On Modified Borel Methods

for x>0. One says that 5-lim 5„ = 5|7?'-lim 5„ = 5] if lim^.,*, B(x; sk) = s[limx^x B'(x; sk)=s]. The relations between these Borel methods B, B', and their behavior under change of index are known [8, p. 183; 6; 7]. Following a suggestion of R. P. Boas, Jr., we intend to study in this paper the modified Borel methods which arise when the continuous variable x in (1.1) is replaced by the discrete sequence of integers n = 1, 2, • • • . The resulting methods shall be denoted by Bi and B{, and our interest is to discuss the relations among the methods B, Bi, B', Bi (which is done in §3) and the behavior of these methods under change of index (cf. §4). The methods Bi, B{ show certain abnormalities in comparison with B, B'. For example, 3-lim sn = s always implies ZJ'-lim sn = s, whereas B/-lim 5„ = 5 implies 5/-lim sn = s if an = 0(Kn) for K<(ir2+iyi2 and not for K = (ir2 + iyi2. Our results are based on two theorems on entire functions (§2). The first allows one to infer f(x)—>s [x—>+ » ] from/(«)—>5 (n = 1, 2, • • • ) if the type oif(z) is less than tt, and is well known; the second allows one to infer /(x)==5e*[x—» + °° ] from/(ra)=sen (n = l, 2, ■ • ■ ) ii the type of f(z) is less than (ir2 + l)1/2.. Finally, in §5 Cesaro-Borel methods are considered but the results there are incomplete, whereas the results in §3 and §4 are in a certain sense best possible.