In this paper, the class of (complex) quasi-Herglotz functions is introduced as the complex vector space generated by the convex cone of ordinary Herglotz functions. We prove characterization theorems, in particular, an analytic characterization. The subclasses of quasi-Herglotz functions that are identically zero in one half-plane as well as rational quasi-Herglotz functions are investigated in detail. Moreover, we relate to other areas such as weighted Hardy spaces, definitizable functions, the Cauchy transform on the unit circle, sum-rule identities and matrix-valued Herglotz functions.
Dans le présent article, la classe des fonctions (complexes) quasi-Herglotz est présentée comme l’espace vectoriel complexe engendré par le cône convexe des fonctions de Herglotz usuelles. Nous démontrons des théorèmes de caractérisation, en particulier une caractérisation analytique. Les sous-classes des fonctions quasi-Herglotz qui sont identiquement nulles dans un demi-plan ainsi que les fonctions quasi-Herglotz rationnelles sont étudiées en détail. De plus, nous faisons le lien avec d’autres domaines tels que les espaces de Hardy avec poids, les fonctions définissables, la transformée de Cauchy sur le cercle unité, les identités « somme-règle » et les fonctions de Herglotz à valeur matricielle.
Revised:
Accepted:
Published online:
Annemarie Luger 1; Mitja Nedic 1
@article{CRMATH_2022__360_G9_937_0, author = {Annemarie Luger and Mitja Nedic}, title = {On {quasi-Herglotz} functions in one variable}, journal = {Comptes Rendus. Math\'ematique}, pages = {937--970}, publisher = {Acad\'emie des sciences, Paris}, volume = {360}, year = {2022}, doi = {10.5802/crmath.364}, language = {en}, }
Annemarie Luger; Mitja Nedic. On quasi-Herglotz functions in one variable. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 937-970. doi : 10.5802/crmath.364. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.364/
[1] The classical moment problem and some related questions in analysis, University Mathematical Monographs, Hafner Publishing Co., 1965 | Zbl
[2] Theory of linear operators in Hilbert space, Dover Publications, 1993, xiv+147, iv+218 pages
[3] Scattering theory for Schrödinger operators with Bessel-type potentials, J. Reine Angew. Math., Volume 666 (2012), pp. 83-113 | DOI
[4] On exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. Anal. Math., Volume 5 (1956), pp. 321-388 | DOI
[5] Canonical de Branges–Rovnyak model transfer-function realization for multivariable Schur-class functions, Hilbert spaces of analytic functions (CRM Proceedings & Lecture Notes), Volume 51, American Mathematical Society, 2010, pp. 1-39 | DOI | MR | Zbl
[6] Spectral shift functions and Dirichlet-to-Neumann maps, Math. Ann., Volume 371 (2018) no. 3-4, pp. 1255-1300 | DOI | MR | Zbl
[7] Sum rules and constraints on passive systems, J. Phys. A, Math. Theor., Volume 44 (2011) no. 14, 145205, 20 pages | DOI | MR | Zbl
[8] Measure theory. Vol. I, II, Springer, 2007 | DOI
[9] The Poisson integral for functions with positive real part, Bull. Am. Math. Soc., Volume 38 (1932) no. 10, pp. 713-717 | DOI | MR | Zbl
[10] The Cauchy transform, Mathematical Surveys and Monographs, 125, American Mathematical Society, 2006, x+272 pages | DOI
[11] Perturbations of orthogonal polynomials with periodic recursion coefficients, Ann. Math., Volume 171 (2010) no. 3, pp. 1931-2010 | DOI | MR | Zbl
[12] Treatise on analysis. Vol. II, Pure and Applied Mathematics, 10-II, Academic Press Inc., 1976 (enlarged and corrected printing, translated by I. G. Macdonald)
[13] Representations of holomorphic operator functions by means of resolvents of unitary or selfadjoint operators in Kreĭn spaces, Operators in indefinite metric spaces, scattering theory and other topics (Bucharest, 1985) (Operator Theory: Advances and Applications), Volume 24, Birkhäuser, 1987, pp. 123-143 | Zbl
[14] Spectral asymptotics for canonical systems, J. Reine Angew. Math., Volume 736 (2018), pp. 285-315 | DOI | MR | Zbl
[15] On a property of the -coefficient of a second-order linear differential equation, J. Lond. Math. Soc., Volume 4 (1971/72), pp. 443-457 | DOI | MR
