Skip to main content
SpringerLink
Log in

A new approach to the convolution operator on a finite interval

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

The convolution operator on a finite interval defined on a space ofL 2 functions is studied by relating it to a singular integral operator acting on a space of functions defined on a system of two parallel straight lines in the complex plane ℂ. The approach followed in the paper applies both to the case where the Fourier transform of the kernel functions is anL ∞ function and to the case where the kernel function is periodic, thus yielding a unified treatment of these two classes of kernel functions. In the non-periodic case it is possible, for a special class of kernel functions, to study the invertibility property of the operator giving an explicit formula for the inverse. An example is presented and generalizations are suggested.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. A. Bastos, A. F. dos Santos; Convolution equations of the first kind on a finite interval in Sobolev spaces, Integr. Equat. Oper. Th., 13, (1990) 638–659.

    Google Scholar 

  2. M. A. Bastos, A. F. dos Santos; Convolution operators on a finite interval with periodic kerned-Fredholm property and invertibility, Integr. Equat. Oper. Th., 16, (1993), 186–223.

    Google Scholar 

  3. M. A. Bastos, A. F. dos Santos, R. Duduchava; Finite interval Convolution Operators on the Bessel potential spacesH 3p , Math. Nachr., 173, (1975), 49–63.

    Google Scholar 

  4. A. Kuijper, I. Spitkovski; On convolution equations with semi-almost periodic symbols on a finite interval, Integr. Equat. Oper. Th., 16, (1993), 530–538.

    Google Scholar 

  5. Y. Karlovich, I. Spitkovski; Factorization of almost periodic matrix-valued functions and the Noether theory for certain classes of equations of convolution type, Math. USSR Izvestia, 34, 2, (1990), 281–316.

    Google Scholar 

  6. W. Rudin; Real and Complex Analysis, McGraw-Hill, 1974.

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1096, Lisboa Codex, Portugal

    P. A. Lopes & A. F. dos Santos

Authors
  1. P. A. Lopes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. F. dos Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lopes, P.A., dos Santos, A.F. A new approach to the convolution operator on a finite interval. Integr equ oper theory 26, 460–475 (1996). https://doi.org/10.1007/BF01191247

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01191247

Keywords

Search

Navigation