The relationship between truncated Fourier series solutions of several different long-wave evolution equations is explored. The models are all designed to represent the same physics in the asymptote of small nonlinearity and frequency dispersion, and yet give different numerical solutions for given values of dimensionless parameters. It is found that the problem of obtaining a steady solution i...

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...

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...

A standard procedure to approximate a function f in an inner product space is to consider the Fourier series of the function with respect to an orthogonal system. The basic general results on this topic can be found in many references in the literature, for example, [2, Chapter VIII]. It is well known that even in the case in which K is a closed interval there always exists a function f in C(K)...

The problem of locating the discontinuities of a function by means of its truncated Fourier series, an interesting question in and of itself, arises naturally from an attempt to overcome the Gibbs phenomenon: the poor approximative properties of the Fourier partial sums of a discontinuous function. In [3], Cai et al. developed the idea already introduced in their previous papers and suggested a...