Spectral Distribution in the Eigenvalues Sequence of Products of g-Toeplitz Structures

Abstract: Starting from the definition of an

n i m e s n

g

-Toeplitz matrix,

T_{n, g} (u) = l e f t [w i d e h a t u_{r - g s} i g h t]_{r, s = 0}^{n - 1},

where

g

is a given nonnegative parameter,

w i d e h a t u_{k}

is the sequence of Fourier coefficients of the Lebesgue integrable function

u

defined over the domain

m a t h b b T = (- p i, p i]

, we consider the product of

g

-Toeplitz sequences of matrices,

T_{n, g} (f_{1}) T_{n, g} (f_{2}),

which extends the product of Toeplitz structures,

T_{n} (f_{1}) T_{n} (f_{2}),

in the case where the symbols

f_{1}, f_{2} i n L^{i n f t y} (m a t h b b T) .

Under suitable assumptions, the spectral distribution in the eigenvalues sequence is completely characterized for the products of

g

-Toeplitz structures. Specifically, for

g g e q 2

our result shows that the sequences

T_{n, g} (f_{1}) T_{n, g} (f_{2})

are clustered to zero. This extends the well-known result, which concerns the classical case (that is,

g = 1

) of products of Toeplitz matrices. Finally, a large set of numerical examples confirming the theoretic analysis is presented and discussed.