Property of the Fourier Series

Question

Suppose that $\sum_{k=-\infty}^{\infty}c_ke^{ikx}$ is a Fourier Series for $f(x)$. To what function does $\sum_{k=-\infty}^{\infty}c_ke^{i4kx}$ converge to?

My best estimate is $f(4x)$ but I'm not sure how to go about solving this problem.

Answer 1

Notice that if $f(x) = \sum_{k=-\infty}^\infty c_k e^{ikx}$ for all $x$, then $f(4x) = \sum_{k=-\infty}^\infty c_k e^{ik(4x)}$, because you can simply plug in $4x$ for $x$ in the original expression. In other words, yes, you're right, provided the series converges for every value of $x$.

More specifically, if $\sum_{k=-\infty}^\infty c_k e^{ikx}$ converges to $f(x)$ when $x=t$, then $\sum_{k=-\infty}^\infty c_k e^{i4kx}$ converges to $f(4x)$ when $x= \frac{1}{4} t$.