Veritasium Fourier transfer video The Remarkable Story Behind The Most Important Algorithm Of All Time explains the adding up (integral) the product of the wave in interest (blue) and a known sine (or cosine) is positive only when the known wave is part of the wave in interest.
When learning Fourier series transform, I see the integral of the inner product of functions out of blue in a text book and has been wondering.
DATA DRIVEN SCIENCE & ENGINEERING Chapter 2
Is it correct to think that the formula is there to suggest that the wave in the interest (blue) is $f(x)$ and sine or cosine wave (red) is $g(x, \theta)$ where $\theta$ is a specific frequency of the red wave. By adding up the dot product of $f(x) \cdot g(x, \theta)$ from (a=-inf, b=inf), it will be positive only if $g(x, \theta)$ is part of $f(x)$?