Functions — Fourier Series And Orthogonal

∫abf(x)g(x)dx=0integral from a to b of f of x g of x space d x equals 0 For Fourier series, the set of functions forms an orthogonal system on the interval

: Represents the average value (DC offset) of the function over one period. Fourier Series -- from Wolfram MathWorld

Harmony in Pieces: The Interplay of Fourier Series and Orthogonal Functions

In linear algebra, two vectors are orthogonal if their dot product is zero. We extend this concept to functions using an integral over a specific interval . Two real-valued functions are orthogonal if:

Because these functions are orthogonal, we can easily extract the specific "amount" (coefficient) of each sine or cosine wave needed to reconstruct a given periodic function . A standard Fourier series is written as: