# Minimal property

of the partial sums of an orthogonal expansion

For any function $f \in L _ {2} [ a , b ]$, any orthonormal system $\{ \phi _ {k} \} _ {k=} 1 ^ \infty$ on $[ a , b ]$ and for any $n$, the equality

$$\inf _ {\{ a _ {k} \} _ {k=} 1 ^ {n} } \ \int\limits _ { a } ^ { b } \left | f ( x) - \sum _ { k= } 1 ^ { n } a _ {k} \phi _ {k} ( x) \ \right | ^ {2} d x =$$

$$= \ \int\limits _ { a } ^ { b } | f ( x) - S _ {n} ( f , x ) | ^ {2} d x$$

holds, where

$$S _ {n} ( f , x ) = \ \sum _ { k= } 1 ^ { n } c _ {k} ( f ) \phi _ {k} ( x)$$

is the $n$- th partial sum of the expansion of $f$ with respect to the system $\{ \phi _ {k} \}$, that is,

$$c _ {k} ( f ) = \ \int\limits _ { a } ^ { b } f ( x) \phi _ {k} ( x) d x .$$

The minimum is attained precisely at the sum $S _ {n} ( f , x )$ and

$$\int\limits _ { a } ^ { b } | f ( x) - S _ {n} ( f , x ) | ^ {2} d x =$$

$$= \ \int\limits _ { a } ^ { b } f ^ { 2 } ( x) d x - \sum _ { k= } 1 ^ { n } | c _ {k} ( f ) | ^ {2} ,\ n = 1 , 2 ,\dots .$$

Bessel's inequality, Parseval's equality for complete systems and also certain other basic properties of orthogonal expansions essentially are corollaries of this equality (cf. Bessel inequality; Parseval equality; Complete system of functions; Orthogonal series; Orthogonal system).

#### References

 [1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) [2] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)