L. P. Castro, R. C. Guerra, and N. M. Tuan
abstract:
We present a detailed study of structural properties for certain algebraic
operators generated by the Fourier transform and a reflection. First, we focus
on the determination of the characteristic polynomials of such algebraic
operators, which, e.g., exhibit structural differences when compared with those
of the Fourier transform. Then, this leads us to the conditions that allow one
to identify the spectrum, eigenfunctions, and the invertibility of this class of
operators. A Parseval type identity is also obtained, as well as the solvability
of integral equations generated by those operators. Moreover, new convolutions
are generated and introduced for the operators under consideration.
Mathematics Subject Classification: 42B10, 43A3, 44A20, 47A05
Key words and phrases: Characteristic polynomials, Fourier transform, reflection, algebraic integral operators, invertibility, spectrum, integral equation, Parseval identity, convolution