Abelian Theorem

A theorem which asserts that if a sequence or function behaves regularly, then some average of it behaves regularly. For example,

implies

for any . The converse is false, but can be made into a correct Tauberian theorem if  is subjected to an appropriate additional condition (Hardy 1999, p. 46).

REFERENCES:

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 46, 1999.