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Inversion of the Lions transmutation operators using generalized wavelets. (English) Zbl 0872.34059

The author considers (Lions) differential operator \[ \Delta\equiv \frac{d^2}{dx^2}+ \frac{A'(x)}{A(x)} \frac{d}{dx}+ \rho^2 \] on \(]0,+\infty[\), with suitable restrictions on the growth of \(A(x)\). There exists a unique isomorphism \({\mathcal H}\), from the space of even \(C^\infty\) functions on \(\mathbb{R}\) onto itself, such that \[ \Delta{\mathcal H}(f)={\mathcal H} \Biggl(\frac{d^2}{dx^2}f\Biggr), \qquad {\mathcal H}(f(0))=f(0). \] The Lions’ transmutation operator \({\mathcal H}\) is a transmutation operator of \(\Delta\) into \(\frac{d^2}{dx^2}\). Using these operators, the paper gives relations between the generalized continuous wavelet transform and the classical continuous wavelet transform on \([0,+\infty]\). This is done after giving results on the eigenfunctions of the operator \(\Delta\) and a harmonic analysis of it (generalized Fourier transform, generalized translation operators and generalized convolution product). Finally, the author obtains formulas which give the inverse operators of the Lions transmutation operator, Riemann-Liouville operator and the Weyl operator.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42A99 Harmonic analysis in one variable
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