In this paper we discuss the asymptotic distribution of the approximation numbers of the finite sections for a Toeplitz operator T (a) ∈ L( p ), 1 < p < ∞, where a is a piecewise continuous function on the unit circle. We prove that the behavior of the approximation numbers of the finite sections T n (a) = P n T (a)P n depends heavily on the Fredholm properties of the operators T (a) and T (ã) (ã(t) = a(1/t)). In particular, if the operators T (a) and T (ã) are Fredholm on p , then the approximation numbers of T n (a) have the so-called k-splitting property. But, in contrast with the case of continuous symbols, the splitting number k is in general larger than dim ker T (a) + dim ker T (ã).
In this paper we present polynomial collocation methods and their modifications for the numerical solution of Cauchy singular integral equations over the interval [−1, 1]. More precisely, the operators of the integral equations have the form A = aI + bµ −1 SµI with piecewise continuous coefficients a and b, and with a Jacobi weight µ. Using the splitting property of the singular values of the collocation methods, we obtain enough stable approximate methods to find the least square solution of our integral equation. Moreover, the modifications of the collocation methods enable us to compute kernel and cokernel dimensions of operators from a C * -algebra, which is generated by operators of the Cauchy singular integral equations.
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