The Inverse of a Two-level Positive Definite Toeplitz Operator Matrix

The inverse of nonsymmetric two-level Toeplitz operator matrices

“…Applying lemma 5 to the factorization (39), yields that 11 , B = S 31 and B 1 = R 31 , we conclude that…”

“…In such a situation, one first needs to find a reasonable guess for these polynomials. In the case that P = Q and R = S (the positive definite case), we outlined a procedure in [11]. This procedure does not easily generalize here, as it involves taking the logarithm of a positive symbol.…”

“…(8) In the scalar case we have that t k ∈ C, while in the operator case we have that t k is a Hilbert space operator. Recently, a formula for the inverse of a positive definite two-level Toeplitz operator matrix was presented in [11]. Two-level Toeplitz matrices appear for instance when one solves a Helmholtz type equation by discretization (see, e.g., [13]).…”

The autoregressive filter problem for multivariable degree one symmetric polynomials

“…Our method to determine other relations rely on the formulas obtained in Proposition 2.7. The inverses in this proposition are obtained via [18,Theorem 1.1] and the ability to find a formula for the inverse of a tridiagonal infinite Toeplitz matrix. If we want to use this method to obtain expressions for Fourier coefficients beyond the indices {−1, 0, 1} 3 , we will need to be able to find manageable expressions for (part of) the inverse of more involved infinite (block) Toeplitz matrices, which is a challenge.…”

“…To prove the first expression for c 011 , by(18) it suffices to prove2π 0 2r cos t + 1 r 2 − 2r cos t − 3 dt = (21) −(r + 3)(r − 1) 3 E( 16r (r+3)(r−1) 3 ) + (r 4 − 2r 2 − 15)K( 16r (r+3)(r−1) 3 ) − 4(r − 3)(r + 1)Π( 4r (r+3)(r−1) , 16r (r+3)(r−1) 3 ) r(r − 1) (r + 3)(r − 1)…”