Published online by Cambridge University Press: 20 January 2009
This note gives a proof of the result:
A necessary and sufficient condition that a trigonometrical series T (x) be the Fourier series of a function is that σn – σm = O(n-n) uniformly in [0, 2π] for all m≤n, where σn is the nth (C, l) mean of T (x).
Page 100 note 1 Poussin, C. de la Vallée, Leçons sur l'approximation des fonctions (Paris, 1919), §41.Google Scholar
Page 100 note 2 Zygmund, , Trigonometrical Series (1935), p. 62, No. 7.Google Scholar