| Abstract: | Decomposition of continuous functions can be accomplished by considering the difference of consecutive interpolation operators. When such a difference is expressed as an infinite series of some "wavelets" basis, the coefficient sequence becomes Donoho's "interpolating wavelet transform." Here, in contrast to the usual $L^2 $-setting, no analyzing wavelet is used to describe the wavelet transform. The objective of this paper is to study the structure of such decomposition spaces, including the formulation of bases and their duals, which lead to the notion of the functional wavelet transform (FWT) using the duals as analyzing wavelets. This transform retains some of the most important properties of the integral wavelet transform of Grossmann and Morlet, such as the property of vanishing moments which has significant applications to engineering problems. |
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