Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?

The problem of computing oscillatory integrals with general oscillators is considered. We employ a Filon-type method, where the interpolation basis functions are chosen in such a way that the moments are in terms of elementary functions and the oscillator only. This allows us to evaluate the moments rapidly and easily without needing to engage hypergeometric functions. The proposed basis functions form a Chebyshev set for any oscillator function even if it has some stationary points in the integration interval. This property enables us to employ the Filon-type method without needing any information about the stationary points if any. Interpolation by the proposed basis functions at the Fekete points (which are known as nearly optimal interpolation points), when combined with the idea of splines, leads to a reliable convergent method for computing the oscillatory integrals. Our numerical experiments show that the proposed method is more efficient than the earlier ones with the same advantages.     

Positive Solutions of the Equation (t) = −c(t)x(t − τ) in the Critical Case

The equation (t) − c(t)x(t − τ) is considered in the critical case. For it, the asymptotic behavior of dominant and subdominant solutions is studied. A generalization is made and connections with known results are discussed.