Quasi wavelets and quasi interpolating wavelets

This Letter introduces two families of new wavelets, quasi wavelets and quasi interpolating wavelets. It is found that the mathematical regularization of Shannonʹs orthonormal interpolating wavelet leads to an interplay between a quasi wavelet and a quasi interpolating wavelet. The resulting quasi wavelet scaling function preserves the interpolation property but it does not satisfy the wavelet normalization requirement. Whereas the quasi interpolating wavelet scaling function satisfies the normalization requirement but it is no longer rigorously interpolative. Both quasi wavelets and quasi interpolating wavelets are functions of the Schwartz class. Their numerical performance is extremely similar to each other for data interpolation, differentiation and for solving partial differential equations.