What are Orthogonal and Orthonormal functions?

Welcome back MechanicaLEi, did you know that several sets of orthogonal and orthonormal functions have become standard bases for approximating mathematical functions? This makes us wonder, what are orthogonal and orthonormal functions? Before we jump in check out the previous part of this series to learn about what Residue theorem is? Now, two functions f of x and g of x are orthogonal over the closed interval of a comma b with weighting function w of x if: inner product of f of x and g of x which is equivalent to integral of f of x into g of x into w of x dx from a to b is equal to zero. If, in addition, integral of f of x the whole square into w of x dx from a to b is equal to 1 and integral of g of x the whole square into w of x dx is equal to 1 also, then the functions f of x and g of x are said to be orthonormal. Given an infinite orthogonal set consisting of values psi j from j equals one to infinity on closed interval of a comma b, an orthogonal series expansion is summation of c j into psi j of x from j equal to one to infinity, where c j are constants. One main consideration is whether a given function f on closed interval of a comma b has an infinite orthogonal expansion that is f of x equals to summation of c j into psi j of x from j equals to one to infinity, for some constant c j, which leads to Fourier transformation. Hence, we first saw what Orthogonal and orthonormal functions are and then went on to see what sum of orthogonal series expressions are?
In the next episode of MechanicaLEi find out what Fourier series are?

Attributions: