The Mathematical Legacy of Leon Ehrenpreis

In the years 1952–1953, I had finished my studies at École Normale Supérieure, and I had a position of research in CNRS, under the supervision of Laurent Schwartz. His book on the theory of distributions had been recently published; this book and his paper on mean periodic functions were full of open problems on linear differential equations, especially with constant coefficients and convolution equations. I was mainly interested in the problem of “elementary solutions”: given a differential polynomial P with constant coefficients, does there exist a distribution f on ℝ ⁿ verifying Pf=δ, δ the Dirac measure?

What made Ehrenpreis’ mathematics so unique was his bold approach to classical problems, and his interest in finding an overarching and unifying framework for a variety of apparently unrelated problems. In this note I will try to highlight this characteristic, by looking at some of Ehrenpreis’ papers which are not, strictly speaking, connected with either the Fundamental Principle or the Radon Transform.

Leon Ehrenpreis was an outstanding world-class mathematician and a wonderful, warm person. I had a privilege to consider myself his friend for the last two decades. It is hard to do justice to his manifold mathematics and personality, but I will try to at least add some recollections to this tribute volume.

Under what conditions can one conclude that a continuous function on a plane domain Ω is holomorphic, given that its restrictions to a collection of Jordan curves in Ω which cover Ω admit holomorphic extensions? We survey progress on this problem over the past 40 years, with an emphasis on recent results.

We consider a class of weighted plane generalized Radon transforms Rf(γ)=∫f(x,u(ξ,η,x))m(ξ,η,x) dx, where the curve γ=γ _(ξ,η) is defined by y=u(ξ,η,x), and m(ξ,η,x) is a given positive weight function. We prove local injectivity for this transform across a given curve γ ⁰ near a given point (x ⁰,y ⁰) on γ ⁰ for classes of curves and weight functions that are invariant under arbitrary smooth coordinate transformations in the plane.

We consider variations on the Pompeiu transform for the Heisenberg group H ⁿ and focus on cases where the transform is known to be injective; in particular the cases of averages over a sphere and a ball, or two balls of appropriate radii. In these cases we develop a method which provides for the reconstruction of the function f from its integrals.

In addition, we consider these issues in connection with the Weyl calculus and group Fourier transform. We furthermore explore issues of convergence for this method of deconvolution and related issues of size of the Gelfand transform near the zero sets. Finally, given a set of deconvolvers which work for Euclidean space C ⁿ , we consider the problem of how to extend the deconvolution to the Heisenberg group, and we provide the extension in special cases.

In this note we consider the Weierstrass points for the linear space of meromorphic functions on a compact Riemann surface whose divisors are multiples of \(\frac{1}{P_{0}^{\alpha}P_{1}\cdots P_{g-1}}\), where P _i are points of the surface, and α is a positive integer for which there is no holomorphic differential on the surface whose divisor is a multiple of \(P_{0}^{\alpha}P_{1}\cdots P_{g-1} \). Thus the dimension of our linear space is precisely α.

The Weierstrass points for our space are those points Q≠P _i for which there is a function in the space which vanishes to order at least α at the point Q. Thus the Weierstrass points are all zeros of the Wronskian determinant of a basis for our space, and the weight of the Weierstrass point is the order of the zero.

We show that all the Weierstrass points are zeros of the Riemann theta function \(\theta(\alpha \varPhi _{P_{0}}(P)-e) \) on the surface where \(e=\varPhi _{P_{0}}(P_{1}\cdots P_{g-1})+K_{P_{0}}\). The question we investigate is whether the order of the zero of the theta function agrees with the order of the zero of the Wronskian. We prove that this is so at least in the case of zeros of order k=1,2.

We consider the Radon transform along lines in an n-dimensional vector space over the two-element field. It is well known that this transform is injective and highly overdetermined. We classify the minimal collections of lines for which the restricted Radon transform is also injective. This is an instance of I.M. Gelfand’s admissibility problem. The solution is in stark contrast to the more uniform cases of the affine hyperplane transform and the projective line transform, which are addressed in other papers (Feldman and Grinberg in Admissible Complexes for the Projective X-Ray Transform over a Finite Field, preprint, 2012; Grinberg in J. Comb. Theory, Ser. A 53:316–320, 1990). The presentation here is intended to be widely accessible, requiring minimum background.

