A Wavelet Plancherel Theory with Application to Multipliers and Sparse Approximations

We introduce an extension of continuous wavelet theory that enables efficient implementation multiplicative operators in the coefficient space. In new theory, signal space is embedded a larger abstract – so called window–signal There canonical transform to isometric isomorphism between and Hence, framework wavelet-Plancherel extended transform. Since isomorphism, any operation can be pulled-back It then possible improve computational complexity methods involve operator space, by performing all computations directly As one example application, we show how multipliers (also Calderón–Toeplitz operators), with polynomial symbols, implemented linear resolution 1D signal. another example, develop for efficiently computing greedy sparse approximations signals based on elements systems.