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TAYLOR & FRANCIS LTD
DOI: 10.1080/00036811.2022.2030725
关键词
Hilbert transform; modified Hilbert transform; compact perturbation
This article investigates the relationship between the Hilbert transform and modified Hilbert transform for time-dependent partial differential equations. It proves that the two transformations differ by a compact perturbation. This result is important for dealing with space-time variational formulations of such equations and implementing related numerical methods.
The Hilbert transform H is a useful tool in the mathematical analysis of time-dependent partial differential equations in order to prove coercivity estimates in anisotropic Sobolev spaces in case of a bounded spatial domain Omega, but an infinite time interval (0, infinity). Instead, a modified Hilbert transform H-T can be used if we consider a finite time interval (0, T). In this note we prove that the classical and the modified Hilbert transformations differ by a compact perturbation, when a suitable extension of a function defined on a bounded time interval (0, T) onto R is used. This result is important when we deal with space-time variational formulations of time-dependent partial differential equations, and for the implementation of related space-time finite and boundary element methods for the numerical solution of parabolic and hyperbolic equations with the heat and wave equations as model problems, respectively.
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