Hi I have a question about Fourier integrals.
Can Fourier cosine integrals represent any function, or just even functions?
2026-04-26 11:01:12.1777201272
Fourier integral representations using only cosine functions.
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An integral of the form $f(t) = \int \cos(\nu t) \hat f(\nu)\,d\nu$ is always an even function of $t$. Indeed, $$f(-t) = \int \cos(-\nu t) \hat f(\nu)\,d\nu = \int \cos( \nu t) \hat f(\nu)\,d\nu = f(t)$$ Therefore, the cosine transform cannot represent a function that is not even.