We develop an orthogonal approach to the construction of the theory of generalized functions of infinitely many variables (without using Jacobi fields) and apply it to the construction and investigation of the Poisson analysis of white noise.
IntroductionIn our previous survey [1], we described the general scheme of the construction of the theory of generalized functions of infinitely many variables based on the notion of generalized translation. In particular, we described the so-called biorthogonal approach, according to which spaces of test functions are constructed on the basis of a certain system of functions, and spaces of generalized functions are associated with the corresponding biorthogonal system.In the present paper, which, in fact, is a continuation of [1], we study the case where the indicated system of functions is orthogonal. This is the so-called orthogonal (or spectral) approach to the construction of the theory of generalized functions. The previous works in this direction were cited in [1].Note that, in the present paper, we do not consider a purely spectral approach related to the spectral theory of Jacobi fields, where a mapping that realizes an isomorphism between spaces from the rigging of the Fock space and constructed spaces is given by the corresponding Fourier transformation in the decomposition in common generalized eigenvectors of the field. Roughly speaking, we assume that the biorthogonal system used in [1] is orthogonal and obtain the corresponding consequences.An example of the orthogonal approach is the classical Brownian analysis of white noise (the corresponding works were cited in [1]). In the present paper, we consider another example, namely, the Poisson analysis of white noise [i.e., the corresponding theory of generalized functions (see [2][3][4][5][6][7][8][9][10])] and show how the results (both known and new) of this theory are obtained on the basis of the general approach described in [1]. Note that this approach to the Poisson analysis was announced in [11].In the present paper, we consider a Poisson measure for which the intensity measure σ is a Lebesgue measure on R 1 . However, the results obtained can easily be generalized to any nonatomic measure σ; for this purpose, it is necessary to use the corresponding generalized Sobolev spaces (for the definition of these spaces, see, e.g., [12,13]).Let us make several additional remarks. Spaces of test and generalized functions in the orthogonal case are constructed in Secs. 3 and 4. In Sec. 5, we study operators of second quantization in terms of the corresponding space L 2 . Note that, in different cases, these operators were studied in [14,15,6,10,16]. The Poisson analysis on the Sobolev space of generalized functions is given in Secs. 7-11, and its modification for the configuration space is presented in Sec. 12.
