Recovering a finite-energy signal from values of some transforms (Q2761891)

The author establishes a method to reconstruct any finite energy signal from values of the integral wavelet transform. A function \(f\in L^2(R)\) is used to represent an analog signal with finite energy and one problem is to recover any finite energy signal from values of some transforms. The main result is that for any \(f\in L^2(R)\), NEWLINE\[NEWLINE f(x)=(C_{\psi })^{-1/2}\int_{-\infty }^{+\infty }\int_{-\infty }^{+\infty }[(W_{\psi }f)(b,a)]\psi_{b,a}(x)a^{-2} da db NEWLINE\]NEWLINE at any point \(x\in R\), where \(f\) is a continuous function and \(\psi _{b,a}(t)=|a|^{-1/2}\psi ((t-b)a^{-1})\). Here \(\psi \) is a basic wavelet such that \(\psi \), \(\widehat \psi \) are window functions and \((W_{\psi }f)(b,a)\) is the integral wavelet transform on \(L^2(R)\) (\(a,b\in R\), \(a\neq 0\)).