| 1. |
- Hedenmalm, Håkan, 1961-, et al.
(author)
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Heisenberg uniqueness pairs and the Klein-Gordon equation
- 2011
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In: Annals of Mathematics. - : Annals of Mathematics. - 0003-486X .- 1939-8980. ; 173:3, s. 1507-1527
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Journal article (peer-reviewed)abstract
- A Heisenberg uniqueness pair (HUP) is a pair (Γ,Λ), where Γ is a curve in the plane and Λ is a set in the plane, with the following property: any finite Borel measure μ in the plane supported on Γ, which is absolutely continuous with respect to arc length, and whose Fourier transform μˆ vanishes on Λ, must automatically be the zero measure. We prove that when Γ is the hyperbola x1x2=1 %, and Λ is the lattice-cross Λ=(αZ×{0})∪({0}×βZ), where α,β are positive reals, then (Γ,Λ) is an HUP if and only if αβ≤1; in this situation, the Fourier transform μˆ of the measure solves the one-dimensional Klein-Gordon equation. Phrased differently, we show that eπiαnt,eπiβn/t,n∈Z, span a weak-star dense subspace in L∞(R) if and only if αβ≤1. In order to prove this theorem, some elements of linear fractional theory and ergodic theory are needed, such as the Birkhoff Ergodic Theorem. An idea parallel to the one exploited by Makarov and Poltoratski (in the context of model subspaces) is also needed. As a consequence, we solve a problem on the density of algebras generated by two inner functions raised by Matheson and Stessin.
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| 2. |
- Aleman, Alexandru, et al.
(author)
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Backward Shift and Nearly Invariant Subspaces of Fock-type Spaces
- 2022
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In: International mathematics research notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; 2022:10, s. 7390-7419
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Journal article (peer-reviewed)abstract
- We study the structure of the backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces ℱWp, whose weight is not necessarily radial. We show that in the spaces ℱWp, which contain the polynomials as a dense subspace (in particular, in the radial case), all nontrivial backward shift invariant subspaces are of the form ℘n, that is, finite-dimensional subspaces consisting of polynomials of degree at most n. In general, the structure of the nearly invariant subspaces is more complicated. In the case of spaces of slow growth (up to zero exponential type), we establish an analogue of de Branges' ordering theorem. We then construct examples that show that the result fails for general Fock-type spaces of larger growth.
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| 3. |
- Bakan, Andrew, et al.
(author)
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Exponential Integral Representations of Theta Functions
- 2020
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In: Computational methods in Function Theory. - : Springer. - 1617-9447 .- 2195-3724. ; 20:3-4, s. 591-621
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Journal article (peer-reviewed)abstract
- Let Θ3(z) : = ∑ n∈Zexp (i πn2z) be the standard Jacobi theta function, which is holomorphic and zero-free in the upper half-plane H:={z∈C|Imz>0}, and takes positive values along i R> 0, the positive imaginary axis, where R> 0: = (0 , + ∞). We define its logarithm log Θ3(z) which is uniquely determined by the requirements that it should be holomorphic in H and real-valued on i R> 0. We derive an integral representation of log Θ3(z) when z belongs to the hyperbolic quadrilateral F□||:={z∈C|Imz>0,-1≤Rez≤1,|2z-1|>1,|2z+1|>1}.Since every point of H is equivalent to at least one point in F□|| under the theta subgroup of the modular group on the upper half-plane, this representation carries over in modified form to all of H via the identity recorded by Berndt. The logarithms of the related Jacobi theta functions Θ4 and Θ2 may be conveniently expressed in terms of log Θ3 via functional equations, and hence get controlled as well. Our approach is based on a study of the logarithm of the Gauss hypergeometric function for a specific choice of the parameters. This has connections with the study of the universally starlike mappings introduced by Ruscheweyh, Salinas, and Sugawa.
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| 4. |
- Bakan, Andrew, et al.
(author)
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Fourier uniqueness in even dimensions
- 2021
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In: Proceedings of the National Academy of Sciences of the United States of America. - : Proceedings of the National Academy of Sciences. - 0027-8424 .- 1091-6490. ; 118:15
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Journal article (peer-reviewed)abstract
- In recent work, methods from the theory of modular forms were used to obtain Fourier uniqueness results in several key dimensions (d = 1, 8, 24), in which a function could be uniquely reconstructed from the values of it and its Fourier transform on a discrete set, with the striking application of resolving the sphere packing problem in dimensions d = 8 and d = 24. In this short note, we present an alternative approach to such results, viable in even dimensions, based instead on the uniqueness theory for the KleinGordon equation. Since the existing method for the Klein-Gordon uniqueness theory is based on the study of iterations of Gauss-type maps, this suggests a connection between the latter and methods involving modular forms. The derivation of Fourier uniqueness from the Klein-Gordon theory supplies conditions on the given test function for Fourier interpolation, which are hoped to be optimal or close to optimal.
