OPERATOR-ADAPTED WAVELETS: CONNECTION WITH THE STRANG–FIX CONDITIONS

摘要

In this paper, we present an explicit method to construct directly in the x-domain compactlynsupported scaling functions corresponding to the wavelets adapted to a sum ofndifferential operators with constant coefficients. Here the adaptation to an operator isntaken to mean that the wavelets give a diagonal form of the operator matrix. We shownthat the biorthogonal compactly supported wavelets adapted to a sum of differentialnoperators with constant coefficients are closely connected with the representation ofnthe null-space of the adjoint operator by the corresponding scaling functions. We considernthe necessary and sufficient conditions (actually the Strang–Fix conditions) onninteger shifts of a compactly supported function (distribution) f ∈ Su0001(R) to representnexactly any function from the null-space of a sum of differential operators with constantncoefficients