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Chains of reproducing kernel Hilbert spaces generated by unimodular functions

Masatoshi Suzuki

TL;DR

This work builds a general framework to generate a chain of reproducing kernel Hilbert spaces from a unimodular function via a lacunary canonical system driven by a positive semidefinite $2\times2$ matrix $H(t)$. It extends Su19_1 by enabling non-diagonal structure Hamiltonians and by formulating the construction through an antilinear operator $\mathsf K$, connecting to de Branges spaces, model spaces, and Krein’s string theory. The paper establishes explicit representations for the kernels, shows how the chain aligns with structure Hamiltonians of de Branges spaces when appropriate inner or meromorphic inner functions are used, and provides rich examples from Paley--Wiener spaces, Hankel transforms, and Selberg-class $L$-functions. It also situates the approach within inverse spectral theory, offering conditional but broad methods for recovering a Hamiltonian from a given generator of a de Branges space, with potential implications for analytic number theory and spectral problems.

Abstract

We present a method to construct a chain of reproducing kernel Hilbert spaces controlled by a first-order system of differential equations from a given unimodular function satisfying several conditions. One of the applications of that method is a conditional but richly general solution to the inverse problem of recovering the structure Hamiltonian from a given de Branges space.

Chains of reproducing kernel Hilbert spaces generated by unimodular functions

TL;DR

This work builds a general framework to generate a chain of reproducing kernel Hilbert spaces from a unimodular function via a lacunary canonical system driven by a positive semidefinite matrix . It extends Su19_1 by enabling non-diagonal structure Hamiltonians and by formulating the construction through an antilinear operator , connecting to de Branges spaces, model spaces, and Krein’s string theory. The paper establishes explicit representations for the kernels, shows how the chain aligns with structure Hamiltonians of de Branges spaces when appropriate inner or meromorphic inner functions are used, and provides rich examples from Paley--Wiener spaces, Hankel transforms, and Selberg-class -functions. It also situates the approach within inverse spectral theory, offering conditional but broad methods for recovering a Hamiltonian from a given generator of a de Branges space, with potential implications for analytic number theory and spectral problems.

Abstract

We present a method to construct a chain of reproducing kernel Hilbert spaces controlled by a first-order system of differential equations from a given unimodular function satisfying several conditions. One of the applications of that method is a conditional but richly general solution to the inverse problem of recovering the structure Hamiltonian from a given de Branges space.

Paper Structure

This paper contains 31 sections, 28 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Construction of the first-order differential systems
  4. Chains of reproducing kernel Hilbert spaces
  5. Specialization to de Branges spaces
  6. Comparison with Kreı̆n's method
  7. Comparison with previous work.
  8. Examples
  9. Paley--Wiener spaces
  10. One dimensional de Branges space
  11. De Branges spaces related to Hankel transform of order zero
  12. De Branges spaces arising from $L$-functions in the Selberg class
  13. Proof of Theorem \ref{['thm_2_1']}
  14. Extension of $\mathsf{K}$ to $S'(\mathbb{R})$
  15. Proof of (1), (2), (3), (4)
  16. ...and 16 more sections

Key Result

Proposition 2.1

For a unimodular function $u$ in $L_{\rm loc}^1(\mathbb{R})$, the map $\mathsf{K}=\mathsf{K}_u$ is an antilinear isometric involution on $L^2(\mathbb{R})$, in other words, $\mathsf{K}$ is an abstract conjugation on $L^2(\mathbb{R})$. Hence, in particular, $\mathsf{K}$ is self-adjoint: $\mathsf{K}=\m

Theorems & Definitions (50)

  • Proposition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Proposition 4.1
  • proof
  • ...and 40 more