ИСТИНА

We consider the problem of recovering, by generalized Fourier formulae, the vector-valued coefficients of series with respect to classical orthogonal systems (in particular Walsh, Haar and trigonometric 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 or Henstock-Pettis integral . 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. A typical example: Theorem 1. For any infinite-dimensional Banach space there exists a function with values in this space such that its Fourier-Henstock-Haar series diverges almost everywhere. Moreover, the rate of growth of the partial sums in the above theorem can be $n^{\frac{1}{2} - \varepsilon}$. And this rate of divergence is close to describing the {\it worst} type of behavior of those partial sums that can occur in an {\it arbitrary} infinite-dimensional Banach space. In fact the growth $O(n^{\frac{1}{2}})$ can be achieved for no Banach space with the Orlicz property, i.e., for spaces on which the identity operator is (2,1)-summing.