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)

References

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