Least-squares function approximation

Functional analysis

A generalization to approximation of a data set is the approximation of a function by a sum of other functions, usually an orthogonal set:

${\displaystyle f(x)\approx f_n(x)=a_1{\varphi }_1(x)+a_2{\varphi }_2(x)+\cdots +a_n{\varphi }_n(x),}$

with the set of functions {${\displaystyle {\varphi }_j(x)}$} an orthonormal set over the interval of interest, say [a, b]: see also Fejér's theorem. The coefficients {${\displaystyle a_j}$} are selected to make the magnitude of the difference ||f − f_n||² as small as possible. For example, the magnitude, or norm, of a function g (x ) over the interval [a, b] can be defined by:

${\displaystyle \Vert g\Vert ={\left({\int }_a^b{g}^{\ast }(x)g(x)dx\right)}^{1/2}}$

where the ‘*’ denotes complex conjugate in the case of complex functions. The extension of Pythagoras' theorem in this manner leads to function spaces and the notion of Lebesgue measure, an idea of “space” more general than the original basis of Euclidean geometry. The { ${\displaystyle {\varphi }_j(x)}$ } satisfy orthonormality relations:

${\displaystyle {\int }_a^b{\varphi }_i^{\ast }(x){\varphi }_j(x)dx={\delta }_{ij},}$

where δ_ij is the Kronecker delta. Substituting function f_n into these equations then leads to the n-dimensional Pythagorean theorem:

${\displaystyle \Vert f_n{\Vert }^2=|a_1{|}^2+|a_2{|}^2+\cdots +|a_n{|}^2.}$

The coefficients {a_j} making ||f − f_n||² as small as possible are found to be:

${\displaystyle a_j={\int }_a^b{\varphi }_j^{\ast }(x)f(x)dx.}$

The generalization of the n-dimensional Pythagorean theorem to infinite-dimensional real inner product spaces is known as Parseval's identity or Parseval's equation. Particular examples of such a representation of a function are the Fourier series and the generalized Fourier series.