[16] The spectral shift operator, Mathematical results in quantum mechanics (Prague, 1998) (Operator Theory: Advances and Applications), Volume 108, Birkhäuser, 1999, pp. 59-90 | DOI | MR | Zbl
[17] On matrix-valued Herglotz functions, Math. Nachr., Volume 218 (2000) no. 1, pp. 61-138 | DOI | MR | Zbl
[18] Mathematical analysis of the two dimensional active exterior cloaking in the quasistatic regime, Anal. Math. Phys., Volume 2 (2012) no. 3, pp. 231-246 | DOI | MR | Zbl
[19] On analytic functions representable by an integral of Cauchy-Stieltjes type, Vestn. Leningr. Univ., Mat. Mekh. Astron., Volume 13 (1958) no. 1, pp. 66-79 (Russian) | MR
[20] Analytic representation of linear functionals in spaces of harmonic and analytic functions which are continuous in a closed region, Dokl. Akad. Nauk SSSR, Volume 151 (1963), pp. 505-508 (Russian) | MR
[21] Passive approximation and optimization using B-splines, SIAM J. Appl. Math., Volume 79 (2019) no. 1, pp. 436-458 | DOI | MR | Zbl
[22] Quasi-Herglotz functions and convex optimization, R. Soc. Open Sci., Volume 7 (2020) no. 1, 191541 | DOI
[23] Operator representations of definitizable functions, Ann. Acad. Sci. Fenn., Math., Volume 25 (2000) no. 1, pp. 41-72 | MR | Zbl
[24] On operator representations of locally definitizable functions, Operator theory in Krein spaces and nonlinear eigenvalue problems (Operator Theory: Advances and Applications), Volume 162, Birkhäuser, 2006, pp. 165-190 | DOI | MR | Zbl
[25] R-functions – analytic functions mapping the upper half-plane into itself, Trans. Am. Math. Soc., Volume 103 (1974) no. 2, pp. 1-18 | Zbl
[26] Introduction to spaces, Cambridge Tracts in Mathematics, 115, Cambridge University Press, 1998, xiv+289 pages (with two appendices by V. P. Havin)
[27] On the trace formula in perturbation theory, Mat. Sb., N. Ser., Volume 33 (1953) no. 75, pp. 597-626 | MR
[28] Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen (German), Math. Nachr., Volume 77 (1977), pp. 187-236 | DOI | Zbl
[29] Superscars in the Šeba billiard, J. Eur. Math. Soc., Volume 19 (2017) no. 10, pp. 2947-2964 | DOI | Zbl
[30] On generalized resolvents and -functions of symmetric linear relations (subspaces) in Hilbert space, Pac. J. Math., Volume 72 (1977) no. 1, pp. 135-165 | DOI | MR | Zbl
[31] Distributional representations of -functions, Math. Nachr., Volume 288 (2015) no. 10, pp. 1127-1149 | DOI | MR | Zbl
[32] Asymptotic completeness and -matrix for singular perturbations, J. Math. Pures Appl., Volume 130 (2019), pp. 36-67 | DOI | MR | Zbl
[33] On a class of functions that can be represented in a domain by an integral of Cauchy-Stieltjes type (Russian), Usp. Mat. Nauk, Volume 52 (1997), pp. 169-170 translated in Russ. Math. Surv. 52 (1997), no. 3, p. 613-614 | DOI
[34] The theory of composites, Cambridge Monographs on Applied and Computational Mathematics, 6, Cambridge University Press, 2002, xxviii+719 pages | DOI
[35] Extending the Theory of Composites to Other Areas of Science, Milton-Patton Publishing, 2016
[36] Topics in Hardy classes and univalent functions, Birkhäuser Advanced Texts. Basler Lehrbücher, Birkhäuser, 1994 | DOI
[37] Integral, measure and derivative: a unified approach, Dover Books on Advanced Mathematics, Dover Publications, 1977 (translated from the Russian and edited by Richard A. Silverman)
[38] Mathematical methods in quantum mechanics, Graduate Studies in Mathematics, 157, American Mathematical Society, 2014 (With applications to Schrödinger operators) | DOI
[39] On integrals of Cauchy-Stieltjes type, Usp. Mat. Nauk, Volume 11 (1956) no. 4(70), pp. 163-166 (Russian) | MR
[40] Holomorphic functions with non-negative imaginary part in a tubular region over a cone (Russian), Mat. Sb., N. Ser., Volume 79 (1969), pp. 128-152 translated in Math. USSR-Sb. 8 (1969), p. 125-146
[41] Generalized functions in mathematical physics, Mir Publishers, 1979 (Translated from the second Russian edition by G. Yankovskiĭ)
[42] Methods of the theory of generalized functions, Analytical Methods and Special Functions, 6, Taylor & Francis, 2002 | DOI
[43] An N-Port Realizability Theory Based on the Theory of Distributions, IEEE Trans. Circuit Theory, Volume 10 (1963) no. 2, pp. 265-274 | DOI
Cited by Sources:
Comments - Policy