Orthogonal functions are a set of functions that satisfy the mathematical condition of orthogonality, which means that the integral of the product of any two distinct functions in the set over a specified interval is zero. This concept is fundamental in various areas of mathematics and engineering, particularly in processamento de sinais, análise de Fourier, and análise funcional.
Em termos mais técnicos, duas funções, f(x) e g(x), são consideradas ortogonais em um intervalo [a, b] se:
∫ab f(x) g(x) dx = 0
This relationship indicates that the functions do not correlate with each other, which is a desirable property in many applications. For instance, orthogonal functions can be used to construct Séries de Fourier, where sine and cosine functions serve as orthogonal bases for representing periodic functions.
Orthogonality is not limited to just two functions; a complete set of orthogonal functions can be used to expand other functions in a way that minimizes redundancy and maximizes efficiency in representation. Examples of such sets include the Legendre polynomials and the Hermite functions. In the context of machine learning, orthogonal functions are also related to various algorithms, including those used for redução de dimensionalidade and feature extraction, where they help in ensuring that the features are independent and non-redundant.