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Differential Equations

, Volume 55, Issue 5, pp 609–619 | Cite as

Uniform Estimates of Remainders in Spectral Analysis of Linear Differential Systems

  • A. M. SavchukEmail author
Ordinary Differential Equations
  • 5 Downloads

Abstract

We study the problem of estimating the expression Υ(λ) = sup{|∫0xf(t)eiλω(t)dt|: x ∈ [0, 1]}, where the derivative of the function ω(t) is positive almost everywhere on [0, 1]. In particular, for fLp[0, 1], p ∈ (1, 2], we prove the estimate ∥Υ(λ)∥ Lq(ℝ) ≤ CfLp, where 1/p + 1/q = 1. The same estimate is obtained in the space Lq(), where is an arbitrary Carleson measure in the open upper half-plane ℂ+. In addition, we estimate more complicated expressions like Υ(λ) that arise when studying the asymptotics of fundamental solution systems for systems of the form y′ = λρ(x)By +A(x)y +C(x, λ)y as |λ| →∞in appropriate sectors of the complex plane.

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Notes

Acknowledgements

This work was supported by a grant from the President of the Russian Federation for support of leading academic schools, project no. NSh-6222.2018.1 “Modern problems in spectral theory, theory of approximations, and harmonic analysis.”

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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

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