet coefficients equal to the corresponding moments. The degree of the polynomial and the number of moments/wavelet coefficients that must coincide are chosen so that the solution is uniquely determined. The moment imputation problem is that of computing the moments/wavelet coefficients of the local polynomials at the next scale and interpolating these moments to define new finer scale polynomials. The process implicitly defines an operator, independent of scale, that maps moments from one scale to the next. This moment interpolation scheme is said to be \(C^{r}\) if the interated refinement applied to a bounded moment sequence at the starting scale produces a \(C^{r}\) limit function. Here the authors show that when the wavelet coefficients are with respect to the Alpert wavelets, this refinement process produces continuous limit functions. More precise regularity estimates are addressed in a related paper authored by \textit{T. P.-Y. Yu} [Parametric families of Hermite subdivision schemes in dimension 1, Available at \url{http://www.rpi.edu/~yut/Papers/hermite1.pdf} (1999)]. In a different direction, Hermite interpolation seeks a smooth function \(f\) such that \(f\) and its derivatives have given values at given points. Polynomial Hermite interpolation specifies a relationship between the degree of the polynomial and the number of neighboring evaluation points to determine the solution. The polynomial imputation problem then is to determine a polynomial solution based on data at a given dyadic scale, compute its values at the next finer scale, then solve the interpolation problem for the finer scale local polynomials. Again, this implicitly defies an interpolation matrix. The main purpose here is to show: (i) moment interpolation and Hermite interpolation are dual in a very natural sense from which one can conclude that (ii) refinement of the moment interpolation of a suitable delta sequence, regarded as Alpert wavelet coefficients, leads to families of wavelets biorthogonal to the Alpert wavelets. The authors conclude in relating these problems to that of optimal recursive dyadic partitioning, that is, of finding a partition of \([0,1]\) into a given number of dyadic pieces, and a function that is piecewise polynomial of a given degree on those pieces, that is as close as possible to a given \(f\in L^{2}[0,1]\) subject to the constraints. A precise algorithm is presented based on these ideas.
scientific article; zbMATH DE number 1512546
scientific article