Let A (u) be of bounded variation over every finite interval of the nonnegative real axis, and let fJoe-u8 dA (u) be summable I C, k I for a given integer k ? 0 and a given s whose real part is negative. Then it is known that the function R(k, w) = (1/r(k+1)) *fw' (u -w)k dA (u) (which certainly exists in the I C, k sense by a well-known summability-factor theorem) satisfies e-wsw-kR(k, w) =o(1) I C, ol (w-> oo). In this paper we extend the above result by showing that if the hypotheses are satisfied with k fractional, then e-wsw-kR(k+8, w) =o(1)| C, 01 for each a>0 and that this is best possible in the sense that a may not be replaced by 0. 1. Let A (w) be of bounded variation over every finite interval of the nonnegative real axis. We write (1) F(a; x) = fZf(u)dA(u) =L + o(1) (C, k) a (read: F(a; x) is summable (C, k) to the limit L, or f' f(u) dA (u) exists in the (C, k) sense and equals L) if r(k + I)x-kFk(a;x) =kf (x u)kf(u)dA(u) -+L a as x0oo. (Stieltjes integrals are to be taken in the Riemann sense.) If in addition X-kFk(a; x) is of bounded variation over [a, oc) we shall write I C, k I instead of (C, k) in the notations above. This paper is concerned with the (C, k) and I C, k I summability of (2) C(x) (=C(O; x)) = f euddA (u)