Integral Equations and Operator Theory Super-Wavelets Versus Poly-Bergman Spaces

We investigate a vector-valued version of the classical continuous wavelet transform. Special attention is given to the case when the analyzing vector consists of the first elements of the basis of admissible functions, namely the functions whose Fourier transform is a Laguerre function. In this case, the resulting spaces are, up to a multiplier isomorphism, poly-Bergman spaces. To demonstrate this fact, we introduce a new map and call it the polyanalytic Bergman transform. Our method of proof uses Vasilevski’s restriction principle for Bergman-type spaces. The construction is based on the idea of multiplexing of signals.

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