7. Almost Summability of Taylor Series

In Chap. 4, we applied the generalized Lototski or [F, d _n ]-summability to study the regions in which this method sums a Taylor series to the analytic continuation of the function which it represents. In the applications of summability theory to function theory it is important to know the region in which the sequence of partial sums of the geometric series is A-summable to \(1/(1 - z)\) for a given matrix A. The well-known theorem of Okada [78] gives the domain in which a matrix A = (a _jk ) sums the Taylor series of an analytic function f to one of its analytic continuations, provided that the domain of summability of the geometric series to \(1/(1 - z)\) and the distribution of the singular points of f are known. In this chapter, we replace the [F, d _n ]-matrix or the general Toeplitz matrix by almost summability matrix to determine the set on which the Taylor series is almost summable to f(z) (see [51]). Most of the basic definitions and notations of this chapter are already given in Chap. 4; in fact, this chapter is in continuation of Chap. 4.