Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting

Agler, J.: On the representation of certain holomorphic functions defined on a polydisk. In: Operator Theory: Advances and Applications, vol. 48, pp. 47–66. Birkhäuser Verlag, Basel (1990)

Alpay, D.: The Schur Algorithm, Reproducing Kernel Spaces and System Theory. American Mathematical Society, Providence (2001). Translated from the 1998 French original by Stephen S. Wilson, Panoramas et Synthèses. [Panoramas and Syntheses]

Alpay D.: A Complex Analysis Problem Book. Springer, Berlin (2011)

Alpay D., Dewilde P., Dym H.: Lossless inverse scattering and reproducing kernels for upper triangular operators. In: Gohberg, I. (eds) Extension and Interpolation of Linear Operators and Matrix Functions. Oper. Theory Adv. Appl., vol. 47, pp. 61–135. Birkhäuser, Basel (1990)

Alpay, D., Dijksma, A., Rovnyak, J., de Snoo, H.: Schur functions, operator colligations, and reproducing kernel Pontryagin spaces. In: Operator Theory: Advances and Applications, vol. 96. Birkhäuser Verlag, Basel (1997)

Alpay D., Dym H.: Hilbert spaces of analytic functions, inverse scattering and operator models, I. Integr. Equ. Oper. Theory 7, 589–641 (1984)

Alpay D., Dym H.: Hilbert spaces of analytic functions, inverse scattering and operator models, II. Integr. Equ. Oper. Theory 8, 145–180 (1985)

Alpay D., Dym H.: On applications of reproducing kernel spaces to the Schur algorithm and rational J-unitary factorization. In: Gohberg, I. (eds) I. Schur Methods in Operator Theory and Signal Processing. Operator Theory: Advances and Applications, vol. 18, pp. 89–159. Birkhäuser Verlag, Basel (1986)

Alpay, D., Kalyuzhnyĭ-Verbovetzkĭ, D.S.: Matrix-J-unitary non-commutative rational formal power series. In: The State Space Method Generalizations and Applications. Oper. Theory Adv. Appl., vol. 161, pp. 49–113. Birkhäuser, Basel (2006)

Alpay D., Levanony D.: Linear stochastic systems: a white noise approach. Acta Appl. Math. 110, 545–572 (2010)

Alpay D., Levanony D., Pinhas A.: Linear stochastic state space theory in the white noise space setting. SIAM J. Control Optim. 48, 5009–5027 (2010)

Alpay D., Mboup M.: Discrete-time multi-scale systems. Integr. Equ. Oper. Theory 68, 163–191 (2010)

Alpay D., Shapiro M.: Reproducing kernel quaternionic Pontryagin spaces. Integr. Equ. Oper. Theory 50, 431–476 (2004)

Alpay D., Shapiro M., Volok D.: Espaces de de Branges Rovnyak: le cas hyper-analytique. Comptes Rendus Mathématiques 338, 437–442 (2004)

Alpay D., Shapiro M., Volok D.: Rational hyperholomorphic functions in R 4. J. Funct. Anal. 221(1), 122–149 (2005)

Alpay, D., Shapiro, M., Volok, D.: Reproducing kernel spaces of series of Fueter polynomials. In: Operator Theory in Krein Spaces and Nonlinear Eigenvalue Problems. Oper. Theory Adv. Appl., vol. 162, pp. 19–45. Birkhäuser, Basel (2006)

Alpay D., Vinnikov V.: Finite dimensional de Branges spaces on Riemann surfaces. J. Funct. Anal. 189(2), 283–324 (2002)

Alpay D., Dijksma A., Volok D.: Schur multipliers and de Branges-Rovnyak spaces: the multiscale case. J. Oper. Theory 61(1), 87–118 (2009)

Bakonyi, M., Constantinescu, T.: Schur’s algorithm and several applications. Pitman Research Notes in Mathematics Series, vol. 261. Longman Scientific & Technical, Harlow (1992)

Ball, J., Bolotnikov, V.: Realization and interpolation for Schur-Agler-class functions on domains with matrix polynomial defining function in $${\mathbb {C}^n}$$ . Preprint

Ball J., Trent T.: Unitary colligations, reproducing kernel Hilbert spaces and Nevanlinna–Pick interpolation in several variables. J. Funct. Anal. 157, 1–61 (1998)

Ball, J., Trent, T., Vinnikov, V.: Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces. In: Proceedings of Conference in honor of the 60th birthday of M.A. Kaashoek. Operator Theory: Advances and Applications, vol. 122, pp. 89–138. Birkhäuser

Ball J.A., Kaliuzhnyi-Verbovetskyi D.S.: Conservative dilations of dissipative multidimensional systems: the commutative and non-commutative settings. Multidimens. Syst. Signal Process. 19(1), 79–122 (2008)

