On uniform convergence of the inverse Fourier transform for differential equations and Hamiltonian systems with degenerating weight

Abstract

We study pseudospectral and spectral functions for Hamiltonian system 𝐽𝑦 ′ − 𝐵(𝑡) = 𝜆Δ(𝑡)𝑦 and differential equation 𝑙[𝑦] = 𝜆Δ(𝑡)𝑦 with matrix-valued coefficients defined on an interval  = [𝑎, 𝑏) with the regular endpoint 𝑎. It is not assumed that the matrix weight Δ(𝑡) ≥ 0 is invertible a.e. on . In this case a pseudospectral function always exists, but the set of spectral functions may be empty. We obtain a parametrization 𝜎 = 𝜎 𝜏 of all pseudospectral and spectral functions 𝜎 by means of a Nevanlinna parameter 𝜏 and single out in terms of 𝜏 and boundary conditions the class of functions 𝑦 for which the inverse Fourier transform 𝑦(𝑡) = ∫ ℝ 𝜑(𝑡, 𝑠) 𝑑𝜎(𝑠)ŷ(𝑠) converges uniformly. We also show that for scalar equation 𝑙[𝑦] = 𝜆Δ(𝑡)𝑦 the set of spectral functions is not empty. This enables us to extend the Kats-Krein and Atkinson results for scalar Sturm-Liouville equation −(𝑝(𝑡)𝑦 ′ ) ′ + 𝑞(𝑡)𝑦 = 𝜆Δ(𝑡)𝑦 to such equations with arbitrary coefficients 𝑝(𝑡) and 𝑞(𝑡) and arbitrary non trivial weight Δ(𝑡) ≥ 0.