I have been trying to understand how I should interpret spatial Fourier series in terms of wavelength. Let me consider Fourier series of a periodic function in $[-\pi,\pi]$,
$$f(x) = \frac{a_{0}}{2}+\sum_{n=1}^{\infty}a_{n}\cos(nx) + +\sum_{n=1}^{\infty}b_{n}\sin(nx).$$
By increasing $n$, I see that the frequency $f$ of the sin and cos that are used to represent the original function gets higher. Equivalently, the period $T$ of the sin and cos gets smaller based on these relationship
$$2\pi f=n, \quad T = \frac{1}{f}.$$
By drawing a figure, I also see that wavelength $\lambda$ of sin and cos gets shorter, which make sense for me because $f \propto \frac{1}{\lambda}$. I started to be confused from this point. I know, a wave function $w(x,t)$ has relationship of $v = f\lambda$, where $v$ is velocity of the wave.
However, in my setting, $f(x)$ is a spatial function. How should I define $\lambda$ for $f(x)$? It seems, from the drawing, to me that the $\lambda = T$, but not sure.
I am asking because, I would like to rewrite the Fourier series in terms of wave number $\frac{1}{\lambda}$or circular wavenumber $\frac{2\pi}{\lambda}$.