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 $ X $ is a normed separable vector space with a scalar product $ ( , ) $, if $ \| \cdot \| $ is the corresponding norm and if $ \{ e _ {n} \} $ is an orthogonal system in $ X $, $ e _ {n} \neq 0 $, $ n = 1, 2 \dots $ then Parseval's equality for an element $ x \in X $ is

$$ \tag{1 } \| x \| ^ {2} = \sum_{n=1} ^ \infty | a _ {n} | ^ {2} \| e _ {n} \| ^ {2} , $$

where $ a _ {n} = ( x, e _ {n} )/( e _ {n} , e _ {n} ) $, $ n = 1, 2 \dots $ are the Fourier coefficients of $ x $ in the system $ \{ e _ {n} \} $. If $ \{ e _ {n} \} $ is orthonormal, then Parseval's equality has the form

$$ \| x \| ^ {2} = \sum_{n=1} ^ \infty | a _ {n} | ^ {2} . $$

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

1) if $ X $ is a separable Hilbert space (cf. Hilbert space) and $ \{ e _ {n} \} $ is an orthogonal basis of it, then Parseval's equality holds for $ \{ e _ {n} \} $ for every $ x \in X $;

2) if $ X $ is a separable Hilbert space, $ x , y \in X $, if $ \{ e _ {n} \} $ is an orthonormal basis of $ X $ and if $ a _ {n} = ( x, e _ {n} ) $ and $ b _ {n} = ( y, e _ {n} ) $ are the Fourier coefficients of $ x $ and $ y $, then

$$ \tag{2 } ( x, y) = \sum_{n=1} ^ \infty a _ {n} \overline{ {b _ {n} }}\; , $$

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 $ \{ e _ \alpha \} $, $ \alpha \in \mathfrak A $( $ \mathfrak A $ is a certain index set), is a complete orthonormal system in a Hilbert space $ X $, then for any element $ x \in X $ Parseval's equality holds:

$$ ( x, x) = \sum _ {\alpha \in \mathfrak A } | ( x, e _ \alpha ) | ^ {2} , $$

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

$$ \sup _ {\mathfrak A _ {0} } \sum _ {\alpha \in \mathfrak A } | ( x, e _ \alpha ) | ^ {2} , $$

where the supremum is taken over all finite subsets $ \mathfrak A _ {0} $ of $ \mathfrak A $.

When $ X = L _ {2} [- \pi , \pi ] $, the space of real-valued functions with Lebesgue-integrable squares on $ [- \pi , \pi ] $, and $ f \in L _ {2} [- \pi , \pi ] $, then one may take the trigonometric system as a complete orthogonal system and

$$ f \sim \frac{a _ {0} }{2} + \sum_{n=1} ^ \infty ( a _ {n} \cos nx + b _ {n} \sin nx), $$

where (1) takes the form

$$ \frac{1} \pi \int\limits _ {- \pi } ^ \pi f ^ { 2 } ( t) dt = \ \frac{a _ {0} ^ {2} }{2} + \sum_{n=1} ^ \infty ( a _ {n} ^ {2} + b _ {n} ^ {2} ), $$

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

If $ g \in L _ {2} [- \pi , \pi ] $ and

$$ g \sim \frac{a _ {0} ^ \prime }{2} + \sum_{n=1} ^ \infty ( a _ {n} ^ \prime \cos nx + b _ {n} ^ \prime \sin nx ), $$

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

$$ \tag{3 } \frac{1} \pi \int\limits _ {- \pi } ^ \pi f( t) g( t) dt = \ \frac{1}{2} a _ {0} a _ {0} ^ \prime + \sum_{n=1} ^ \infty ( a _ {n} a _ {n} ^ \prime + b _ {n} b _ {n} ^ \prime ). $$

Two classes $ K $ and $ K ^ \prime $ of real-valued functions defined on $ [- \pi , \pi ] $ and such that for all $ f \in K $ and $ g \in K ^ \prime $ Parseval's equality (3) holds are called complementary. An example of complementary classes are the spaces $ L _ {p} [- \pi , \pi ] $ and $ L _ {q} [- \pi , \pi ] $, $ p ^ {-} 1 + q ^ {-} 1 = 1 $, $ 1 < p < + \infty $.

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