Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


Gleason's problem associated to the fractional Cauchy-Riemann operator, Fueter series, Drury-Arveson space and related topics

Authors: D. Alpay, P. Cerejeiras and U. Kähler
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 30G35; Secondary 26A33, 30A05, 31B05
Published electronically: May 30, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we present the building blocks for a function theory based on fractional Cauchy-Riemann operators. We are going to construct basic monogenic powers and Fueter series. With these tools we are going to study Gleason's problem and reproducing kernel spaces, like the Drury-Arveson space and de Branges-Rovnyak spaces. We end with a statement on Schur multipliers in this setting.

References [Enhancements On Off] (What's this?)

  • [1] Jim Agler, On the representation of certain holomorphic functions defined on a polydisc, Topics in operator theory: Ernst D. Hellinger memorial volume, Oper. Theory Adv. Appl., vol. 48, Birkhäuser, Basel, 1990, pp. 47–66. MR 1207393
  • [2] Daniel Alpay, Aad Dijksma, James Rovnyak, and Hendrik de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Operator Theory: Advances and Applications, vol. 96, Birkhäuser Verlag, Basel, 1997. MR 1465432
  • [3] Daniel Alpay, Michael Shapiro, and Dan Volok, Rational hyperholomorphic functions in ℝ⁴, J. Funct. Anal. 221 (2005), no. 1, 122–149. MR 2124899,
  • [4] Daniel Alpay, Michael Shapiro, and Dan Volok, Reproducing kernel spaces of series of Fueter polynomials, Operator theory in Krein spaces and nonlinear eigenvalue problems, Oper. Theory Adv. Appl., vol. 162, Birkhäuser, Basel, 2006, pp. 19–45. MR 2240272,
  • [5] Joseph A. Ball and Vladimir Bolotnikov, Realization and interpolation for Schur-Agler-class functions on domains with matrix polynomial defining function in ℂⁿ, J. Funct. Anal. 213 (2004), no. 1, 45–87. MR 2069781,
  • [6] M. Caputo, The Green function of the diffusion of fluids in porous media with memory, Rend. Fis. Acc. Lincei (Ser. 9), 7, (1996), 243-250.
  • [7] P. Cerejeiras, U. Kähler, Monogenic signal theory, in D. Alpay, Operator Theory, Springer References, Springer, 2015, 1701-1724.
  • [8] Louis de Branges and James Rovnyak, Square summable power series, Holt, Rinehart and Winston, New York-Toronto, Ont.-London, 1966. MR 0215065
  • [9] R. Delanghe, F. Sommen, and V. Souček, Clifford algebra and spinor-valued functions, Mathematics and its Applications, vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1992. A function theory for the Dirac operator; Related REDUCE software by F. Brackx and D. Constales; With 1 IBM-PC floppy disk (3.5 inch). MR 1169463
  • [10] A. O. Gel′fond and A. F. Leont′ev, On a generalization of Fourier series, Mat. Sbornik N.S. 29(71) (1951), 477–500 (Russian). MR 0045812
  • [11] J. Gutiérrez-Vega, Fractionalization of optical beams: I. Planar analysis, Optics Letters 32 (2007), 1521-1523.
  • [12] S. Held, Monogenic wavelet frames for image analysis, PhD thesis, TU Munich and Helmholz-Center Munich, 2012.
  • [13] Richard Herrmann, Fractional calculus, 2nd ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. An introduction for physicists. MR 3243574
  • [14] Richard Hotzel and Michel Fliess, On linear systems with a fractional derivation: introductory theory and examples, Math. Comput. Simulation 45 (1998), no. 3-4, 385–395. Delay systems (Lille, 1996). MR 1622950,
  • [15] Tadeusz Kaczorek, Selected problems of fractional systems theory, Lecture Notes in Control and Information Sciences, vol. 411, Springer-Verlag, Berlin, 2011. MR 2798773
  • [16] U. Kähler and N. Vieira, Fractional Clifford analysis, Hypercomplex Analysis: New perspectives and applications, Trends in Mathematics, S. Bernstein, U. Kähler, I. Sabadini and F. Sommen (eds.), Birkhäuser, Basel, (2014), 191-201.
  • [17] Virginia Kiryakova, Generalized fractional calculus and applications, Pitman Research Notes in Mathematics Series, vol. 301, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994. MR 1265940
  • [18] V. Lakshminarayanan, M. L. Calvo, T. Alieva, Mathematical Optics - Classical, Quantum and Computational Methods, CRC Press, 2013.
  • [19] H. Malonek, Zum Holomorphiebegriff in höheren Dimensionen, Habilitation thesis, PH Halle, 1987.
  • [20] Emilio Marmolejo-Olea and Salvador Pérez-Esteva, An analytic Radon-Nikodym property related to systems of vector-valued conjugate harmonic functions and Clifford analysis, Bol. Soc. Mat. Mexicana (3) 13 (2007), no. 1, 131–144. MR 2468030
  • [21] Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
  • [22] F. Sommen, A product and an exponential function in hypercomplex function theory, Applicable Anal. 12 (1981), no. 1, 13–26. MR 618518,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30G35, 26A33, 30A05, 31B05

Retrieve articles in all journals with MSC (2010): 30G35, 26A33, 30A05, 31B05

Additional Information

D. Alpay
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, POB653, Beer-Sheva 84105, Israel
Address at time of publication: Foster G. and Mary McGaw Professorship in Mathematical Sciences, Department of Mathematics, von Neumann Hall, Chapman University, Orange, California 92866

P. Cerejeiras
Affiliation: CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro Campus Universitário de Santiago 3810-193 Aveiro, Portugal

U. Kähler
Affiliation: CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro Campus Universitário de Santiago 3810-193 Aveiro, Portugal

Received by editor(s): July 27, 2016
Received by editor(s) in revised form: December 9, 2016
Published electronically: May 30, 2017
Additional Notes: The first author thanks the Earl Katz family for endowing the chair which supported his research. The work of the second and third authors was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”), within project UID/MAT/ 0416/2013.
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2017 American Mathematical Society