B-spline quarklets and biorthogonal multiwavelets

Abstract

In this paper, we show that B-spline quarks and the associated quarklets fit into the theory of biorthogonal multiwavelets. Quark vectors are used to define sequences of subspaces [Formula: see text] of [Formula: see text] which fulfill almost all conditions of a multiresolution analysis. Under some special conditions on the parameters, they even satisfy all those properties. Moreover, we prove that quarks and quarklets possess modulation matrices which fulfill the perfect reconstruction condition. Furthermore, we show the existence of generalized dual quarks and quarklets which are known to be at least compactly supported tempered distributions from [Formula: see text]. Finally, we also verify that quarks and quarklets can be used to define sequences of subspaces [Formula: see text] of [Formula: see text] that yield non-orthogonal decompositions of [Formula: see text].