Bart H., Gohberg I., Kaashoek M.A.: Minimal factorization of matrix and operator functions. Operator Theory: Advances and Applications, vol. 1. Birkhäuser Verlag, Basel (1979)

Bochner, S., Chandrasekharan, K.: Fourier transforms. Princeton University Press, 1949. Reprinted with permission of the original publishers, Kraus Reprint Corporation, New York (1965)

Bolotnikov V.: On higher order boundary derivatives of an analytic self-map of the unit disk. J. Approx. Theory 163(4), 568–589 (2011)

Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis, vol. 76. Pitman Research Notes (1982)

de Branges L., Rovnyak J.: Canonical models in quantum scattering theory. In: Wilcox, C. (eds) Perturbation Theory and its Applications in Quantum Mechanics, pp. 295–392. Wiley, New York (1966)

de Branges L., Rovnyak J.: Square Summable Power Series. Holt, Rinehart and Winston, New York (1966)

Brodskĭ, M.S.: Triangular and Jordan representations of linear operators. American Mathematical Society, Providence (1971). Translated from the Russian by J.M. Danskin, Translations of Mathematical Monographs, vol. 32

Brodskĭ M.S., Livšic M.S: Spectral analysis of non-self-adjoint operators and intermediate systems. Uspehi Mat. Nauk (N.S.) 13(179), 3–85 (1958)

Bruckstein A.M., Kailath T.: Inverse scattering for discrete transmission-line models. SIAM Rev. 29(3), 359–389 (1987)

Colombo F., Gentili G., Sabadini I., Struppa D.: Extension results for slice regular functions of a quaternionic variable. Adv. Math. 222(5), 1793–1808 (2009)

Colombo F., Gentili G., Sabadini I., Struppa D.C.: Non commutative functional calculus: bounded operators. Complex Anal. Oper. Theory 4(4), 821–843 (2010)

Colombo, F., Gonzalez-Cervantes, J.O., Sabadini, I.: A non constant coefficients differential operator associated to slice monogenic functions. Trans. Am. Math. Soc. (to appear)

Colombo, F., Sabadini, I.: The $${\mathcal {F}}$$ -spectrum and the $${\mathcal {SC}}$$ -functional calculus. Proc. R. Soc. Edinb. Section A (to appear)

Colombo F., Sabadini I.: On some properties of the quaternionic functional calculus. J. Geom. Anal. 19(3), 601–627 (2009)

Colombo, F., Sabadini, I.: A structure formula for slice monogenic functions and some of its consequences. In: Hypercomplex Analysis. Trends Math., pp. 101–114. Birkhäuser Verlag, Basel (2009)

Colombo F., Sabadini I.: On the formulations of the quaternionic functional calculus. J. Geom. Phys. 60(10), 1490–1508 (2010)

Colombo F., Sabadini I.: The Cauchy formula with s-monogenic kernel and a functional calculus for noncommuting operators. J. Math. Anal. Appl. 373(2), 655–679 (2011)

Colombo F., Sabadini I.: The quaternionic evolution operator. Adv. Math. 227(5), 1772–1805 (2011)

Colombo F., Sabadini I., Sommen F., Struppa D.C.: Analysis of Dirac Systems and Computational Algebra. Progress in Mathematical Physics, vol. 39. Birkhäuser Boston Inc., Boston (2004)

Colombo F., Sabadini I., Struppa D.C.: A new functional calculus for noncommuting operators. J. Funct. Anal. 254(8), 2255–2274 (2008)

Colombo F., Sabadini I., Struppa D.C.: Slice monogenic functions. Isr. J. Math. 171, 385–403 (2009)

Colombo F., Sabadini I., Struppa D.C.: Duality theorems for slice hyperholomorphic functions. J. Reine Angew. Math. 645, 85–105 (2010)

Colombo F., Sabadini I., Struppa D.C.: An extension theorem for slice monogenic functions and some of its consequences. Isr. J. Math. 177, 369–389 (2010)

Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative functional calculus. Progress in Mathematics, vol. 289. Theory and Applications of Slice Hyperholomorphic Functions. Birkhäuser/Springer Basel AG, Basel (2011)

Constantinescu T.: Schur parameters, factorization and dilation problems. Operator Theory: Advances and Applications, vol. 82. Birkhäuser Verlag, Basel (1996)

Constantinescu, T., Sayed, A., Kailath, T.: Inverse scattering experiments, structured matrix inequalities, and tensor algebra. Linear Algebra Appl. 343/344, 147–169 (2002). Special issue on structured and infinite systems of linear equations

Donoghue W.F.: Monotone matrix functions and analytic continuation. Die Grundlehren der mathematischen Wissennschaften, vol. 207. Springer, Berlin (1974)

