arXiv subjects

Haakan Hedenmalm

Publications and source records attributed to Haakan Hedenmalm.

At least 19 recordsLinked to original sources

arXiv2026

Dirichlet symbols and the nonlinear wave equation

We study the operator symbols of Dirichlet type introduced by Hedenmalm and Shimorin (2020), in connection with a given contraction on $L^2$ of the unit disk. They are always holomorphic functions on the bidisk. Such Dirichlet symbols associated with the Grunsky operator of a univalent function on the disk or exterior disk are of particular significance. From the work of Hedenmalm and Shimorin, we know they are characterized as solutions of a certain nonlinear wave equation. We perform a local analysis of such symbols near the diagonal on the bidisk, and in so doing, we provide alternative chart coordinates for the infinite-dimensional manifolds of univalent functions of the (exterior) disk. Those coordinates allow us to characterize $\logψ'$ for $ψ$ in the class $Σ$ of normalized univalent functions without explicitly touching the univalence property. Moreover, that manifold extends the universal Teichmüller space of Lipman Bers beyond the quasicircle boundary setting, allowing for even more fractality. The fractality of harmonic measure for the domain associated with the given univalent function can be studied in terms of the asymptotic variance introduced by McMullen (2008). The asymptotic variance captures the $L^2$ average amplitude of the nonlinearity. We here introduce the new concept of Schwarzian asymptotic variance, which measures the average amplitude of the Schwarzian derivative in place of the nonlinearity. For this new Schwarzian asymptotic variance, we find that the effective average amplitude of $(1-|z|^2)^4|\Sop(\vp)|^2$ on the disk in the hyperbolic metric sense is at most $9.07735\ldots$, considerably smaller than the maximum amplitude of $36$. Here, $\Sop(\vp)$ is the Schwarzian derivative of $φ\in\mathscr{S}$, and the analogous statement is valid for $ψ\inΣ$ as well.

math.CV↗

arXiv2026

Spectral interpretation of Riemann zeta zeros

It is a well-known problem to identify the nontrivial zeros of the Riemann zeta function in terms of an eigenvalue problem. We here find such an eigenvalue problem for second order differential operators on the half-line. In a sense, our analysis pushesthe analysis of the zeta function over to the study of the Jacobi theta function, which may be thought of as the fundamental solution of the heat (or Schrödinger) equation on the unit circle (or the semi-infinite cylinder, if time is added). The eigenvalue problem takes the form $LD u+αLu=0$, where $L$ and $D$ are first-order differential operators, of which only $L$ involves the theta function. In a formal sense, then, $α$ is an eigenvalue of the twisted operator $-LDL^{-1}$. Based on this formal thinking, we develop the notion of self-adjointness of the pair $(LD,L)$, to adapt the Hilbert-Pólya idea to the spectral problem at hand.

math.CA↗

arXiv2026

Liouville phenomenon for the Klein-Gordon equation in 1+1 dimensions

We study the Klein-Gordon equation in one spatial and one temporal dimension. Physically, this equation describes the wave function of a relativistic spinless boson with positive rest mass. Mathematically, this is the most elementary hyperbolic partial differential equation, after the wave equation itself. Relative to the origin, the spacetime splits according to the light cones, and we find four quarter-planes, two of which are timelike while the remaining two are spacelike. Not unexpectedly, the solutions behave quite differently in the two types of quarter-planes. It turns out that the spacelike quarter-planes exhibit a Liouville phenomenon, where insufficient growth forces the solutions to display a certain kind of symmetry, where the values on the two linear edges are in a one-to-one relation. This phenomenon shares features with the classical Liouville theorem as well as the Phragmen-Lindelof principle for harmonic functions.

math.AP↗

arXiv2026

Discrete Gaussian Free Field via Hadamard's formula

We present a novel way of constructing the Gaussian Free Field on a weighted graph via a dynamical expansion of the Green function along an expanding family of subgraphs. Along the way we obtain the discrete analogue of the classical Hadamard variational formula regarding the variation of the Green function under infinitesimal variations of the domain. In order to develop necessary machinery we construct expanding bases of the naturally associated energy spaces. An interesting observation is that both our discrete Hadamard variation formula and and the related construction of the discrete Gaussian Free Field are completely dimension-free and do not require smoothness of any kind. The graph model contains geometric information via the edges which supply the discrete topological information, and by conductances which give metric information. Going to a continuum limit, we would then obtain continuous version of the Hadamard variational formula and the associated Hadamard operator in e.g. fractal geometries of arbitrary dimension.

