Hardy criterion

for uniform convergence of series of functions

If a sequence of real-valued functions $ a _ {n} ( x) $, $ n = 1, 2 \dots $ is monotone for every $ x \in E $, where $ E $ is a certain set, and converges uniformly to zero on $ E $, and if the sequence of partial sums of a series $ \sum _ {n = 1 } ^ \infty b _ {n} ( x) $ is bounded on $ E $( the functions $ b _ {n} ( x) $ may take complex values), then the series $ \sum _ {n = 1 } ^ \infty a _ {n} ( x) b _ {n} ( x) $ converges uniformly on $ E $.

This criterion was established by G.H. Hardy [1].

References

[1]	G.H. Hardy, "Some theorems connected with Abel's theorem on the continuity of power series" Proc. London. Math. Soc. (2) , 4 (1907) pp. 247–265

Comments

This criterion can be proved using the Euler–MacLaurin formula (cf. the proof of Thm. 3.42 in [a1]).

References

[a1]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108