I am currently stuck with the following sum:
$$f(x)=\sum\limits_{k=1}^{N-1}a_k\cos(\frac{kx\pi}{N})$$
Here $x\in[0,N)$. The coefficients $a_k$ are decreasing ($a_1\geq a_2\geq...\geq a_{N-1}$).
I am looking for:
a) An identity to simplify the sum (in the best case)
b) The behaviour of $f(x)$ on $[0,N)$. From some experiments I have tried, it seems that f(x) is decreasing on $x\in[0,N)$ whenever the coefficients $a_k$ are decreasing. I would be interested in the exact relation. For example if $a_k\approx 1/k$, is $f(x)\approx 1/x$?
What I tried so far:
a) Lagrange trigonometric identities and Dirichlet kernels look similar to the sum, but I could not find any result using weights. I also looked at phase addition formulars, but they only seem to work if there is a constant phase shift between the terms ($\cos(x+a)+\cos(x+b)$ instead of $\cos(x)+\cos(2x)$ for example).
b)If we only use integer values for $x$, then this sum can be written as the Discrete Fourier Transform or the Cosine Transform of a signal. But I couldn't find anything about the behaviour of the Fourier coefficients of a decreasing series.
Any tips / links / references would be helpfull. Thanks in advance!
It depends on what $x$ and $a_k$ are here's what you can do, $$a_{k}\cos\left(\frac{kx\pi}{N}\right)=\frac{a_{k}e^{i\frac{kx\pi}{N}}+a_{k}e^{-i\frac{kx\pi}{N}}}{2}$$ $$\implies\sum_{k=1}^{N-1} a_{k}\cos\left(\frac{kx\pi}{N}\right)=\frac{\sum _{k=1}^{N-1}a_{k}e^{i\frac{kx\pi}{N}}+\sum_{k=1}^{N-1}a_{k}e^{-i\frac{kx\pi}{N}}}{2}$$ Now, you can approxiamete $\sum _{k=1}^{N-1}a_{k}e^{i\frac{kx\pi}{N}}$ and $\sum _{k=1}^{N-1}a_{k}e^{-i\frac{kx\pi}{N}}$ using a geometric series based on what $a_{k}$ are, for a abitary sequence of $a_{k}$ but for some special sequnces you will be able evaluate the exact sum but in genral you can only arrive at a approximation for the series.