math.PR↗

arXiv2025

A Bombieri-type inequality and equidistribution of points

In recent work, Etayo introduces a new Bombieri-type inequality for monic polynomials. Here we reinterpret this new inequality as a more general integral inequality involving the Green function for the sphere. This rather geometric interpretation allows for generalizations of the basic inequality, involving fractional zeros while also opening up the possibility to extend the setting to general compact Riemann surfaces. We derive a sharp form of these generalized Bombieri-type inequalities for the case of the sphere and the torus. These inequalities involve a quantity we call the packing number, which in turn is inspired by the geometric zero packing problems considered by Hedenmalm in the context of the asymptotic variance of the Bergman projection of a bounded function. As for the torus, we introduce analogs of polynomials (pseudopolynomials) based on the classical Weierstrass $σ$ function, and we explain how such pseudopolynomials fit in with the extended geometric Bombieri-type inequality. The sharpness of the packing number bound on the torus involves the construction of a lattice configuration on the torus for any given integer number of points. The corresponding bound for the sphere instead relies on the existence of well-conditioned polynomials in the sense of Shub and Smale.

math.CA↗

arXiv2023

Berezin density and planar orthogonal polynomials

We introduce a nonlinear potential theory problem for the Laplacian, the solution of which characterizes the Berezin density $B(z,\cdot)$ for the polynomial Bergman space, where the point $z\in\mathbb{C}$ is fixed. When $z=\infty$, the Berezin density is expressed in terms of the squared modulus of the corresponding normalized orthogonal polynomial $P$. We use an approximate version of this characterization to study the asymptotics of the orthogonal polynomials in the context of exponentially varying weights. This builds on earlier works by Its-Takhtajan and by the first author on a soft Riemann-Hilbert problem for planar orthogonal polynomials, where in place of the Laplacian we have the $\bar\partial$-operator. We adapt the soft Riemann-Hilbert approach to the nonlinear potential problem, where the nonlinearity is due to the appearance of $|P|^2$ in place of $\overline{P}$. Moreover, we suggest how to adapt the potential theory method to the study of the asymptotics of more general Berezin densities $B(z,w)$ in the off-spectral regime, that is, when $z$ is fixed outside the droplet. This is a first installment in a program to obtain an explicit global expansion formula for the polynomial Bergman kernel, and, in particular, of the one-point function of the associated random normal matrix ensemble.

math.CV↗

arXiv2023

Hyperbolic Fourier series

In this article we explain the essence of the interrelation described in [PNAS 118, 15 (2021)] on how to write explicit interpolation formula for solutions of the Klein-Gordon equation by using the recent Fourier pair interpolation formula of Viazovska and Radchenko from [Publ Math-Paris 129, 1 (2019)]. We construct explicitly the sequence in $L^1 (\mathbb{R} )$ which is biorthogonal to the system $1$, $\exp ( i πn x)$, $\exp ( i πn/ x)$, $n \in \mathbb{Z} \setminus \{0\}$, and show that it is complete in $L^1 (\mathbb{R})$. We associate with each $f \in L^1 (\mathbb{R}, (1+x^2)^{-1} d x)$ its hyperbolic Fourier series $h_{0}(f) + \sum_{n \in \mathbb{Z}\setminus \{0\}}(h_{n}(f) e^{ i πn x} + m_{n}(f) e^{-i πn / x} )$ and prove that it converges to $f$ in the space of tempered distributions on the real line. Applied to the above mentioned biorthogonal system, the integral transform given by $U_φ (x, y):= \int_{\mathbb{R}} φ(t) \exp \left( i x t + i y / t \right) d t $, for $φ\in L^{1} (\mathbb{R})$ and $(x, y) \in \mathbb{R}^{2}$, supplies interpolating functions for the Klein-Gordon equation.

math.AP↗

arXiv2022

Quasicircles and hyperbolic zero packing

We look at the work of Oleg Ivrii connected with the dimension of quasicircles for asymptotically small quasiconformality parameter $k$. We intend to make this work more easily accessible. Our main focus is the integral means spectrum associated with normalized conformal mappings of the exterior disk which have quasiconformal extensions to the whole plane with small dilatation parameter $k$. Moreover, we address the estimates from above only, not the sharpness from below.

math.CV↗

arXiv2022

Deep zero problems

We introduce a novel collection of uniqueness problems, with related sampling and interpolation issues. We call them deep zero problems, as they are concerned with local properties at a small number of given points.