We first show that the WKB-theoretic canonical form of an M2P1T (merging two poles and one turning point) Schrödinger equation is given by the algebraic Mathieu equation. We further show that, in analyzing the structure of WKB solutions of a Mathieu equation near fixed singular points relevant to simple poles of the equation, we can focus our attention on the pole part of the equation so that we may reduce it to the Legendre equation. The Borel transformation of WKB-theoretic transformations thus obtained gives rise to microdifferential relations, which lead to the microlocal analysis of the Borel transformed WKB solutions of an M2P1T equation near their fixed singular points. The fully detailed account of the results will be given in Kamimoto et al. (Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point—its relevance to the Mathieu equation and the Legendre equation, 2011).

We study the automorphism group action on a bounded domain in ℂ ⁿ . In particular, we consider boundary orbit accumulation points, and what geometric properties they must have. These properties are formulated in the language of Levi geometry.

For case (a), we characterize the Radon transform as a Fourier integral operator associated to a fold and blowdown. This has implications on how the operator adds singularities, how backprojection reconstructions will show those singularities, and in comparison of the strengths of the original and added singularities in a Sobolev sense.

For case (b), we show that this Radon transform has similar structure to case (a): it is a Fourier integral operator associated to a fold and blowdown. This case is related to previous results of authors one and three. We characterize singularities that are added by the reconstruction operator, and we present reconstructions from the authors’ algorithm that illustrate the microlocal properties.

The article provides a brief survey of the mathematics of newly being developed so-called “hybrid” (also called “multi-physics” or “multi-wave”) imaging techniques.

This paper concerns entire and meromorphic solutions to functional and nonlinear partial differential equations of the form a ₁ f ^m +a ₂ g ⁿ =a ₃ with function coefficients a _k , k=1,2,3, where f and g are unknown functions or partial derivatives of an unknown function. We will discuss some recent results and also give, among other things, some new results on these equations.

For transverse parallelisms without first integral, I give a result similar to the Galois correspondence in the differential Galois theory of Kolchin.

Carathéodory’s theorem for compact convex sets K⊂ℝ ^m shows that every point x of K lies in the convex hull of m+1 extreme points of K; that is, in the m-simplex with vertices at m+1 extreme points. However, it need not be the case that if x is a positive distance away from the boundary of K, then x is a positive distance away from the boundary of one of these simplices. Here, we show that if K has only finitely many extreme points, then there are a finite set F⊂∂K and a constant c>0 such that if x∈K is of distance δ>0 from the boundary of K, then x belongs to one of the m-simplices with vertices from F and is of distance at least cδ from its boundary.

In this paper we suggest that, under suitable conditions, supervised learning can provide the basis to formulate at the microscopic level quantitative questions on the phenotype structure of multicellular organisms. The problem of explaining the robustness of the phenotype structure is rephrased as a real geometrical problem on a fixed domain. We further suggest a generalization of path integrals that reduces the problem of deciding whether a given molecular network can generate specific phenotypes to a numerical property of a robustness function with complex output, for which we give heuristic justification. Finally, we use our formalism to interpret a pointedly quantitative developmental biology problem on the allowed number of pairs of legs in centipedes.

In this paper we expand on some ideas originally put forward by Ehrenpreis in his monograph (Fourier Analysis in Several Complex Variables, Wiley Interscience, New York, 1970), and we show how to extend approximate solutions to the Cauchy–Fueter system in n variables.

We study, from different points of view, the two series \(\chi_{+}(z)= \sum_{n\geq0} z^{2^{n}}\) and \(\chi_{-}(z)= \sum_{n\geq0}(-1)^{n} z^{2^{n}}\). We show that the first series is related to the Jacobi theta function and the second is related to the Dedekind eta function and to the modular curve X ₀(14). We also present another approach to a celebrated identity of Hardy.

For a class of PT-symmetric operators with small random perturbations, the eigenvalues obey Weyl asymptotics with probability close to 1. Consequently, when the principal symbol is nonreal, there are many nonreal eigenvalues.

Let \(\widetilde{M}\) be a complex manifold of complex dimension n+k. We say that the functions u ₁,…,u _k and the vector fields ξ ₁,…,ξ _k on \(\widetilde{M}\) form a complex gradient system if ξ ₁,…,ξ _k ,Jξ ₁,…,Jξ _k are linearly independent at each point \(p\in \widetilde{M}\) and generate an integrable distribution of \(T \widetilde{M}\) of dimension 2k and du _α (ξ _β )=0, d ^c u _α (ξ _β )=δ _αβ for α,β=1,…,k. We prove a Cauchy theorem for such complex gradient systems with initial data along a CR-submanifold of type (n,k). We also give a complete local characterization for the complex gradient systems which are holomorphic and abelian, which means that the vector fields \(\xi _{\alpha }^{c}=\xi _{\alpha }-iJ \xi _{\alpha }\), α=1,…,k, are holomorphic and satisfy \([\xi _{\alpha }^{c},\bar{\xi _{\beta }^{c}} ]=0\) for each α,β=1,…,k.