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| 7. |
- Hedenmalm, Håkan, 1961-
(author)
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A Beurling-Rudin theorem for H^\infty
- 1987
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In: Illinois Journal of Mathematics. - 0019-2082 .- 1945-6581. ; 31, s. 629-644
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Journal article (peer-reviewed)
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| 10. |
- Hedenmalm, Håkan, 1961-, et al.
(author)
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A critical topology for L^p Carleman classes with 0
- 2018
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In: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 371:3-4, s. 1803-1844
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Journal article (peer-reviewed)abstract
- In this paper, we explore a sharp phase transition phenomenon which occurs for (Formula presented.)-Carleman classes with exponents (Formula presented.). These classes are defined as for the standard Carleman classes, only the (Formula presented.)-bounds are replaced by corresponding (Formula presented.)-bounds. We study the quasinorms (Formula presented.)for some weight sequence (Formula presented.) of positive real numbers, and consider as the corresponding (Formula presented.)-Carleman space the completion of a given collection of smooth test functions. To mirror the classical definition, we add the feature of dilatation invariance as well, and consider a larger soft-topology space, the (Formula presented.)-Carleman class. A particular degenerate instance is when (Formula presented.) for (Formula presented.) and (Formula presented.) for (Formula presented.). This would give the (Formula presented.)-Sobolev spaces, which were analyzed by Peetre, following an initial insight by Douady. Peetre found that these (Formula presented.)-Sobolev spaces are highly degenerate for (Formula presented.). Indeed, the canonical map (Formula presented.) fails to be injective, and there is even an isomorphism (Formula presented.)corresponding to the canonical map (Formula presented.) acting on the test functions. This means that e.g. the function and its derivative lose contact with each other (they “disconnect”). Here, we analyze this degeneracy for the more general (Formula presented.)-Carleman classes defined by a weight sequence (Formula presented.). If (Formula presented.) has some regularity properties, and if the given collection of test functions is what we call (Formula presented.)-tame, then we find that there is a sharp boundary, defined in terms of the weight (Formula presented.): on the one side, we get Douady–Peetre’s phenomenon of “disconnexion”, while on the other, the completion of the test functions consists of (Formula presented.)-smooth functions and the canonical map (Formula presented.) is correspondingly well-behaved in the completion. We also look at the more standard second phase transition, between non-quasianalyticity and quasianalyticity, in the (Formula presented.) setting, with (Formula presented.).
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| 21. |
- Hedenmalm, Håkan, 1961-
(author)
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BLOCH FUNCTIONS, ASYMPTOTIC VARIANCE, AND GEOMETRIC ZERO PACKING
- 2020
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In: American Journal of Mathematics. - : JOHNS HOPKINS UNIV PRESS. - 0002-9327 .- 1080-6377. ; 142:1, s. 267-321
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Journal article (peer-reviewed)abstract
- Motivated by a problem in quasiconformal mapping, we introduce a problem in complex analysis, with its roots in the mathematical physics of the Bose-Einstein condensates in superconductivity. The problem will be referred to as geometric zero packing, and is somewhat analogous to studying Fekete point configurations. The associated quantity is a density, denoted pc in the planar case, and pH in the case of the hyperbolic plane. We refer to these densities as discrepancy densities for planar and hyperbolic zero packing, respectively, as they measure the impossibility of atomizing the uniform planar and hyperbolic area measures. The universal asymptotic variance Sigma(2) associated with the boundary behavior of conformal mappings with quasiconformal extensions of small dilatation is related to one of these discrepancy densities: Sigma(2) = 1- rho H. We obtain the estimates 3.2 x 10(-5) < rho H <= 0.12087, where the upper estimate is derived from the estimate from below on Sigma(2) obtained by Astala, Ivrii, Perala, and Prause, and the estimate from below is more delicate. In particular, it follows that Sigma(2) < 1, which in combination with the work of ivrii shows that the maximal fractal dimension of quasicircles conjectured by Astala cannot be reached. Moreover, along the way, since the universal quasiconformal integral means spectrum has the asymptotics B(k, t) similar to 1/4 Sigma(2)vertical bar t vertical bar(2) for small t and k, the conjectured formula B(k, t) = 1/4 k(2)vertical bar t vertical bar(2) is not true. As for the actual numerical values of the discrepancy density rho(C), we obtain the estimate from above rho(C) <= 0.061203 ... by using the equilateral triangular planar zero packing, where the assertion that equality should hold can be attributed to Abrikosov. The value of pH is expected to be somewhat close to that of rho(C).
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| 24. |
- Hedenmalm, Håkan, 1961-
(author)
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Closed ideals in the ball algebra
- 1989
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In: Bulletin of the London Mathematical Society. - 0024-6093 .- 1469-2120. ; 21, s. 469-474
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Journal article (peer-reviewed)
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| 29. |
- Hedenmalm, Håkan, 1961-
(author)
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Cyclicity in Bergman-type spaces
- 1995
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In: International mathematics research notices. - : Duke Univ. Press. - 1073-7928 .- 1687-0247. ; 5, s. 253-262
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Journal article (peer-reviewed)
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| 32. |
- Hedenmalm, Håkan, 1961-, et al.
