We now start considering discrete–time signals. A discrete–time signal is a function (real or complex valued) whose argument runs over the integers, rather than over the real line. We shall use square brackets, as in x[n], for discrete–time signals and round parentheses, as in x(t), for continuous–time signals. This is the notation used in EECE 359 and EECE 369. Discrete–time signals arise in t...

Despite the many models of saccadic eye movements, little attention has been paid to the shape of saccade trajectories. Some investigators have argued that saccades are driven by a rectangular "bang-bang" neural control signal, whereas others have emphasized the similarity to fast arm movement trajectories, such as the "minimum jerk" profile. However, models have not been tested rigorously agai...

We discuss the effects of several sequence acceleration methods on the partial sums of Fourier series. For a large set of functions we show that these methods fail. 2000 Mathematics Subject Classification: 65B10, 65T10, 42A20

A general summability method of different orthogonal series is given with the help of an integrable function θ. As special cases the trigonometric Fourier, Walsh-, Walsh-Kaczmarz-, Vilenkinand Ciesielski-Fourier series and the Fourier transforms are considered. For each orthonormal system a different Hardy space is introduced and the atomic decomposition of these Hardy spaces are presented. A s...

A function or a real variable f is said to be periodic with period P if f(x+ P ) = f(x) holds for all x. Hence, if we know the values of f on an interval of length P , we know its values everywhere. If f is a function defined on an interval [a, b), we can extend f to a function defined for all x which is periodic of period b− a. We simply define f(x) to be f(x+ n(b− a)), where n is the integer ...

Every function f(x) which is of period 1 and Lebesgue integrable on [0, 1 ] may be expanded in a Walsh-Fourier series(3), f(x)~ ?.?=n ak\pk(x), where ak=fof(x)ypk(x)dx, k=0, 1, 2, • • • . Fine exhibited some of the basic similarities and differences between the trigonometric orthonormal system and the Walsh system. He identified the Walsh functions with the full set of characters of the dyadic ...

The purpose of this paper is to explore the basic question of the convergence of Fourier series. This paper will not delve into the deeper questions of convergence that measure theory illuminates, but requires only the basic principles set out by introductory real and complex analysis.