So, before we proceed to another approach to the pseudospectral method using Chebyshev polynomials, I'd like to explain a concept in connection with the convolution theorem that helps us to understand finite difference type operators in a quite general and elegant way. Now remember, the operator in the wave number domain for the first derivative was IK, and we denoted it as just a function in the k domain, D of k. So, in the wave number domain or spectral domain, that's equivalent. To calculate the derivative is we multiply P of k which is the spectrum of the pressure field by big capital D of k. So, that's a multiplication in the wave number domain. Now, as you probably remember the convolution theorem says that a multiplication in the wave number domain is a convolution in the space domain. So, the remaining question is actually, so. What is this differential operator D of k, what does it look like in the space domain? We know very well what the pressure field looks like, it's just a space dependent function of pressure wave that's propagating, but what does big capital D of k look like in the space domain? That's going to be the next step. So, what do we need to do is we need to inverse transform capital D of k to a space domain to obtain small d of x, we can do this analytically. Now, our wave number space is limited. The reason is because we have a regular grid spacing with dx, so we have a maximum wave number which is actually K Nyquist, which we call K-max. That limits our space, so, in the end it turns out we can use the Heaviside function to describe. capital D of k as is given here. So, it's relatively straightforward to inverse transform this analytically to obtain a dx and that's given here. It actually turns out to be the derivative of a sinc function, and we will look at this graphically in a second. This inverse transform returns a continuous function in x. However, we live again in a discrete world. So, what happens if we discretize basically this function with dx, and that's given here. This dn of x suddenly turns out to be a very very simple function as you see here in this in this equation. Now, we look at this graphically. So, what you see here is basically the differential operator, the Fourier differential operator in space, once in a continuous form and the dots denote the discrete form. Actually, it contains the original simplest finite difference operator plus one and minus one which we denote here in colors. So, this is a way of understanding finite difference operators as well, because the finite difference operator basically is an ingredient of this operator, as long as your physical domain here. The decay of that operator d of x with space away from the central point where we want to calculate the derivative is actually reminiscent of the Taylor operators. However if you remember the Taylor operators decayed much faster, but that raises the question. Could we not maybe use that approach to calculate limited operators using the Fourier method? It's actually possible simply by multiplying this d of x with a Gaussian function which you see here. That kind of tapers off the weight away from the central point and you can use then basically the inner part of that operator as finite difference type operator. Now, you can see this here in another representation on one side because it's anti-symmetric, on one side you can see the behavior of the description of a finite difference operator as a limited Fourier type operator and the entire Fourier operator. What that tells us actually is we can turn the whole story around. Why can we now simply take the Fourier transform of each of those discrete operators and then compare those in the frequency domain? Because in the frequency wave number domain, we know that the difference operator has to be IK. IK is the exact opera which is, if we plot the imaginary part in that space is simply a straight line. Now, we can take any finite difference operator, calculate a Fourier transform, and then compare it with the exact operator. That's done in this graph here and actually shows that actually all are quite similar or almost identical and the left part of the spectrum and that's the part where we have many many grid points per wavelength. If we approach Nyquist, You can see that the finite difference operators or the limited Fourier operator are going away from, or departing from the actual exact IK operator, which is- we're going to make an error. So, that's just a very elegant way of understanding finite difference type operators also called convolutional operators for once. The other concept is that we can use this concept from the convolution theorem to investigate the accuracy of a difference operator in the spectral domain.
