ИСТИНА

It is a popular theme in vector-valued Fourier analysis during the last decades to investigate if classical results about scalar-valued functions remain valid if the functions considered take values in some Banach space. Some results remain true for any Banach space. But the most frequently observed case is that it depends on the structure and geometry of the Banach spaces considered whether a result can be carried over to the vector-valued setting. Here we consider the problem of recovering, by generalized Fou-rier formulae, the vector-valued coefficients of series with respect to classical orthogonal systems (in particular Walsh, Haar and trigono-metric systems). To solve this problem some generalizations of Bochner and Pettis integrals are introduced and investigated. In the case of Walsh and Haar series a suitable integral is a dyadic version of the Henstock integral (see [2]). In the simplest case of convergence everywhere this integral solves the problem with coefficients from any Banach space. At the same time some nice properties of Fourier series in the sense of these generalized integrals remain valid only for functions with values in finite dimensional spaces.