B-Spline Quarklets and Biorthogonal Multiwavelets

12/05/2022

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by   Marc Hovemann, et al.

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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 V_p,j of L_2(ℝ) 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 𝒮'(ℝ). Finally we also verify that quarks and quarklets can be used to define sequences of subspaces W_p,j of L_2(ℝ) that yield non-orthogonal decompositions of L_2(ℝ).