Four Fourier transforms are usually defined, the Integral Fourier transform, the Discrete-Time Fourier transform (DTFT), the Discrete Fourier transform (DFT) and the Integral Fourier transform for periodic functions. However, starting from their definitions, we show that all four Fourier transforms can be reduced to actually only one Fourier transform, the Fourier transform in the distributiona...

A new method for analysing nonlinear and non-stationary data has recently been developed, namely the partial-differential-equation (PDE) transform. The PDE transform is based on the use of coupled arbitrarily high order PDEs to decompose signals and images into various functional frequency modes, which allow perfect reconstruction of the original signals and images. The PDE transform is compare...

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 paper derives sharp estimates of the error arising from explicit and implicit approximations of the constant-coefficient 1D convection–diffusion equation with Dirac initial data. The error analysis is based on Fourier analysis and asymptotic approximation of the integrals resulting from the inverse Fourier transform. This research is motivated by applications in computational finance and t...