Suppose I have some data, given by $\boldsymbol{x}$ and $\boldsymbol{y}$ pairs, for instance$\{(\boldsymbol{x_1}, \boldsymbol{y_1}), (\boldsymbol{x_2},\boldsymbol{y_2}),...,(\boldsymbol{x_n},\boldsymbol{y_n})\}$ for $n\in\mathbb{N}$.

I assume that the data comes from some function $f(\boldsymbol{x})$ with some noise, I don't know the function, so we don't know whether it is periodic (very likely, it is NOT periodic).

This means $\boldsymbol{y_i} = f(\boldsymbol{x_i}) + \boldsymbol{\epsilon_i}$ where $\boldsymbol{\epsilon_i} \sim \textbf{N}(\textbf{0}, \boldsymbol{\sigma}^2))$.

Now I know that the Fourier Series is defined only for periodic functions. However (here) it is possible to "extend" a non-periodic function defined on an interval, to a periodic function.

The amount of data that I have is finite, so will lie in an interval.

So my question is:

Does it make sense (is it even possible) to try and fit a Fourier Series to the data, by considering the data (which is multi-dimensional, both x and y are vectors) in a multi-dimensional interval , by finding the coefficients that minimize some loss function? (for instance Mean Squared Error) How would someone define this fourier series for a multi-dimensional, non-periodic, noisy function?

(by using only a finite number of fourier terms!)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.