Orthogonal functions

Summary

In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval: : \langle f,g\rangle = \int \overline{f(x)}g(x),dx . The functions f and g are orthogonal when this integral is zero, i.e. \langle f, , g \rangle = 0 whenever f \neq g. As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector dot product; two vectors are mutually independent (orthogonal) if their dot-product is zero. Suppose { f_0, f_1, \ldots} is a sequence of orthogonal functions of nonzero L2-norms \left| f_n \right| _2 = \sqrt{\langle f_n, f_n \rangle} = \left(\int f_n ^2 \ dx \right) ^

Official source

https://en.wikipedia.org/wiki/Orthogonal_functions

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A New Design Algorithm for Two-Band Orthonormal Rational Filter Banks and Orthonormal Rational Wavelets

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