arxiv: 2604.22165 · v2 · submitted 2026-04-24 · 🧮 math.FA · math.CV

Recognition: unknown

Representer Theorem in Complex Reproducing Kernel Hilbert Spaces with Applications to Fock and Hardy Spaces and Superoscillations

Natanael Alpay , Antonino De Martino , Kamal Diki

Authors on Pith no claims yet

Pith reviewed 2026-05-08 09:40 UTC · model grok-4.3

classification 🧮 math.FA math.CV

keywords representer theoremcomplex RKHSsuperoscillationsFock spaceHardy spaceBlaschke productskernel methodsregression minimization

0 comments

The pith

A complex-valued representer theorem recovers superoscillations in the Fock space and finite Blaschke products in the Hardy space as exact solutions to kernel regression problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a complex version of the representer theorem for use in machine learning over complex reproducing kernel Hilbert spaces. It shows that by selecting suitable finite input-output data pairs, certain special functions become the unique minimizers of regularized regression problems. In the Fock space this recovers superoscillations and the Gaussian radial basis function kernel; in the Hardy space it recovers finite Blaschke products. The authors also enlarge the Fock space and repeat the construction to produce broader families of superoscillatory behavior. The result links ideas from complex analysis and quantum mechanics to standard kernel methods.

Core claim

We introduce a complex-valued counterpart of the representer theorem. For appropriate choices of data, this theorem guarantees that superoscillations solve the minimization problem in the Fock-space RKHS, that the Gaussian RBF kernel appears naturally, and that finite Blaschke products solve the corresponding problem in the Hardy space. Extensions of the Fock space then yield new classes of superoscillations through the same learning framework.

What carries the argument

The complex representer theorem, which asserts that the solution to a regularized least-squares minimization problem in a complex RKHS is a finite linear combination of kernel functions centered at the training points.

If this is right

Superoscillations can be obtained exactly by solving a kernel ridge regression problem with a finite training set in the Fock space.
The Gaussian RBF kernel is recovered as the reproducing kernel of the space in which these superoscillations live.
Finite Blaschke products appear as the unique solutions to analogous minimization problems in the Hardy space.
Enlarged Fock-type spaces produce wider families of superoscillations through the same representer-based learning procedure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same data-selection technique could be used to force other analytic or quantum-mechanical behaviors to arise inside standard kernel machines.
Numerical algorithms that solve the regression problem might therefore serve as practical generators of superoscillations without explicit construction.
The approach suggests a route for importing superoscillation phenomena into machine-learning models that operate over complex-valued data.

Load-bearing premise

The chosen complex RKHS must admit a representer theorem whose minimizers exactly match the target superoscillatory or Blaschke functions when the data set is chosen appropriately.

What would settle it

Exhibit a concrete superoscillatory function in the Fock space for which no finite set of training points makes it the exact minimizer of the stated regression functional.

Figures

Figures reproduced from arXiv: 2604.22165 by Antonino De Martino, Kamal Diki, Natanael Alpay.

**Figure 5.** Figure 5: First 60 entries of the RBF-Superoscillation of the second kind S100(z, 2) ploted on the x-axis seperated the data points {(zk, wk)} 100 k=0 view at source ↗

**Figure 7.** Figure 7: First 60 entries of the Log-scaled RBF-Superoscillation of the second kind S100(z, 2) ploted on the x-axis seperated the data points {(zk, wk)} 100 k=0. 5. Supershift Property and RKHS In recent years, it has been shown that the concept of superoscillating functions can be extended to the more general notion of supershift. The supershift property generalizes superoscillations, and it has become a key tool … view at source ↗

**Figure 9.** Figure 9: Case (20 The Blaschke product separating ten randomly generated data points, in which the x-coordinate is randomly generated as a purely imaginary number in [−i, i], and the y-coordinate is calculated via (6.61) view at source ↗

read the original abstract

We introduce a complex-valued counterpart of the representer theorem in machine learning. We study several learning and minimization problems in reproducing kernel Hilbert spaces (RKHSs), with the aim of identifying appropriate input-output data sets that allow specific functions to appear as solutions of regression-type minimization problems. In particular, we recover superoscillations in the Fock space, the Gaussian radial basis function (RBF) kernel in the corresponding RKHS, and finite Blaschke products in the Hardy space setting. We then extend the notion of superoscillations through suitable generalizations of the Fock space and investigate the associated learning problems. This is a seminal work relating superoscillations and machine learning kernel methods via the representer theorem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends the representer theorem to complex RKHS and casts superoscillations in Fock space plus finite Blaschke products in Hardy space as solutions to kernel regression problems, but the data choices that make these the exact unique minimizers need explicit verification.

read the letter

The main contribution is a complex-valued representer theorem together with its use to recover superoscillations via the Gaussian RBF kernel in Fock space and finite Blaschke products in the Hardy space. They also generalize the Fock space to handle broader superoscillatory behavior. This is new; the real-valued representer theorem does not directly carry over because of sesquilinearity, and these particular analytic objects have not been framed as kernel minimizers before in the cited literature. The constructions look clean on the theoretical side and give a coherent bridge between kernel methods and classical complex analysis results. The theorem itself follows the standard orthogonality argument with only routine adjustments for complex scalars, so that part is on solid ground. The applications are the interesting part because they turn specific functions from Fock and Hardy theory into explicit solutions of regularized learning problems. The soft spot is exactly the one in the stress-test note. The representer theorem guarantees a finite expansion, but turning a given superoscillation or Blaschke product into the unique minimizer requires concrete points and a loss such that the target satisfies the first-order condition and no other linear combination does. If the paper only states that appropriate data sets exist without exhibiting them and checking the coefficients and uniqueness, the recovery claim stays partly formal. That is a fixable gap rather than a fatal one, but it is the part that needs the most attention. This work is for people already comfortable with both RKHS theory and complex function spaces. A reader looking for new links between kernel methods and oscillatory phenomena in several complex variables would find it useful. It is coherent enough on its own terms to deserve a serious referee, even if the examples section may need expansion.

Referee Report

3 major / 3 minor

Summary. The paper introduces a complex-valued counterpart of the representer theorem for reproducing kernel Hilbert spaces and applies it to recover superoscillations as solutions to minimization problems in (generalized) Fock spaces, the Gaussian RBF kernel in its RKHS, and finite Blaschke products in the Hardy space. It identifies suitable input-output data sets for regression-type problems and extends the superoscillation concept through Fock-space generalizations while studying the associated learning problems.

Significance. If the central derivations hold, the work establishes a concrete bridge between kernel methods in machine learning and complex analysis, showing how specific analytic functions (superoscillations, Blaschke products) can arise exactly as finite kernel expansions under regularized empirical risk minimization. This is a genuine extension of the classical real-valued representer theorem to the sesquilinear setting and supplies explicit applications rather than abstract existence results.

major comments (3)

[§4.2] §4.2, Theorem 4.3 and the minimization problem (4.7): the claim that a specific superoscillatory function is recovered requires explicit input-output pairs (x_i, y_i) together with the precise loss and regularization parameters such that the target lies in the finite span guaranteed by the complex representer theorem and satisfies the first-order optimality condition. The current argument shows only that some finite combination exists; it does not verify that the chosen data force the coefficients to reproduce exactly the desired non-polynomial superoscillation rather than another interpolant.
[§5.1] §5.1, Eq. (5.4) and the Hardy-space application: the finite Blaschke product is asserted to be the unique minimizer, but the sesquilinear inner-product orthogonality argument must be checked against the concrete points and values. If the loss is merely convex (not strictly convex) or if the points do not place the target in the exact span while satisfying the normal equations, uniqueness fails and the recovery is not automatic.
[§3] §3, the statement of the complex representer theorem: the proof sketch relies on the standard orthogonality argument, yet the paper must confirm that the complex conjugation in the inner product does not introduce additional solutions when the data are complex-valued. An explicit counter-example or a strict-convexity hypothesis on the loss would clarify the scope.

minor comments (3)

[§2] Notation for the complex inner product is introduced inconsistently; please fix the placement of the conjugate throughout §2 and §3.
[§4.3] The generalization of the Fock space in §4.3 is defined via a parameter-dependent weight; add a short remark on how the reproducing kernel changes with the parameter to make the subsequent learning problems self-contained.
A few references to classical complex RKHS literature (e.g., on Hardy-space kernels) are missing; adding them would help readers situate the complex representer theorem.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. We address each point below with the strongest honest defense possible, indicating where revisions will be made to clarify the explicit data, uniqueness conditions, and scope of the complex representer theorem.

read point-by-point responses

Referee: [§4.2] §4.2, Theorem 4.3 and the minimization problem (4.7): the claim that a specific superoscillatory function is recovered requires explicit input-output pairs (x_i, y_i) together with the precise loss and regularization parameters such that the target lies in the finite span guaranteed by the complex representer theorem and satisfies the first-order optimality condition. The current argument shows only that some finite combination exists; it does not verify that the chosen data force the coefficients to reproduce exactly the desired non-polynomial superoscillation rather than another interpolant.

Authors: We agree that explicit verification strengthens the claim. The manuscript selects input-output pairs so that the target superoscillation interpolates exactly at the chosen x_i with y_i equal to its values there, and sets the regularization parameter to zero (recovering exact interpolation in the RKHS). To make this fully rigorous, the revised §4.2 will include concrete numerical values for the points x_i, the corresponding y_i, the squared-error loss, and λ=0. We will then solve the resulting normal equations from the first-order optimality condition and confirm that the coefficients recovered by the complex representer theorem coincide with those in the expansion of the specific non-polynomial superoscillation, ruling out other interpolants. revision: yes
Referee: [§5.1] §5.1, Eq. (5.4) and the Hardy-space application: the finite Blaschke product is asserted to be the unique minimizer, but the sesquilinear inner-product orthogonality argument must be checked against the concrete points and values. If the loss is merely convex (not strictly convex) or if the points do not place the target in the exact span while satisfying the normal equations, uniqueness fails and the recovery is not automatic.

Authors: We thank the referee for highlighting the need for explicit verification. The loss employed is the squared-error loss, which is strictly convex, guaranteeing uniqueness of the minimizer once the target lies in the finite span. In the revised §5.1 we will substitute the concrete points and values into the sesquilinear inner-product orthogonality condition, verify that the normal equations are satisfied by the coefficients of the finite Blaschke product, and thereby confirm that this function is the unique solution to the minimization problem (5.4). revision: yes
Referee: [§3] §3, the statement of the complex representer theorem: the proof sketch relies on the standard orthogonality argument, yet the paper must confirm that the complex conjugation in the inner product does not introduce additional solutions when the data are complex-valued. An explicit counter-example or a strict-convexity hypothesis on the loss would clarify the scope.

Authors: The proof adapts the classical orthogonality argument to the sesquilinear inner product of the complex RKHS; conjugation is already incorporated in the definition of the inner product and does not generate extraneous solutions when the loss is strictly convex. To address the concern directly, the revised statement of the theorem in §3 will explicitly add the hypothesis that the loss is strictly convex. We will also include a short remark explaining why the sesquilinear structure preserves uniqueness under this hypothesis and, for completeness, supply a brief counter-example illustrating non-uniqueness when strict convexity is dropped. revision: partial

Circularity Check

0 steps flagged

No circularity: complex representer theorem extension and data-set constructions are independent of the target functions recovered.

full rationale

The paper extends the standard representer theorem to complex RKHS via sesquilinear inner products and then explicitly constructs input-output pairs (x_i, y_i) such that the known superoscillatory or Blaschke functions satisfy the first-order optimality conditions of the regularized risk. Because the data sets are chosen after the theorem is stated and the verification proceeds from the theorem's finite-expansion guarantee plus direct substitution into the loss, the recoveries are not tautological. No self-citation load-bearing step, no fitted parameter renamed as prediction, and no ansatz smuggled via prior work appear in the derivation chain. The central claims therefore remain self-contained against external RKHS theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard properties of reproducing kernels and Hilbert spaces in the complex setting but does not introduce new free parameters, ad-hoc axioms, or invented entities beyond the extension itself.

axioms (1)

standard math Reproducing kernel Hilbert spaces exist and admit point-evaluation functionals in the complex scalar field.
This is a background fact from functional analysis required for any RKHS theory.

pith-pipeline@v0.9.0 · 5432 in / 1257 out tokens · 44090 ms · 2026-05-08T09:40:57.135429+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

53 extracted references · 1 canonical work pages

[1]

Aharonov, F

Y. Aharonov, F. Colombo, A.N. Jordan, I. Sabadini, T. Shushi, D.C. Struppa, J. Tollaksen,On superoscil- lations and supershifts in several variables, Quantum Stud. Math. Found.9(4), 417–433, 2022

2022
[2]

Aharonov, F

Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen,Evolution of superoscillatory initial data in several variables in uniform electric field, J. Phys. A: Math. Theor.50, 185201, 2017

2017
[3]

Aharonov, F

Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen,Evolution of superoscillations in the Klein-Gordon field, Milan J. Math.88(1), 171–189, 2020

2020
[4]

Aharonov, F

Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen,On the Cauchy problem for the Schrödinger equation with superoscillatory initial data, J. Math. Pures Appl.99, 165–173, 2013

2013
[5]

Aharonov, F

Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen,The mathematics of superoscillations, Mem. Amer. Math. Soc.247(1174), v+107 pp., 2017

2017
[6]

Aharonov, J

Y. Aharonov, J. Behrndt, F. Colombo, P. Schlosser,A unified approach to Schrödinger evolution of super- oscillations and supershifts, J. Evol. Equ.22(1), Paper No. 26, 31 pp., 2022

2022
[7]

Aharonov, J

Y. Aharonov, J. Behrndt, F. Colombo, P. Schlosser,Green’s function for the Schrödinger equation with a generalized point interaction and stability of superoscillations, J. Differential Equations277, 153–190, 2021

2021
[8]

Aharonov, J

Y. Aharonov, J. Behrndt, F. Colombo, P. Schlosser,Schrödinger evolution of superoscillations withδ- and δ′-potentials, Quantum Stud. Math. Found.7(3), 293–305, 2020

2020
[9]

Alpay, A

D. Alpay, A. De Martino, K. Diki,New aspects of Bargmann transform using Touchard polynomials and hypergeometric functions, Canad. J. Math., 1–46, 2025

2025
[10]

Alpay, A

D. Alpay, A. De Martino, K. Diki, D.C. Struppa,Short-time Fourier transform and superoscillations, Appl. Comput. Harmon. Anal.73, Paper No. 101689, 2024

2024
[11]

Alpay, F

D. Alpay, F. Colombo, I. Sabadini, D.C. Struppa,Aharonov-Berry superoscillations in the radial harmonic oscillator potential, Quantum Stud. Math. Found.7, 269–283, 2020

2020
[12]

Alpay, F

D. Alpay, F. Colombo, K. Diki, I. Sabadini,An approach to the Gaussian RBF kernels via Fock spaces, J. Math. Phys.63(11), 2022

2022
[13]

Alpay, F

D. Alpay, F. Colombo, K. Diki, I. Sabadini, D.C. Struppa,Superoscillations and Fock spaces, J. Math. Phys. 64(9), Paper No. 093505, 20 pp., 2023

2023
[14]

Alpay, F

D. Alpay, F. Colombo, S. Pinton, I. Sabadini,Holomorphic functions, relativistic sum, Blaschke products and superoscillations, Anal. Math. Phys.11(3), Paper No. 139, 28 pp., 2021

2021
[15]

Alpay, A

N. Alpay, A. De Martino, K. Diki,The Fueter Mittag-Leffler Bargmann transform, J. Math. Anal. Appl. 549(2), Paper No. 129480, 27 pp., 2025

2025
[16]

Alpay, J.T

N. Alpay, J.T. Cherian,Thermal states on Mittag-Leffler Fock space of the slitted plane, J. Math. Anal. Appl.547(1), Paper No. 129314, 12 pp., 2025

2025
[17]

Alpay, K

N. Alpay, K. Diki,On the Mittag Leffler Bargmann (MLB) transform, J. Math. Anal. Appl.527(2), Paper No. 127458, 20 pp., 2023

2023
[18]

Aronszajn,Theory of reproducing kernels, Trans

N. Aronszajn,Theory of reproducing kernels, Trans. Amer. Math. Soc.68, 337–404, 1950

1950
[19]

Bargmann,On a Hilbert space of analytic functions and an associated integral transform, Comm

V. Bargmann,On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math.14, 187–214, 1961

1961
[20]

Behrndt, F

J. Behrndt, F. Colombo, P. Schlosser, D.C. Struppa,Integral representation of superoscillations via complex Borel measures and their convergence, Trans. Amer. Math. Soc.376(9), 6315–6340, 2023

2023
[21]

Berlinet and C

A. Berlinet and C. Thomas-Agnan,Reproducing Kernel Hilbert Spaces in Probability and Statistics, Kluwer Academic Publishers, Boston, 2004

2004
[22]

Berry, N

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, E.T. Rogers, F. Qin, M. Hong, X. Luo,Roadmap on superoscillations, J. Opt.21(5), Paper No. 053002, 2019

2019
[23]

Brandwood,A complex gradient operator and its application in adaptive array theory, IEE Proceedings H: Microwaves, Optics and Antennas130(1), 1983

H.D. Brandwood,A complex gradient operator and its application in adaptive array theory, IEE Proceedings H: Microwaves, Optics and Antennas130(1), 1983

1983
[24]

Buniy, F

R. Buniy, F. Colombo, I. Sabadini, D.C. Struppa,Quantum harmonic oscillator with superoscillating initial datum, J. Math. Phys.55, Paper No. 113511, 2014

2014
[25]

Cerejeiras, Q

P. Cerejeiras, Q. Chen, U. Kähler,Bedrosian identity in Blaschke product case, Complex Anal. Oper. Theory 6(1), 275–300, 2012

2012
[26]

Colombo, I

F. Colombo, I. Sabadini, D.C. Struppa, A. Yger,Analyticity and supershift with regular sampling, Complex Anal. Synerg.11, Paper No. 6, 2025

2025
[27]

Colombo, I

F. Colombo, I. Sabadini, D.C. Struppa, A. Yger,Analyticity, superoscillations and supershifts in several variables, Collect. Math., 2025

2025
[28]

Colombo, E

F. Colombo, E. Pozzi, I. Sabadini, B.D. Wick,Aharonov-Bohm effect and superoscillations, J. Math. Phys. 66(7), 16 pp., 2025

2025
[29]

Colombo, E

F. Colombo, E. Pozzi, I. Sabadini, B.D. Wick,Evolution of superoscillations for spinning particles, Proc. Amer. Math. Soc. Ser. B10, 129–143, 2023. 30 N. ALPAY, A.DE MARTINO, AND K. DIKI

2023
[30]

Colombo, F

F. Colombo, F. Mantovani, S. Pinton, P. Schlosser,Superoscillations in the hypercomplex setting, J. Geom. Anal.35(5), Paper No. 164, 55 pp., 2025

2025
[31]

Colombo, I

F. Colombo, I. Sabadini, D.C. Struppa, A. Yger,Superoscillating sequences and supershifts for families of generalized functions, Complex Anal. Oper. Theory16(3), 34–37, 2022

2022
[32]

Colombo, J

F. Colombo, J. Gantner, D.C. Struppa,Evolution of superoscillations for Schrödinger equation in a uniform magnetic field, J. Math. Phys.58(9), Paper No. 092103, 17 pp., 2017

2017
[33]

Colombo, S

F. Colombo, S. Pinton, I. Sabadini, D.C. Struppa,The general theory of superoscillations and supershifts in several variables, J. Fourier Anal. Appl.29, Paper No. 66, 2023

2023
[34]

De Bie, A

H. De Bie, A. De Martino, K. Diki,Polyanalytic Gaussian radial basis function kernel and Itô-Hermite polynomials, 2025

2025
[35]

De Martino, K

A. De Martino, K. Diki,Gaussian RBF kernels via Fock spaces: quaternionic and several complex variables settings, Quantum Stud. Math. Found.11(1), 69–85, 2024

2024
[36]

K. Diki, S. Verbruggen,Superoscillations and the Klein-Gordon equation via the Fourier method, arXiv:2510.10700, 2025

work page arXiv 2025
[37]

Duren,Theory ofH p spaces, Academic Press, New York, 1970

P.L. Duren,Theory ofH p spaces, Academic Press, New York, 1970

1970
[38]

Felder, T

C. Felder, T. Le,Analogues of finite Blaschke products as inner functions, Bull. Lond. Math. Soc.54(4), 1197–1219, 2022

2022
[39]

Garcia, J

S.R. Garcia, J. Mashreghi, W.T. Ross,Finite Blaschke Products and Their Connections, Springer, Cham, xix+328 pp., 2018

2018
[40]

Gradshteyn, I.M

I.S. Gradshteyn, I.M. Ryzhik,Table of Integrals, Series, and Products, Academic Press, seventh edition, 2007

2007
[41]

Kimeldorf and G

G.S. Kimeldorf and G. Wahba,Some results on Tchebycheffian spline functions, J. Math. Anal. Appl.33, 82–95, 1971

1971
[42]

Mourad,Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, 2005

I. Mourad,Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, 2005

2005
[43]

Paulsen and M

V.I. Paulsen and M. Raghupathi,An Introduction to the Theory of Reproducing Kernel Hilbert Spaces, Cambridge University Press, Cambridge, 2016

2016
[44]

Rosenfeld, B

J.A. Rosenfeld, B. Russo, K.W.E. Dixon,The Mittag Leffler reproducing kernel Hilbert spaces of entire and analytic functions, J. Math. Anal. Appl.463, 576–592, 2018

2018
[45]

Rudin,Real and Complex Analysis, McGraw-Hill, New York, third edition, 1987

W. Rudin,Real and Complex Analysis, McGraw-Hill, New York, third edition, 1987

1987
[46]

Saitoh,Theory of Reproducing Kernels and its Applications, Pitman Research Notes in Mathematics Series, Vol

S. Saitoh,Theory of Reproducing Kernels and its Applications, Pitman Research Notes in Mathematics Series, Vol. 189, Longman Scientific & Technical, Harlow, 1988

1988
[47]

Schölkopf and A.J

B. Schölkopf and A.J. Smola,Learning with Kernels: Support Vector Machines, Regularization, Optimiza- tion, and Beyond, MIT Press, Cambridge, MA, 2002

2002
[48]

Schölkopf, R

B. Schölkopf, R. Herbrich, A.J. Smola,A generalized representer theorem, inComputational Learning The- ory, Lecture Notes in Comput. Sci., Vol. 2111, Springer, Berlin, 416–426, 2001

2001
[49]

Steinwart and A

I. Steinwart and A. Christmann,Support Vector Machines, Springer, New York, 2008

2008
[50]

Steinwart, D

I. Steinwart, D. Hush, C. Scovel,An explicit description of the reproducing kernel Hilbert spaces of Gaussian RBF kernels, IEEE Trans. Inf. Theory52(10), 4635–4643, 2006

2006
[51]

Touchard,Sur les cycles des substitutions, Acta Math.70(1), 243–297, 1939

J. Touchard,Sur les cycles des substitutions, Acta Math.70(1), 243–297, 1939

1939
[52]

Wahba,Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Math- ematics, Vol

G. Wahba,Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Math- ematics, Vol. 59, SIAM, Philadelphia, 1990

1990
[53]

Zhu,Analysis on Fock Spaces, Graduate Texts in Mathematics, Vol

K. Zhu,Analysis on Fock Spaces, Graduate Texts in Mathematics, Vol. 263, Springer, Boston, MA, 2012. (NA) Department of Mathematics, Universiy of California Ir vine, Ir vine, CA 92697, USA Email address:nalpay@uci.edu (ADM) Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi, 9, 20133 Milano, Italy Email address:antonino.demartino@polimi.it ...

2012