On the Theory of Fourier Transforms
Einar Hille, J. D. Tamarkin · Bulletin of the American Mathematical Society · 1933
We have slightly modified Titchmarsh's notation inasmuch as he deals with cosine-and sine-transforms, while we use exponential transforms.* Contrary to our unfounded statement in a recent paper.See E, Hille and J. D. Tamarkin, On the summahility of Fourier series, III, Mathematische Annalen, vol.108 (1933), pp.525-577 (p.530).According to results of the present note, however, this statement is correct.f The case p=l goes back to H. Hahn, Ueber die Darstellung gegebener Funktionen durch singuldre Intergrale, II, Denkschriften Wiener Akademie, Math.-Nat.Kl., vol.93 (1916), p. 557-692 (p.681).For p>\ see M. Riesz, Sur les fonctions conjuguées, Mathematische Zeitschrift, vol.27 (1927), pp.218-244 (p.230).M. Riesz states at the end of his paper that to his results concerning Fourier series there correspond similar results for Fourier integrals.Î We refer for these properties to the recent book by S, Banach, Théorie des Opérations Linéaires, Warsaw, 1932.