(author)
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Gaussian analytic functions and operator symbols of Dirichlet type
- 2020
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In: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 372
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Journal article (peer-reviewed)abstract
- Let M = {m(j,k)}(j,k=1)(+infinity)be an infinite complex-valued matrix which acts contractively on l(2). For the weighted short diagonal sums S-M (l) := Sigma(j,k:j+k=l) (l/jk)(1/2) m(j,k), we obtain the estimate Sigma(+infinity)(l=2) sl/l vertical bar S-M (l)vertical bar(2) <= 2s log e/1 - s, 0 <= s < 1. Expressed more vaguely, vertical bar S-M(l)vertical bar(2) (sic) 2 holds in the sense of averages. Concerning the optimality of the above bound, a construction due to Zachary Chase shows that the statement does not hold if the number 2 is replaced by the smaller number 1.72. In the construction, Mis a permutation matrix. We interpret our bound in terms of the correlation E Phi(z)Psi(z) of two copies of a Gaussian analytic function with possibly intricate Gaussian correlation structure between them. The Gaussian analytic function we study arises in connection with the classical Dirichlet space, which is naturally Mobius invariant. The study of the correlations E Phi(z)Psi(z) leads us to introduce a new space, the mock-Bloch space(or Blochish space) which is slightly bigger than the standard Bloch space. Our bound has an interpretation in terms of McMullen's asymptotic variance, originally considered for functions in the Bloch space. Finally, we show that the correlations E Phi(z)Psi(w) may be expressed as Dirichlet symbols of contractions on L-2(D), and show that the Dirichlet symbols of Grunsky operators associated with univalent functions find a natural characterization in terms of a nonlinear wave equation. (C) 2020 Elsevier Inc. All rights reserved.
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| 40. |
- Hedenmalm, Håkan, 1961-
(author)
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My Recollections of Serguei Shimorin
- 2019
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In: Analysis of Operators on Function Spaces. - Cham : Springer International Publishing. ; , s. 1-4
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Book chapter (peer-reviewed)abstract
- The author shares his personal reminiscences of Serguei Shimorin.
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| 41. |
- Hedenmalm, Håkan, 1961-, et al.
(author)
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Off-Spectral Analysis of Bergman Kernels
- 2020
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In: Communications in Mathematical Physics. - : Springer. - 0010-3616 .- 1432-0916. ; 373:3, s. 1049-1083
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Journal article (peer-reviewed)abstract
- The asymptotic analysis of Bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention. Such interfaces typically occur when the associated limiting Bergman density function vanishes on a portion of the plane, the off-spectral region. This type of behavior is observed when the metric is negatively curved somewhere, or when we study partial Bergman kernels in the context of positively curved metrics. In this work, we cover these two situations in a unified way, for exponentially varying weights on the complex plane. We obtain a uniform asymptotic expansion of the coherent state of depthn rooted at an off-spectral point, which we also refer to as the root function at the point in question. The expansion is valid in the entire off-spectral component containing the root point, and protrudes into the spectrum as well. This allows us to obtain error function transition behavior of the density of states along the smooth interface. Previous work on asymptotic expansions of Bergman kernels is typically local, and valid only in the bulk region of the spectrum, which contrasts with our non-local expansions.
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| 48. |
- Hedenmalm, Håkan, 1961-, et al.
(author)
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RIEMANN-HILBERT HIERARCHIES FOR HARD EDGE PLANAR ORTHOGONAL POLYNOMIALS
- 2024
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In: American Journal of Mathematics. - : Johns Hopkins University Press. - 0002-9327 .- 1080-6377. ; 146:2, s. 371-403
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Journal article (peer-reviewed)abstract
- We obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain D with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly positive, and for any given precision κ, the expansion holds with an O(N−κ−1) error in N-dependent neighborhoods of the exterior region as the degree N tends to infinity. The main ingredient is the derivation and analysis of Riemann-Hilbert hierarchies—sequences of scalar Riemann-Hilbert problems—which allows us to express all higher order correction terms in closed form. Indeed, the expansion may be understood as a Neumann series involving an explicit operator. The expansion theorem leads to a semiclassical asymptotic expansion of the corresponding hard edge probability wave function in terms of distributions supported on ∂D.
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| 49. |
- Hedenmalm, Håkan, 1961-, et al.
(author)
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Sharpening Holder's inequality
- 2018
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In: Journal of Functional Analysis. - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0022-1236 .- 1096-0783. ; 275:5, s. 1280-1319
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Journal article (peer-reviewed)abstract
- We strengthen Holder's inequality. The new family of sharp inequalities we obtain might be thought of as an analog of the Pythagorean theorem for the L-p-spaces. Our treatment of the subject matter is based on Bellman functions of four variables.
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