The proof of existence of this class of functions is due to Ingrid Daubechies.
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It was introduced in this context in 1988/89 by Stephane Mallat and Yves Meyer and has predecessors in the microlocal analysis in the theory of differential equations (the ironing method) and the pyramid methods of image processing as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James Crowley.
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Specifically, he collaborated with Yves Meyer to develop the Multiresolution Analysis (MRA) construction for compactly supported wavelets, which made the implementation of wavelets practical for engineering applications by demonstrating the equivalence of wavelet bases and conjugate mirror filters used in discrete, multirate filter banks in signal processing.