Abstract
In this paper, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters. A series expansion for the kernel of the resulting integral transform is derived. In the case of even dimension, also an explicit expression for the kernel in terms of Bessel functions is obtained. Finally, the analytic properties of this new integral transform are studied in detail.
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References
Bargmann V.: On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math. 14, 187–214 (1961)
Ben Saïd, S., Kobayashi, T., Ørsted, B.: Laguerre semigroup and Dunkl operators. Compositio Math. arXiv:0907.3749 (to appear)
Brackx F., De Schepper N., Sommen F.: The Clifford-Fourier transform. J. Fourier Anal. Appl. 11, 669–681 (2005)
Brackx F., De Schepper N., Sommen F.: The two-dimensional Clifford-Fourier transform. J. Math. Imaging Vis. 26, 5–18 (2006)
Brackx F., De Schepper N., Sommen F.: The Fourier transform in Clifford analysis. Adv. Imaging Electron Phys. 156, 55–203 (2008)
Colombo, F., De Bie, H.: The S-spectrum and the Clifford-Fourier transform (in preparation)
Condon E.U.: Immersion of the Fourier transform in a continuous group of functional transformations. Proc. Nat. Acad. Sci. USA 23, 158–164 (1937)
De Bie, H., De Schepper, N.: Fractional Fourier transforms of hypercomplex signals. Signal Image Video Process 2012 (accepted for publication)
De Bie, H., De Schepper, N., Sommen, F.: The class of Clifford-Fourier transforms. J. Fourier Anal. Appl. 17, 1198–1231 (2011)
De Bie, H., Xu, Y.: On the Clifford-Fourier transform. Int. Math. Res. Not. IMRN 2011, 5123–5163 (2011)
Delanghe, R., Sommen, F., Souček, V.: Clifford algebra and spinor-valued functions. Mathematics and its applications, vol. 53. Kluwer Academic Publishers Group, Dordrecht
Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G.: Higher transcendental functions, vol. II. McGraw-Hill, New York (1953)
Folland G.B.: Harmonic analysis in phase space. Annals of Mathematics Studies, vol. 122. Princeton University Press, Princeton (1989)
Gradshteyn I.S., Ryzhik I.M.: Table of integrals, series, and products. Academic Press, New York (1980)
Howe, R.: The oscillator semigroup. In: The mathematical heritage of Hermann Weyl (Durham, NC, 1987). Proc. Sympos. Pure Math., vol. 48, pp. 61–132. Am. Math. Soc., Providence (1988)
Mehler F.G.: . J. Reine Angew. Math. 66, 161–176 (1866)
Mustard D.: Fractional convolution. J. Aust. Math. Soc. Ser. B 40, 257–265 (1998)
Namias V.: The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Math. Appl. 25(3), 241–265 (1980)
Ozaktas H., Zalevsky Z., Kutay M.: The fractional Fourier transform. Wiley, Chichester (2001)
Sommen F.: Special functions in Clifford analysis and axial symmetry. J. Math. Anal. Appl. 130(1), 110–133 (1988)
Szegő G.: Orthogonal Polynomials, 4th edn. Am. Math. Soc. Colloq. Publ, Providence (1975)
Thangavelu, S.: Hermite and Laguerre semigroups: some recent developments. CIMPA-Venezuela lecture notes (to appear)
Walters S.: Periodic integral transforms and C *-algebras. C. R. Math. Acad. Sci. Soc. R. Can. 26, 55–61 (2004)
Watson G.N.: A treatise on the theory of Bessel Functions. Cambridge University Press/The Macmillan Company, Cambridge/New York (1944)
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Communicated by Fred Brackx.
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De Bie, H., De Schepper, N. The Fractional Clifford-Fourier Transform. Complex Anal. Oper. Theory 6, 1047–1067 (2012). https://doi.org/10.1007/s11785-012-0229-7
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DOI: https://doi.org/10.1007/s11785-012-0229-7