Cosine Products, Fourier Transforms, and Random Sums
نویسنده
چکیده
one gets the first actual formula for 7r that mankind ever discovered, dating from 1593 and due to Frangois Viete (1540-1603), whose Latinized name is Vieta. (Was any notice taken of the formula's 400th anniversary, perhaps by the issue of a postage stamp?) From the samples of a function t(x) at equally spaced points x", n E z, one can reconstruct the complete function with the aid of sin x/x, provided t is "band-limited" and the spacing of the samples is small enough. This is the content of the Sampling Theorem, which lends its name to sin x/x as the sampling function. Its importance in signal processing, where it is also known as sinc x, is the result of its Fourier transform being the characteristic function of the interval [-1, 1] (modulo a scalar factor). In Section 2 we prove the infinite product expansion for sin x/x and derive Viete's formula. In Section 3 we transform the product expansion with the Fourier transform and use convolution and delta distributions to prove it in a way that reveals a host of similar identities. Section 4 puts these identities into a probabilistic setting, and in Section 5 we alter the probability experiments in order to make connections between infinite cosine products, Cantor sets, and sums of series with random signs, particularly the harmonic series. This leaves us with some interesting unsolved problems and conjectures for further work.
منابع مشابه
The relationship between Fourier and Mellin transforms, with applications to probability
The use of Fourier transforms for deriving probability densities of sums and differences of random variables is well known. The use of Mellin transforms to derive densities for products and quotients of random variables is less well known. We present the relationship between the Fourier and Mellin transform, and discuss the use of these transforms in deriving densities for algebraic combination...
متن کاملNumerical stability of fast trigonometric and orthogonal wavelet transforms
Fast trigonometric transforms and periodic orthogonal wavelet transforms are essential tools for numerous practical applications. It is very important that fast algorithms work stable in a floating point arithmetic. This survey paper presents recent results on the worst case analysis of roundoff errors occurring in floating point computation of fast Fourier transforms, fast cosine transforms, a...
متن کاملPreservation of whiteness in spectral and time-frequency transforms of second order processes Préservation de la blancheur dans les transformations spectrales et temps- fréquences de processus du second ordre
In many signal processing applications, recent techniques often rely on the estimation of a probabilistic model. Many times, this model does not focus on the observed data itself, but rather on a spectral or time-frequency transform of this data, such as the discrete Fourier transform (DFT) or the short-time Fourier transform (STFT). A common statistical assumption regarding these transforms is...
متن کاملFractional cosine, sine, and Hartley transforms
In previous papers, the Fourier transform (FT) has been generalized into the fractional Fourier transform (FRFT), the linear canonical transform (LCT), and the simplified fractional Fourier transform (SFRFT). Because the cosine, sine, and Hartley transforms are very similar to the FT, it is reasonable to think they can also be generalized by the similar way. In this paper, we will introduce sev...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004