Dubovoj V., Fritzsche B., Kirstein B.: Matricial version of the classical Schur problem. Teubner–Texte zur Mathematik, vol. 129. B.G. Teubner Verlagsgesellschaft, Stuttgart (1992)

Dym, H.: J-contractive matrix functions, reproducing kernel Hilbert spaces and interpolation. Conference Board of the Mathematical Sciences, Washington, DC (1989)

Fillmore P.A., Williams J.P.: On operator ranges. Adv. Math. 7, 254–281 (1971)

Fliess M.: Matrices de Hankel. J. Math. Pures Appl. 53(9), 197–222 (1974)

Fritzsche, B., Kirstein, B. (eds): Ausgewählte Arbeiten zu den Ursprüngen der Schur-Analysis. Teubner–Archiv zur Mathematik, vol. 16. B.G. Teubner Verlagsgesellschaft, Stuttgart (1991)

Gentili G., Struppa D.C.: A new approach to Cullen-regular functions of a quaternionic variable. C.R. Acad. Sci. Paris 324, 741–744 (2006)

Gentili G., Struppa D.C.: A new theory of regular functions of a quaternionic variable. Adv. Math. 216(1), 279–301 (2007)

Ghiloni R., Perotti A.: Slice regular functions on real alternative algebras. Adv. Math. 226(2), 1662–1691 (2011)

Gohberg, I. (eds): I. Schur methods in operator theory and signal processing. Operator Theory: Advances and Applications, vol. 18. Birkhäuser Verlag, Basel (1986)

Hazewinkel M., Lewis J., Martin C.: Symmetric systems with semisimple structure algebra: the quaternionic case. Syst. Control Lett. 3(3), 151–154 (1983)

Holden H., Øksendal B., Ubøe J., Zhang T.: Stochastic Partial Differential Equations. Birkhäuser Boston Inc., Boston (1996)

Kailath T.: A theorem of I. Schur and its impact on modern signal processing. In: Gohberg, I. (eds) I. Schur methods in operator theory and signal processing. Oper. Theory Adv. Appl., vol. 18, pp. 9–30. Birkhäuser, Basel (1986)

Kalyuzhniy-Verbovetzky D.S.: On J-conservative scattering system realizations in several variables. Integr. Equ. Oper. Theory 43(4), 450–465 (2002)

Lax, P.D.: Translation invariant spaces. In: Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pp. 299–306. Jerusalem Academic Press, Jerusalem (1961)

Livšic M.S.: Isometric operators with equal deficiency indices, quasi-unitary operators. Mat. Sbornik N.S. 26(68), 247–264 (1950)

Livs̆ic, M.S.: On a class of linear operators in Hilbert spaces. Math. USSR-Sb. 61, 239–262 (1946). English translation in: American mathematical society translations (2), vol. 13, pp. 61–84 (1960)

Livs̆ic, M.S.: Operator Colligations, Waves, Open Systems. Transl. Math. Monog. AMS. AMS, Providence (1973)

Pereira R., Rocha P., Vettori P.: Algebraic tools for the study of quaternionic behavioral systems. Linear Algebra Appl. 400, 121–140 (2005)

Pereira R., Vettori P.: Stability of quaternionic linear systems. IEEE Trans. Automat. Control 51(3), 518–523 (2006)

Potapov, V.P.: The multiplicative structure of J-contractive matrix–functions. Trudy Moskow. Mat. Obs. 4, 125–236 (1955). English translation in: American Mathematical Society translations (2), vol. 15, pp. 131–243 (1960)

Sayed A.H., Kailath T.: Fast algorithms for generalized displacement structures and lossless systems. Linear Algebra Appl. 219, 49–78 (1995)

Sayed A.H., Kailath T.: A look-ahead block Schur algorithm for Toeplitz-like matrices. SIAM J. Math. Anal. 16, 388–414 (1995)

Schur, I.: Über die Potenzreihen, die im Innern des Einheitkreises beschränkten sind, I. Journal für die Reine und Angewandte Mathematik 147, 205–232 (1917). English translation in: I. Schur methods in operator theory and signal processing. (Operator theory: Advances and Applications OT 18 (1986), Birkhäuser Verlag), Basel

Schur, I.: Über die Potenzreihen, die im Innern des Einheitkreises beschränkt sind, II. Journal für die Reine und Angewandte Mathematik 148, 122–145 (1918). English translation in: I. Schur methods in operator theory and signal processing. (Operator theory: Advances and Applications OT 18 (1986), Birkhäuser Verlag), Basel

Weiss G.: Representation of shift-invariant operators on L 2 by H ∞ transfer functions: an elementary proof, a generalization, and a counterexample for L ∞*. Math. Control Signals Syst. 4, 193–203 (1991)