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Transmutation operators and Paley-Wiener theorem associated with a Cherednik type operator on the real line. (English) Zbl 1200.42003

Summary: We consider a singular differential-difference operator \(\Lambda \) on the real line which generalizes the one-dimensional Cherednik operator. We construct transmutation operators between \(\Lambda \) and first-order regular differential-difference operators on \(\mathbb R\). We exploit these transmutation operators, firstly to establish a Paley-Wiener theorem for the Fourier transform associated with \(\Lambda \), and secondly to introduce a generalized convolution on \(\mathbb R\) tied to \(\Lambda \).

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
39A70 Difference operators
34L99 Ordinary differential operators
46F12 Integral transforms in distribution spaces
44A05 General integral transforms
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
44A35 Convolution as an integral transform
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