math.CV↗

arXiv2021

Soft Riemann-Hilbert problems and planar orthogonal polynomials

Riemann-Hilbert problems are jump problems for holomorphic functions along given interfaces. They arise in various contexts, e.g. in the asymptotic study of certain nonlinear partial differential equations and in the asymptotic analysis of orthogonal polynomials. Matrix-valued Riemann-Hilbert problems were considered by Deift et al. in the 1990s with a noncommutative adaptation of the steepest descent method. For orthogonal polynomials on the line or on the circle with respect to exponentially varying weights, this led to a strong asymptotic expansion in the given parameters. For orthogonal polynomials with respect to planar exponentially varying weights, the corresponding asymptotics was obtained by Hedenmalm and Wennman (2017), using a technically involved construction of an invariant foliation for the orthogonality. Planar orthogonal polynomials are characterized in terms of a matrix dbar-problem (Its, Takhtajan), which we call a soft Riemann-Hilbert problem. Here, we use this perspective to offer a simplified approach based not on foliations but instead on the ad hoc insertion of an algebraic ansatz for the Cauchy potential in the soft Riemann-Hilbert problem. This allows the problem to decompose into a hierarchy of scalar Riemann-Hilbert problems along the interface, which appears as the free boundary for an obstacle problem. Inspired by microlocal analysis, the method allows for control of the solution in such a way that for real-analytic weights, the asymptotics holds in the $L^2$ sense with error $O(e^{-δ\sqrt{m}})$ in a fixed neighborhood of the closed exterior of the interface, for some constant $δ>0$, where $m\to+\infty$. Here, $m$ is the degree of the polynomial, and in terms of pointwise asymptotics, the expansion dominates the error term in the exterior domain and across the interface a distance proportional to $m^{-\frac14}$.

math.CV↗

arXiv2020

Riemann-Hilbert hierarchies for hard edge planar orthogonal polynomials

We obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain $\mathscr{D}$ with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly positive, and for any given precision $\varkappa$, the expansion holds with an $\mathrm{O}(N^{-\varkappa-1})$ 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. In fact, 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 $\partial\mathscr{D}$.

math.CV↗

arXiv2020

Planar orthogonal polynomials and boundary universality in the random normal matrix model

We show that the planar normalized orthogonal polynomials $P_{m,n}(z)$ of degree $n$ with respect to an exponentially varying planar measure $\mathrm{e}^{-2mQ}\mathrm{dA}$ enjoy an asymptotic expansion \[ P_{m,n}(z)\sim m^{\frac{1}{4}}\sqrt{ϕ_τ'(z)}[ϕ_τ(z)]^n \mathrm{e}^{m\mathcal{Q}_τ(z)}\left(\mathcal{B}_{τ, 0}(z) +m^{-1}\mathcal{B}_{τ, 1}(z)+m^{-2} \mathcal{B}_{τ,2}(z)+\ldots\right), \] as $n,m\to\infty$ while the ratio $τ=\frac{n}{m}$ is fixed. Here $\mathcal{S}_τ$ denotes the droplet, the boundary of which is assumed to be a smooth simple closed curve, and $ϕ_τ$ is a conformal mapping from the complement $\mathcal{S}_τ^c$ to the exterior disk $\Bbb{D}_\mathrm{e}$. The functions $\mathcal{Q}_τ$ and $\mathcal{B}_{τ, j}$ are bounded holomorphic functions which may be expressed in terms of $Q$ and $\mathcal{S}_τ$. We apply these results to obtain boundary universality in the random normal matrix model for smooth droplets, i.e., that the limiting rescaled process is the random process with correlation kernel \[ \mathrm{k}(ξ,η)= \mathrm{e}^{ξ\barη\,-\frac12(\lvertξ\rvert^2+\lvert η\rvert^2)} \,\mathrm{erf}\,(ξ+\barη). \] A key ingredient in the proof of the asymptotic expansion of the orthogonal polynomials is the construction of an orthogonal foliation -- a smooth flow of closed curves near $\partial\mathcal{S}_τ$, on each of which $P_{m,n}$ is appropriately orthogonal to lower order polynomials. To compute the coefficient functions, we develop an algorithm which determines the coefficients $\mathcal{B}_{τ, j}$ successively in terms of inhomogeneous Toeplitz kernel conditions. These inhomogeneous Toeplitz kernel conditions may be understood in terms of scalar Riemann-Hilbert problems.

math.CV↗

arXiv2020

Backward shift and nearly invariant subspaces of Fock-type spaces

We study the structure of the backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces $\mathcal{F}_W^p$, whose weight $W$ is not necessarily radial. We show that in the spaces $\mathcal{F}_W^p$ which contain the polynomials as a dense subspace (in particular, in the radial case) all nontrivial backward shift invariant subspaces are of the form $\mathcal{P}_n$, i.e., 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 which show that the result fails for general Fock-type spaces of larger growth.

