headshot photo of Dr. Daniel Alpay

Dr. Daniel Alpay

Professor, Foster G. and Mary McGaw Professorship in Mathematical Sciences
Schmid College of Science and Technology; Mathematics
Fowler School of Engineering; Electrical Engineering and Computer Science
Expertise: Schur Analysis; Slice-Hyperholomorphic functions; Signal Processing; Linear Systems; Wavelet Filters; White Noise Space
Office Location: Keck Center for Science and Engineering 381
Phone: 714-628-7266
Scholarly Works:
Digital Commons



  • 1980-1986 PhD, Dept. of Theoretical Mathematics, Weizmann Institute of Sciences, Rehovot, Israel (Mentor: Prof. Harry Dym)
  • 1978-1980 MSc, Dept. of Theoretical Mathematics, Weizmann Institute of Sciences, Rehovot, Israel (Mentor: Prof. Harry Dym)
  • 1975-1978 Electrical Engineering Degree, ‘Ecole Nationale Sup’erieure des T’el’ecommunications, Paris, France


  • Dr. Alpay’s research interests are along five main directions, namely Schur analysis in the setting of slice-hyperholomorphic functions, infinite dimensional analysis and white noise space theory, linear systems and applications to wavelet filters and signal processing, functions of a bicomplex variable, and function theory on a compact Riemann surface. A number of the underlying projects are at the interface between mathematics, signal processing, and models for linear systems appearing in electrical engineering. The notions of positive definite function and of the associated reproducing kernel Hilbert space play an important role in these various directions of research.

KEYWORDS (meta tags): Schur analysis, slice-hyperholomorphic function, signal processing

Recent Creative, Scholarly Work and Publications

D. Alpay, M.E. Luna-Elizarraràs, M. Shapiro and D.C. Struppa. Gleason problem, rational functions and spaces of regular functions: The split-quaternion case. Israel Journal of Mathematics (2018), no. 1, 319-349.
D. Alpay, A. Vajiac and M. Vajiac. Gleason's problem associated to a real ternary algebra and applications. Advances in Applied Clifford Algebras, vol. 28 (2018), no. 2, Art. 43, 16 pp.
D. Alpay and M. Porat. Generalized Fock spaces and the Stirling numbers. Journal of Mathematical Physics, vol. 59 (2018) no 6, 063509, 12 pp.
D. Alpay and I.Sabadini. Beurling-Lax type theorems in the complex and quaternionic setting. Linear Algebra and its Applications, vol. 530 (2017) pp 15-46.
D. Alpay, M.E. Luna-Elizarraràs and M. Shapiro. Kolmogorov's axioms for probabilities with values in hyperbolic numbers. Advances in Applied Clifford Algebras, vol. 27 (2017) pp. 913-929
D. Alpay, F. Colombo, and I. Sabadini. On a class of quaternionic positive definite functions and their derivatives. Journal of Mathematical Physics, vol. 58 (2017), no. 3, 033501, 15 pp.
D. Alpay, P. Cerejeiras and U. Kaehler. Gleason's problem associated to the fractional Cauchy-Riemann operator, Fueter series, Drury-Arveson space and related topics. Proceedings of the AMS, vol 145 (2017), pp. 4821-4835.
D. Alpay, F. Colombo, I. Sabadini and T. Qian. Adaptative decomposition: The case of the Drury-Arveson space. Journal of Fourier Analysis, vol. 23 (2017) pp. 1426-1444
I am the editor in chief of the journal "Complex analysis and operator theory".
D. Alpay, F. Colombo and I. Sabadini. Slice hyperholomorphic Schur analysis. Operator Theory: Advances and Applications, vol. 256, 2016.