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Hardy spaces of almost periodic functions

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Hardy spaces of almost periodic functions ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze J. P. MILASZEWICZ Hardy spaces of almost periodic functions Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 24, no 3 (1970), p. 401-428. <> © Scuola Normale Superiore, Pisa, 1970, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » ( implique l’accord avec les conditions générales d’utilisation ( Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques HARDY SPACES OF ALMOST PERIODIC FUNCTIONS by J. P. MILASZEWICZ 1. Introduction. In the classic context, Hardy spaces can be defined in two equivalent ways : first as the subspaces of LP of the unit circle with vanishing nega- tive Fourier coefficients; second as spaces of holomorphic functions in the unit disc such that the p-th means in each circle of radius less than one are uniformly bounded. One departure point for a generalization is to substitute Z, the inte- gers, by any subgroup g of the real line with the discrete topology. This leads to the construction of the so called « big disc » (the closed unit disc if g = Z) and of a generalized Poisson kernel. As in the classic context two directions can be chosen to define the Hardy classes ; one dealing witb, the « boundary of this generalized disc, that is the dual group of g (see [X]), and another dealing with its « interior » (see [VIII]). This note deals mainly with the second definition. In the first part we investigate in which measure some classic results can be extend

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