in numerical analysis we students are obligated to implement code in R that given a function f(x) finds its Fourier interpolation tN(x) and computes the interpolation error
$||f(x)-t_{N}(x)||=\int_{0}^{2\pi}$ $|f(x)-t_{N}(x)|^2$
or a variety of different $N$ I first tried to compute the d-coefficients according to this formular:
$d = \frac 1N M y$
with M denoting the DFT matrix and y denoting a series of equidistant function values with
$y_j = f(x_j)$ and
$x_j = e^{\frac{2*pi*i}N*j}$
for $j = 1,..,N-1$.
My goal was to come up with a sum that can be described by:
$t_{N}(x) = \Sigma_{k=0}^{N-1} d_k * e^{i*k*x}$
Which would be easier to later integrate in sort of a subsequently additive notation.
f <- function(x) 3/(6+4*cos(x)) #first function to compare with
g <- function(x) sin(32*x) #second one
xj <- function(x,n) 2*pi*x/n
M <- function(n){
w = exp(-2*pi*1i/n)
m = outer(0:(n-1),0:(n-1))
return(w^m)
}
y <- function(n){
f(xj(0:(n-1),n))
}
transformFunction <- function(n, f){
d = 1/n * t(M(n)) %*% f(xj(0:(n-1),n))
script <- paste(d[1])
for(i in 2:n)
script <- paste0(script,paste0("+",d[i],"*exp(1i*x*",i,")"))
#trans <- sum(d[1:n] * exp(1i*x*(0:(n-1))))
return(script)
}
The main purpose of the transform function was, initially, to return a function - or rather: a mathematical expression - which could then be used in order to declarate my Fourier Interpolation Function. Problem is, based on my fairly limited knowledge, that I cannot integrate functions that still have sums nested in them (which is why I commented the corresponding line in the code). Out of absolute desperation I then tried to paste each of the summands in form of text subsequently, only to parse them again as an expression. So the main question that remains is: how do I return mathmatical expressions in a manner that allow me to use them as a function and later on integrate them? I am sincerely sorry for any misunderstanding or confusion, as well as my seemingly amateurish coding. Thanks in advance!