Minimal property

of the partial sums of an orthogonal expansion

For any function , any orthonormal system on and for any , the equality

holds, where

is the -th partial sum of the expansion of with respect to the system , that is,

The minimum is attained precisely at the sum and

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)

References

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