“orthogonal functions”的概念、定义、翻译、参考文献-科学参考

orthogonal functions

Computer

单词	orthogonal functions
释义	orthogonal functions Computer Let $f_{1} (x), f_{2} (x), \dots, f_{n} (x)$ be a set of functions defined on the interval (a,b); also let w(x) be a given positive function (a weight function) on (a,b). The functions f_i(x), $i = 1, 2, \dots, n$ are said to be orthogonal with respect to the interval (a,b) and weight function w(x), if $\begin{array}{l} \int_{a}^{b} w (x) f_{i} (x) f_{j} (x) d x = 0, \\ i \neq j, i, j = 1, 2, \dots, n \end{array}$ If, for i = j, $\begin{array}{l} \int_{b}^{a} w (x) f_{i}^{2} (x) d x = 1, \\ i = 1, 2, \dots, n \end{array}$ then the functions are said to be orthonormal. A similar property is defined when (a,b) is replaced by the set of points $x_{1}, x_{2}, \dots, x_{n}$ and the integral is replaced by a sum, $\begin{array}{l} Σ_{k = 1}^{N} w (x_{k}) f_{i} (x_{k}) f_{i} (x_{k}) = 0, \\ i \neq j \end{array}$ Orthogonal functions play an important part in the approximation of functions and data.
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