Parseval equality

An equality expressing the square of the norm of an element in a vector space with a scalar product in terms of the square of the moduli of the Fourier coefficients of this element in some orthogonal system. Thus, if is a normed separable vector space with a scalar product , if is the corresponding norm and if is an orthogonal system in , , then Parseval's equality for an element is

(1)

where , are the Fourier coefficients of in the system . If is orthonormal, then Parseval's equality has the form

The validity of Parseval's equality for a given element is a necessary and sufficient condition for its Fourier series in the orthogonal system to converge to in the norm of . The validity of Parseval's equality for every element is a necessary and sufficient condition for the orthogonal system to be complete in (cf. Complete system). This implies, in particular, that:

1) if is a separable Hilbert space (cf. Hilbert space) and is an orthogonal basis of it, then Parseval's equality holds for for every ;

2) if is a separable Hilbert space, , if is an orthonormal basis of and if and are the Fourier coefficients of and , then

(2)

the so-called generalized Parseval equality. In a fairly-definitive form the question of the completeness of a system of functions that are the eigen functions of differential operators was studied by V.A. Steklov in [1].

Parseval's equality can also be generalized to the case of non-separable Hilbert spaces: If , ( is a certain index set), is a complete orthonormal system in a Hilbert space , then for any element Parseval's equality holds:

and the sum on the right-hand side is to be understood as

where the supremum is taken over all finite subsets of .

When , the space of real-valued functions with Lebesgue-integrable squares on , and , then one may take the trigonometric system as a complete orthogonal system and

where (1) takes the form

which is called the classical Parseval equality. It was proved in 1805 by M. Parseval.

If and

then an equality similar to (2) looks as follows:

(3)

Two classes and of real-valued functions defined on and such that for all and Parseval's equality (3) holds are called complementary. An example of complementary classes are the spaces and , , .

References

[1]	V.A. Steklov, "Sur certaines égalités générales communes à plusieurs séries de fonctions souvent employées dans l'analyse" Zap. Nauchn. Fiz.-Mat. Obshch. Ser. 8 , 157 (1904) pp. 1–32
[2]	S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian)
[3]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 2 , MIR (1982) (Translated from Russian)
[4]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[5]	A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)
[6]	A.A. Kirillov, A.D. Gvishiani, "Theorems and problems in functional analysis" , Springer (1982) (Translated from Russian)

References

[a1]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)