In the present paper, we describe the recent approach to residue currents by Andersson, Björk, and Samuelsson (Andersson in Ann. Fac. Sci. Toulouse Math. Sér. 18(4):651–661, 2009; Björk in The Legacy of Niels Henrik Abel, pp. 605–651, Springer, Berlin, 2004; Björk and Samuelsson in J. Reine Angew. Math. 649:33–54, 2010), focusing primarily on the methods inspired by analytic continuation (that were initiated in a quite primitive form in Berenstein et al. in Residue Currents and Bézout Identities. Progress in Mathematics, vol. 114, Birkhäuser, Basel, 1993). Coleff–Herrera currents (with or without poles) play indeed a crucial role in Lelong–Poincaré-type factorization formulas for integration currents on reduced closed analytic sets. As revealed by local structure theorems (which can also be understood as global when working on a complete algebraic manifold due to the GAGA principle), such objects are of algebraic nature (antiholomorphic coordinates playing basically the role of “inert” constants). Thinking about division or duality problems instead of intersection ones (especially in the “improper” setting, which is certainly the most interesting), it happens then to be necessary to revisit from this point of view the multiplicative inductive procedure initiated by Coleff and Herrera (Lecture Notes in Mathematics, vol. 633, Springer, Berlin, 1978), this being the main objective of this presentation. In homage to the pioneering work of Leon Ehrenpreis, to whom we are both deeply indebted, and as a tribute to him, we also suggest a currential approach to the so-called Nœtherian operators that remain the key stone in various formulations of Leon’s Fundamental Principle.

In this paper we study the question when a linear partial differential operator P(D) with constant coefficients admits a continuous linear right inverse in the space A(ℝ ⁿ ) of real analytic functions on ℝ ⁿ (or, more generally, in A(Ω) where Ω is a open subset of ℝ ⁿ ). To obtain a necessary condition, we investigate when P(D) admits solvability “with real analytic parameter” in A(Ω) and solve it completely for convex Ω, using a different approach from the one used in Domański (Funct. Approx. 44:79–109, 2011). To obtain a sufficient condition, we show that the global real analytic Cauchy problem is solvable if and only if the principal part of P(D) is hyperbolic. In this way we get a complete solution of our main problem for A(ℝ²) and, in the homogeneous case, for A(Ω) where Ω is the open unit ball in ℝ ⁿ .

Trace formulas (Lagrange, Jacobi–Kronecker, Bergman–Weil) play a key role in division problems in analytic or algebraic geometry (including arithmetic aspects, see, for example, Berenstein and Yger in Am. J. Math. 121(4):723–796, 1999). Unfortunately, they usually hold within the restricted frame of complete intersections. Besides the fact that it allows one to carry local or semi-global analytic problems to a global geometric setting (think about Crofton’s formula), averaging the Cauchy kernel (from ℂ ⁿ ∖{z ₁…z _n =0}⊂ℙ ⁿ (ℂ)), in order to get the Bochner–Martinelli kernel (in ℂ ⁿ⁺¹∖{0}⊂ℙ ⁿ⁺¹(ℂ)=ℂ ⁿ⁺¹∪ℙ ⁿ (ℂ)), leads to the construction of explicit candidates for the realization of Grothendieck’s duality, namely BM residue currents (Passare et al. in Publ. Mat. 44:85–117, 2000; Andersson in Bull. Sci. Math. 128(6):481–512, 2004; Andersson and Wulcan in Ann. Sci. École Norm. Super. 40:985–1007, 2007), extending thus the cohomological incarnation of duality which appears in the complete intersection or Cohen–Macaulay cases. We will recall here such constructions and, in parallel, suggest how far one could take advantage of the multiplicative inductive construction introduced by Coleff and Herrera (Lecture Notes in Mathematics, vol. 633, Springer, Berlin, 1978), by relating it to the Stückrad–Vogel algorithm developed in Stückrad and Vogel (Queen Pap. Pure Appl. Math. 61:1–32, 1982), Tworzewski (Ann. Pol. Math. 62:177–191, 1995), Andersson et al. (arXiv:​1009.​2458, 2010) toward improper intersection theory. Results presented here were initiated all along my long-term collaboration with Carlos Berenstein. To both of us, the mathematical work of Leon Ehrenpreis certainly remained a constant and very stimulating source of inspiration. This presentation relies also deeply on my collaboration over the past years with M. Andersson, H. Samuelsson, and E. Wulcan in Göteborg.