Inspired by the Abel-Goncharov interpolation problem, we consider a collection of uniqueness problems, with related interpolation and sampling issues. We call them deep zero problems, as they are concerned with local properties at a small number of given points.

We introduce a nonlinear potential theory problem for the Laplacian, the solution of which characterizes the Berezin density B(z,·) for the polynomial Bergman space, where the point z ∈ C is fixed. When z = ∞, 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 ∂̄-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 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.

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.

Riemann-Hilbert problems are jump problems for holomorphic functions along given interfaces. They arise in various contexts, for example, 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 exponentially varying weights in the plane, the corresponding asymptotics was obtained by Hedenmalm and Wennman (2017), based on the technically involved construction of an invariant foliation for the orthogonality. Planar orthogonal polynomials are characterized in terms of a certain matrix 𝜕-problem (Its, Takhtajan), which we refer to as 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 (the free boundary for a related 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 L² sense with error O(e^{−𝛿√𝑚}). in a fixed neighborhood of the closed exterior of the interface, for some constant 𝛿 > 0, where 𝑚 → +∞. 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 (by a distance proportional to 𝑚^−1/4). In particular, the zeros of the orthogonal polynomial are located in the interior of the spectral droplet, away from the droplet boundary by a distance at least proportional to 𝑚^−1/4.

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.

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.

The impetus to this work is the need to characterize when the system {ϕ1m, ϕ2n} where m, n = 0, 1, 2,…., is complete in the weak-star topology of H∞ on the unit disk (or the half-plane). Here, ϕ1 and ϕ2 are two atomic inner functions, of the form ϕ1(z)=exp(λ1z+1z−1)andϕ2(z)=exp(λ2z−1z+1), where λ1, λ2 are positive reals. Our main result asserts that the system of non-negative integral powers {ϕ1m, ϕ2n} is weak-star dense in H∞ of the unit disk if and only if λ1λ2 ≤ π2. In earlier work in the L∞ setting on the unit circle all the integer powers were considered, and the corresponding result was obtained (Hedenmalm and Montes-Rodríguez, 2011). The approach was to first transfer the completeness problem to the real line via the Cayley transform, and to then connect with the dynamics of Gauss-type transformations on the interval [−1, 1]. Indeed, the nonexistence of nontrivial finite absolutely continuous invariant measures for the Gauss-type map was the key ingredient of the analysis. Moreover, it was shown that the answer to the completeness problem has striking consequences for the Klein-Gordon equation. Here, the analysis is much more subtle as a result of the required finer topology. To appreciate the difference, we observe that the standard quotient space L1/H1 used as the predual of H∞ is not appropriate for our purposes. Instead we model the predual in the real way, as L1 plus the Hilbert transform of L1, in analogy with the decomposition of BMO. The next step is to analyze carefully the iterates of the transfer operator applied to the Hilbert kernel. The approach involves a splitting of the Hilbert kernel which is induced by the transfer operator. The careful analysis of this splitting involves detours to the Hurwitz zeta function as well as to the theory of totally positive matrices. 

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).

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&gt;0}, and takes positive values along i R&gt; 0, the positive imaginary axis, where R&gt; 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&gt; 0. We derive an integral representation of log Θ3(z) when z belongs to the hyperbolic quadrilateral F□||:={z∈C|Imz&gt;0,-1≤Rez≤1,|2z-1|&gt;1,|2z+1|&gt;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.