| Wiener's Generalized Harmonic Analysis (GHA) extends harmonic analysis to functions not accessible to the L1, L2, and Fourier series theories. Even with this extension, the spectral analysis of a large class of functions, e.g., nonstationary processes and in particular, fractal noise, is not completely resolved. Using Wiener's work as a starting point, we apply harmonic analysis, distribution theory, and wavelet theory to adapt GHA to these functions.;Wiener defined the deterministic autocorrelation of a function on the real line where the analogous probability concept is the stochastic autocorrelation of a random process. The deterministic autocorrelation Rf of a function f:R→C is defined as Rf t=lim T→infinity12T -TTft+u fudu. 1 The right side of (1) is the mean of the convolution f*f&d5; , where f˜ is the involution defined as f&d5;t =f-t . If the limit in (1) exists, the Herglotz-Bochner Theorem implies that there is a bounded Radon measure S such that the Fourier transform of Rf equals S. In this setting, S is called the power spectrum of f. Conversely, the Wiener-Wintner theorem asserts that for a given positive bounded Radon measure S, there exists a function f such that the Fourier transform of the autocorrelation Rf of f is equal to S.;My dissertation considers an extension of this relationship between f and its power spectrum to functions whose spectra S behave like 1g k , where 0 < k < 1. For 0 < k < 1, 1g k is a positive measure but is not bounded. Thus, classical GHA and in particular the Wiener-Wintner theorem do not apply. This 1f -type behavior occurs in the spectra of a family of processes commonly referred to as 1f processes. As the spectral analysis of such processes has been an area of extensive statistically-based research, we first consider the statistical perspective of our problem by reviewing three relatively recent contributions in the area. We then turn to our deterministic approach where, specifically, the task is to find a function f such that Rf&d14; g=1 gk . 2 We give a solution to (2) which generalizes techniques introduced by Wiener and Wintner [18] and extended to higher dimensions by Benedetto and Kerby [2, 13]. We also briefly discuss the extension of our solution to higher dimensions.;From a theoretical point of view, our work extends the Wiener-Wintner theorem to spectra S which are not positive bounded Radon measures, and in so doing, validates equation (2), a formula analogous to (1), but in a more general setting. From an applications point of view, our result provides information from a mathematical and deterministic perspective about the structure of signals which give rise to the widely occurring physical phenomena termed fractal noise, where here the term fractal refers to the spectra's power law decay of fractional order, e.g., see [9]. |
