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 \(\Lambda \).


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
Full Text: DOI


[1] DOI: 10.1016/S0764-4442(00)88570-5 · Zbl 1019.47036 · doi:10.1016/S0764-4442(00)88570-5
[2] DOI: 10.1007/BF01243918 · Zbl 0725.20012 · doi:10.1007/BF01243918
[3] Gallardo L., Adv. Pure Appl. Math. 1 pp 163–
[4] Heckmann G. J., Compositio Math. 64 pp 329–
[5] Heckmann G. J., Harmonic Analysis and Special Functions on Symmetric Spaces (1994)
[6] DOI: 10.1007/BF02386203 · Zbl 0303.42022 · doi:10.1007/BF02386203
[7] DOI: 10.1142/S0219530503000090 · Zbl 1140.42302 · doi:10.1142/S0219530503000090
[8] DOI: 10.1007/BF02392487 · Zbl 0836.43017 · doi:10.1007/BF02392487
[9] Opdam E., MJS Memoirs 8, in: Dunkl Operators for Real and Complex Reflection Groups (2000)
[10] DOI: 10.1007/s00039-008-0658-7 · Zbl 1147.33004 · doi:10.1007/s00039-008-0658-7
[11] Trimèche K., J. Math. Pures Appl. 60 pp 51–
[12] Trimèche K., Adv. Pure Appl. Math. 1 pp 293–
[13] Trimèche K., Adv. Pure Appl. Math.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.