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
Website: http://www1.chapman.edu/~alpay
Scholarly Works:
Digital Commons

Biography

EDUCATION/TRAINING

  • 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

RESEARCH INTERESTS

  • 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

Alpay, Daniel (ed.); Behrndt, Jussi (ed.); Colombo, Fabrizio (ed.); Sabadini, Irene (ed.); Struppa, Daniele C. (ed.) Recent developments in operator theory, mathematical physics and complex analysis. Proceedings of the 32nd international workshop on operator theory and its applications, IWOTA 2021, Chapman University, Orange, CA, USA, August 9–13, 2022. (English) Operator Theory: Advances and Applications 290. Cham: Springer viii, 420 p. (2023).
D. Alpay, K. Diki and M. Vajiac: A note on the complex and bicom- plex neural networks. Applied Mathematics and Computation, vol. 445 (2023) Paper No. 127864, 12pp.
D. Alpay, A. Pinhas and V. Vinnikov. Commuting operators over Pontryagin spaces with applications to system theory. Journal of functional analysis, vol. 284 No 10, Article ID 109864, 66 p. (2023).
D. Alpay, P. Cerejeiras, U. Kaehler and T. Kling. Commutators on Fock spaces. J. Math. Phys. 64, 042102 (2023)
D. Alpay, F. Colombo, K. Diki, I. Sabadini, and D. Struppa Hörmander L2- method, partial-problem and polyanalytic function theory in one complex variable. Com- plex analysis and operator theory, vol. 17 (2023) No. 3, Paper No. 41.
D. Alpay, I. Lewkowicz and M. Vajiac. Interpolation with symmetry and a Herglotz Theorem in the bicomplex setting. Journal of Mathematical Analysis and Applications, vol. 524 (2023), no. 2, Paper No. 127201
D. Alpay and I. Cho. Dynamical systems of operators induced by scaled hypercomplex rings. Advances in Applied Clifford Algebras, 33, no 3, paper no 33, 39 pages
D. Alpay and I. Cho. Operators induced by certain hypercomplex systems. Opuscula Mathematica, 43, No. 3, 275-333 (2023).
D. Alpay, F. Colombo, K. Diki, I. Sabadini, and D. Struppa. A Hormander- Fock space. Complex Variables and Elliptic Functions, 17, No. 3, Paper No. 41, 25 p. (2023).
D. Alpay, I. Lewkowicz and M. Vajiac. Discrete Wiener algebra in the bicom- plex setting, spectral factorization with symmetry, and superoscillations. Analysis and Mathematical Physics,13, No. 3, Paper No. 54, 32 p. (2023).
D. Alpay, T. Bhattacharyya, A. Jindal and P. Kumar. A dilation theoretic ap- proach to approximation by inner functions. Bulletin of the London Mathematical Society, 55, No. 6, 2840-2855 (2023).
D. Alpay and D. Volok. Schur analysis and discrete analytic functions: Ra- tional functions and co-isometric realizations. Linear Algebra and Applications, 675, 179-202 (2023).
D. Alpay and I. Cho. Certain invertible operator-block matrices induced by C * -algebras and scaled hypercomplex numbers. Research in the Mathematical Sci- ences, 10, No. 4, Paper No. 47, 44 p. (2023)
D. Alpay, F. Colombo, K. Diki, I. Sabadini, and D. Struppa. Superoscillations and Fock space. Journal of Mathematical Physics, . 64, No. 9, Article ID 093505, 20 p. (2023)
D. Alpay, P. Cerejeiras and U. Kaehler. Generalized Grassman algebras and applications to stochastic processes. Mathematical Methods in the Applied Sciences, volume 45,issue 1. 19 pages
D. Alpay and D. Volok. Discrete analytic Schur functions. Proceedings of the AMS. 150, No. 5, 2145-2152 (2022) Articles in this Issue .
Alpay, Daniel; Jorgensen, Palle mu-Brownian motion, dualities, diffusions, transforms, and reproducing kernel Hilbert spaces. (English) Zbl 07621028 J. Theor. Probab. 35, No. 4, 2757-2783 (2022).
Alpay, Daniel; Cerejeiras, Paula; Kähler, Uwe Generalized white noise analysis and topological algebras. (English) Zbl 1498.60294 Stochastics 94, No. 6, 926-958 (2022).
Alpay, Daniel, Jorgensen, Palle Reflection positivity via Krein space analysis. Adv. Appl. Math. 141, Article ID 102411, 45 p. (2022).
Alpay, Daniel, Colombo, Fabrizio; Diki, Kamal; Sabadini, Irene; Volok, Dan Discrete analytic functions, structured matrices and a new family of moment problems. Bull. Sci. Math. 179, Article ID 103175, 70 p. (2022).
Alpay, Daniel, Colombo, Fabrizio; Sabadini, Irene; Schneider, Baruch Beurling-Lax type theorems and Cuntz relations. (English) Zbl 07432147 Linear Algebra Appl. 633, 152-212 (2022).
D. Alpay, P. Jorgensen and I. Lewkowicz. Representation theory and multilevel filters. Journal of Applied Mathematics and Computing, (2022), 59 pages
D. Alpay, F. Colombo, K. Diki and I. Sabadini. An approach to the Gaussian RBF kernels via Fock spaces J. Math. Phys. 63, 113506 (2022);
D. Alpay, R. Peretz, D. Shoikhet, and M.B. Vajiac, New Directions in Function Theory: From Complex to Hypercomplex to Non-Commutative Chapman University, November 2019 (2021). Operator Theory: Advances and Applications, vol. 286 (in press)
