2. Unbounded Grids: The Semidiscrete Fourier Transform
Society for Industrial and Applied Mathematics eBooks · 2000
We now derive our first spectral method, as given by the doubly infinite matrix of (1.4). This scheme applies to a discrete, unbounded domain, so it is not a practical method. However, it does introduce the mathematical ideas needed for the derivation and analysis of the practical schemes we shall see later. Our infinite grid is denoted by hℤ, with grid points for j ∊ ℤ, the set of all integers: j ∈ ℤ, the set of all integers. We shall derive (1.4) by various methods based on the key ideas of the semidiscrete Fourier transform and band-limited sinc function interpolation. Before discretizing, we review the continuous case [DyMc86, Kat76, Kör90]. The Fourier transform of a function u(x), x ∈ ℝ, is the function û(k) defined by û (k) = ∫ −∞ ∞ e−ikx u (x) ⅆ x , k∈ℝ. 2.1 The number û(k) can be interpreted as the amplitude density of u at wavenumber k, and this process of decomposing a function into its constituent waves is called Fourier analysis.