math.CV↗

arXiv2020

Fourier uniqueness in $\mathbb{R}^4$

We show an interrelation between the uniqueness aspect of the recent Fourier interpolation formula of Radchenko and Viazovska and the Heisenberg uniqueness study for the Klein-Gordon equation and the lattice-cross of critical density, studied by Hedenmalm and Montes-Rodriguez. This has been known since 2017.

math.FA↗

arXiv2020

Bloch functions, asymptotic variance, and geometric zero packing

We study a new type of extremal problem in complex analysis, referred to as "geometric zero packing", which is the hyperbolic analogue of a problem considered by Abrikosov in the 1950s concerning Bose-Einstein condensates. We relate the corresponding minimal discrepancy density with the asymptotic variance for Bloch functions of the form "Bergman projection of bounded functions" and obtain a corresponding identity. Together with related work of Ivrii, this gives the asymptotic behavior of the universal quasiconformal integral means spectrum for small values of quasiconformality k and small exponents t. In particular, the conjectured behavior is shown to be smaller than conjectured by Prause and Smirnov, which also shows that there are no quasidisks with dimension 1+k^2, at least for small k.

math.CV↗

arXiv2020

The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps

A pair $(Γ,Λ)$, where $Γ\subset\mathbb{R}^2$ is a locally rectifiable curve and $Λ\subset\mathbb{R}^2$ is a {\em Heisenberg uniqueness pair} if an absolutely continuous (with respect to arc length) finite complex-valued Borel measure supported on $Γ$ whose Fourier transform vanishes on $Λ$ necessarily is the zero measure. Recently, it was shown by Hedenmalm and Montes that if $Γ$ is the hyperbola $x_1x_2=M^2/(4π^2)$, where $M>0$ is the mass, and $Λ$ is the lattice-cross $(α\mathbb{Z}\times\{0\}) \cup (\{0\}\timesβ\mathbb{Z})$, where $α,β$ are positive reals, then $(Γ,Λ)$ is a Heisenberg uniqueness pair if and only if $αβM^2\le4π^2$. The Fourier transform of a measure supported on a hyperbola solves the one-dimensional Klein-Gordon equation, so the theorem supplies very thin uniqueness sets for a class of solutions to this equation. The case of the semi-axis $\mathbb{R}_+$ as well as the holomorphic counterpart remained open. In this work, we completely solve these two problems. As for the semi-axis, we show that the restriction to $\mathbb{R}_+$ of the above exponential system spans a weak-star dense subspace of $L^\infty(\mathbb{R}_+)$ if and only if $0<αβ<4$, based on dynamics of Gauss-type maps. This has an interpretation in terms of dynamical unique continuation. As for the holomorphic counterpart, we show that the above exponential system with $m,n\ge0$ spans a weak-star dense subspace of $H^\infty_+(\mathbb{R})$ if and only if $0<αβ\le1$. To obtain this result, we need to develop new harmonic analysis tools for the dynamics of Gauss-type maps, related to the Hilbert transform. Some details are deferred to a separate publication.

math.DS↗

arXiv2019

Off-spectral analysis of Bergman kernels

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 behaviour 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 planar measures on the complex plane. We obtain uniform asymptotic expansions of root functions, which are essentially normalized partial Bergman kernels at an off-spectral point, valid in the entire off-spectral component and protruding into the spectrum as well, which allows us to show error function transition behaviour of the original kernel along the interface. In contrast, previous work on asymptotic expansions of Bergman kernels is typically local, and valid only in the bulk region of the spectrum.

math.CV↗

arXiv2019

Gaussian analytic functions and operator symbols of Dirichlet type

Let $\calH$ be a separable infinite-dimensional $\C$-linear Hilbert space, with sesquilinear inner product $\langle\cdot,\cdot\rangle_\calH$. Given any two orthonormal systems $x_1,x_2,x_3,\ldots$ and $y_1,y_2,y_3,\ldots$ in $\calH$, we show that the weighted sums \[ S(l):=\sum_{j,k:j+k=l}\bigg(\frac{l}{jk}\bigg)^{\frac12}\, \langle x_j,y_k\rangle_{\calH} \] satisfy $|S(l)|^2\lessapprox2$ holds in an average sense. A construction due to Zachary Chase shows that this would not be true if the number $2$ is replaced by the smaller number $1.72$. In the construction, the system $y_1,y_2,y_3,\ldots$ is a permutation of the system $x_1,x_2,x_3,\ldots$. We interpret our bound in terms of the correlation $\expect Φ(z)Ψ(z)$ 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 Möbius invariant. The study of the correlations $\expectΦ(z)Ψ(z)$ leads us to introduce a new space, the \emph{mock-Bloch 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 $\expectΦ(z)Ψ(w)$ 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.

math.CV↗