Um estudo da invertibilidade do operador de convolucao num intervalo limitado

A study of the invertibility of the finite interval convolution operator is conducted using several different approaches. By establishing an equivalence to an operator acting on a space of functions defined on the real axis and using the corona theorem, the invertibility of the operator is studied for kernel functions for which a solution of the correspondent homogeneous Riemann-Hilbert problem is known thus enabling the study of invertibility for a class of symbols which includes the products of exponential polynomials by functions with Fourier spectra in (0,1). Here, the main purpose is to enlarge the number of classes of symbols for which the invertibility of the operator can be completely studied. By establishing an equivalence to an operator acting on a space of functions defined on a system of two straight lines parallel to the real axis, the same ideas (the corona theorem) are used to study the invertibility for a wide class of kernel functions which satisfy a condition of the same type as above. Using this method include the products of exponential polynomials with Fourier spectra in (-1,1) by functions with Fourier spectra in (-1,1) but under conditions quite different from the previous class. These results are extended to Bessel potential spaces since these spaces often appear in the applications. As a last approach an algebraic perspective of the convolution operator is given by establishing an equivalence to an operator which is an element on an algebra generated by three idempotents with relations binding them which admits a matricial symbol. Here the main purpose is to give a different approach to the study of this operator which can, in the near future, allow for different classes to be studied and, above all, to give an example of an algebra with symbol which may be of some interest in the study of other operators or other operator algebrasAvailable from Fundacao para a Ciencia e a Tecnologia, Servico de Informacao e Documentacao, Av. D. Carlos I, 126, 1249-074 Lisboa, Portugal / FCT - Fundação para o Ciência e a TecnologiaSIGLEPTPortuga