D. Alpay, F. Colombo, K. Diki and I. Sabadini. On a polyanalytic approach to noncommutative de Branges-Rovnyak spaces and Schur analysis. Integral Equations and Operator Theory, vol. 93, no. 4, paper no. 38, 63 p. (2021).
D. Alpay, K. Diki, and I. Sabadini. On the global operator and Fueter mapping theorem for slice polyanalytic functions. Analysis and applications, vol. 19, pp. 941-964 (2021)
D. Alpay, F. Colombo, S. Pinton and I. Sabadini. Holomorphic functions, relativistic sum, Blaschke products and superoscillations. Analysis and Mathematical Physics, vol. 11, paper 139, 28p. (2021)
D. Alpay, K. Diki, and I. Sabadini. Fock and Hardy spaces: the Clifford Appell case. Mathematische Nachrichten. Volume 295, Issue 5 27 pages
D. Alpay and I. Lewkowicz. Quantitatively Hyper-Positive Real Functions. Linear Algebra and Its Applications, vol. 623 (2021) 316-334.
D. Alpay, F. Colombo and I. Sabadini. Superoscillations and analytic exten- sion. Journal of Fourier Analysis and Applications, vol. 27 (2021), Paper No. 28, 19pp.
D. Alpay and H.T. Kaptanoglu. Shift operators on harmonic Hilbert function spaces on real balls and von Neumann inequality. J. Funct. Anal. 281 (2021), no. 4, 109058.
D. Alpay, F. Colombo, S. Pinton and I. Sabadini. Holomorphic functions, rel- ativistic sum, Blaschke products and superoscillations. Analysis and Mathematical Physics, 11 (2021), 28pp.
D. Alpay and A. Yger. Cauchy-Weil formula, Schur-Agler classes, new Hardy spaces of the polydisk and interpolation problems. Journal of mathematical analysis and applications, 504 (2021), 29pp.
D. Alpay, F. Colombo, S. Pinton, I. Sabadini and D. Struppa. Infinite order differential operators acting on entire hyperholomorphic functions. Journal of Geo- metric Analysis, 31 (2021) 253-288
D. Alpay, I. Paiva and D.C. Struppa. A general setting for functions of Fueter variables: differentiability, rational functions, Fock module and related topics. Israel Journal of Mathematics, vol. 236 (2020), no. 1, pp. 207-246.
D. Alpay and I. Paiva. On the extension of positive definite kernels to topological algebras. Journal of Mathematical Physics, vol. 61 (2020) 063507, 10pp.
D. Alpay, V. Bolotnikov and D. Shoikhet. Pseudo-contractions, rigidity, fixed points and related questions. Analysis and Mathematical Physics, vol. 10 (2020), no 1, Art. 7, 26 pp.
D. Alpay, F. Colombo, I. Sabadini and D. Struppa. Aharonov- Berry superoscillations in the radial harmonic oscillator potential. Quantum Studies: Mathematics and Foundations, vol. 7 (2020), no 3, 269-283.
D. Alpay, A. Pinhas and V. Vinnikov. de Branges spaces on compact Riemann surfaces and a Beurling-Lax type theorem. Advances in Mathematics, vol. 373 (2020), 107315, 67 pp.
Daniel Alpay and Mihaela Vajiac. Linear systems, signal processing and hypercomplex analysis. Chapman University, 2017. Operator Theory: Advances and Applications, vol. 275 (2019)
D. Alpay and A. Yger. About a Non-Standard Interpolation Problem. Computional Methods and Function Theory, vol. 18 (2019), no.1, 97-115
D. Alpay, A. Dijksma and J. Rovnyak. Corrigendum to ``Notes on interpolation in the generalized Schur class. II. Nudel'man's problem''. Transactions of the AMS, vol. 371 (2019), 3743-3745
D. Alpay, F. Colombo, and I. Sabadini. Herglotz functions: The Fueter variables case. Linear Algebra and Its Applications, vol. 563 (2019), 75-97.
D. Alpay, K. Diki, and I. Sabadini. On slice polyanalytic functions of a quaternionic variable. Results in Mathematics, vol. 74 (2019), no. 1, Art. 17, 25 pp
D. Alpay, F. Colombo, I. Lewkowicz and I. Sabadini. Positive and generalized positive real lemma for slice hyperholomorphic functions. Integral Equations and Operator Theory, vol. 91 (2019), no. 1, Art. 4, 22 pp.
D. Alpay, I. Paiva and D.C. Struppa. Distribution spaces and a new construction of stochastic processes associated with the Grassmann algebra. Journal of Mathematical Physics. 60 (2019), no. 1, 013508, 21 pp
D. Alpay, I. Paiva and D.C. Struppa. Positivity, rational Schur functions, Blaschke factors, and other related results in the Grassmann algebra. Integral Equations and Operator Theory, vol. 91 (2019), no. 2, 91:8.
D. Alpay and I. Lewkowicz. Composition of Rational Function: State-Space Realization and Applications. Linear Algebra and Its Applications vol. 580 (2019) 359-383
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, 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
D. Alpay, F. Colombo and I. Sabadini. Slice hyperholomorphic Schur analysis. Operator Theory: Advances and Applications, vol. 256, 2016.

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