We show that the answer to the question in the title is “very well indeed.” In particular, we prove that, throughout the maximum possible range, the finite Fourier coefficients provide a good approximation to the Fourier coefficients of a piecewise continuous function. For a continuous periodic function, the size of the error is estimated in terms of the modulus of continuity of the function. T...

This brief explores the connections between nonuniform sampling of a certain function and the almost periodic extension of its Fourier transform. It is shown that the Fourier transform of a bandlimited function can be extended (as a weighted sum of translates) as a Stepanoff almost periodic function, to the whole frequency axis. This result leads to a generalized nonuniform sampling theorem whi...

In this paper, some design criteria for a Tubular Linear Induction Motor (TLIM) as a fast actuator are considered. The influence of geometrical and physical parameters on the operating conditions of a TLIM are investigated by means a quasi-analytic model. The model is based on the application of the Fourier Transform both in space and in time. The Fourier transform in space is introduced to tak...

where Î denotes the Fourier transform, and ω denotes frequency. Proof 1 Taking the Fourier transform of the Heat Equation: { ∂tu(t, x) = ∆u(t, x) x ∈ R u(0, x) = I(x) t = 0 yields: ∂tû(t, ω) = (iω) · (iω)û(t, ω) = −|ω|û(t, ω), where û(t, ω) is the Fourier transform of u. Solving this differential equation yields û(t, ω) = e−|ω| 2tÎ(ω). We note that a = 0 when I ∈ L since Î(0) = ∫ R2 I(x) dx is ...

This brief explores the connections between nonuniform sampling of a certain function and the almost periodic extension of its Fourier transform. It is shown that the Fourier transform of a band-limited function can be extended (as a weighted sum of translates) as a Stepanoff almost periodic function to the whole frequency axis. This result leads to a generalized nonuniform sampling theorem whi...

It is now nearly a decade since the theories of perverse sheaves and of the /adic Fourier transform came into being and gave fundamental new insights into the behavior of additive character sums over finite fields. Most of these insights were spellings out of the basic fact that the Fourier transform of a perverse sheaf is itself a perverse sheaf, a statement that amounts to a succession of van...

We propose a matrix approach to the computation of BattleLemarié's wavelets. The Fourier transform of the scaling function is the product of the inverse F(x) of a square root of a positive trigonometric polynomial and the Fourier transform of a B-spline of order m . The polynomial is the symbol of a bi-infinite matrix B associated with a B-spline of order 2m . We approximate this bi-infinite ma...

Fourier analysis started even before Fourier, but in the context of Fourier’s work on the Analytic Theory of Heat, it began with a claim regarding expansions of functions as trigonometrical series, what we now call Fourier series. Such expansions form the framework for writing down solutions to the heat equation in terms of initial temperature distributions. [refer to Enrique A. Gonzalez-Velasc...