# Mean-square approximation of a function

An approximation of a function $f( t)$ by a function $\phi ( t)$, where the error measure $\mu ( f; \phi )$ is defined by the formula

$$\mu _ \sigma ( f; \phi ) = \int\limits _ { a } ^ { b } [ f( t) - \phi ( t)] ^ {2} d \sigma ( t),$$

where $\sigma ( t)$ is a non-decreasing function on $[ a, b]$ different from a constant.

Let

$$\tag{* } u _ {1} ( t), u _ {2} ( t) \dots$$

be an orthonormal system of functions on $[ a, b]$ relative to the distribution $d \sigma ( t)$. In the case of a mean-square approximation of the function $f( t)$ by linear combinations $\sum _ {k=} 1 ^ {n} \lambda _ {k} u _ {k} ( t)$, the minimal error for every $n = 1, 2 \dots$ is given by the sums

$$\sum _ { k= } 1 ^ { n } c _ {k} ( f ) u _ {k} ( t),$$

where $c _ {k} ( f )$ are the Fourier coefficients of the function $f( t)$ with respect to the system (*); hence, the best method of approximation is linear.

#### References

 [1] V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian) [2] G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)