Note on 1D quarklet approximation

Abstract

On the example of the simplest <em>C⁰</em> spline quarklet construction of [1], we demonstrate the possible reduction of the complexity estimates for the approximation of singularity functions on the unit interval given in [3], to come closer to the complexity estimates known for hp-methods. To come up with a simplified argument, we explore the fact that the CDF biorthogonal spline wavelets [2] used in the construction of quarklets can be obtained by the lifting scheme [4]. Similar results are also possible for spline quarklet systems built from smooth B-splines with higher order of vanishing moments.