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)
[a1]	K. Yosida, "Functional analysis" , Springer (1978) pp. Sect. III.4