The generalized analytic signal

AbstractThis paper is an explication of the analytic signal in the generalized case, i.e., the analytic signal of a generalized function and of a generalized stochastic process. The contributions of the author are: (1) an L2-theory of distributions which, in the study of the analytic signal, has an advantage over the usual Schwartz-Itô-Gel'fand theory because the Cauchy representation is defined; (2) a proof (Theorem 2.5) that the Schwartz distributions δ, δ+, δ− and ϱ may be extended to the L2-case, expressions (Theorems 2.6 and 2.7) for their Hilbert and Fourier transforms in the L2-case, and expressions (Section 2.1) for their analytic signals; (3) a proof (Theorem 3.3) that an orthogonal L2-process, and therefore the Fourier transform of a second-order stationary stochastic process (Theorem 3.4), is strictly generalized; (4) a representation theorem (Theorem 3.5) which extends the Itô spectral representation theorem for stationary random distributions to the nonspectral, nonstationary, L2-case; (5) expressions for the Cauchy representation (Theorem 3.6) and the analytic signal (Theorem 3.7) of an L2-process; (6) an expression for and the covariance kernel of the analytic signal of white noise (Section 3.4). The word application in the text refers to the application of previously developed concepts