Matrices concentrated in the origin and dome applications in matricial harmonic analysis
by
Nicolae Popa
Institute of Mathematics, Bucharest, Romania

If A is an infinite matrix, then we denote the k-th diagonal, k in Z by A_k and call it "The Fourier coefficient of A". With this convention it is a natural analogy between the periodical functions defined on the one dimensional torus and infinite matrices. We are interested to develop a theory parallel to classical harmonic analysis and in order to do this we intreoduce some new Banach spaces as C(l₂) "the space of all continuous matrices", the L¹(l₂) "the space of all summable matirces" or L²(l₂) "the space of all 2-summable matrices". For instance the space C(l₂) is the closure of Cesaro sums associated to psrtial finite sums of diagonals associated to a matrix A in B(l₂). In this theory the Schur product of matrices plays an important role and can be viewed as an analogue of the usual convolution. It is also possible to intyroduce the notion of a value for a matrix and the notion of support and apply them to give some expansions of infinite matrices in series of more simple matrices. All the results extend to matrices the known results from classical harmonic analysis.

Date received: May 10, 2002