Bessel inequality

The inequality

$$ \| f \| ^ {2} = (f, f) \geq \ \sum _ {\alpha \in A } \frac{| (f, \phi _ \alpha ) | ^ {2} }{( \phi _ \alpha , \phi _ \alpha ) } = $$

$$ = \ \sum _ {\alpha \in A } \left | \left ( f, \frac{\phi _ \alpha }{\| \phi _ \alpha \| } \right ) \right | ^ {2} , $$

where $ f $ is an element of a (pre-) Hilbert space $ H $ with scalar product $ (f, \phi ) $ and $ \{ {\phi _ \alpha } : {\alpha \in A } \} $ is an orthogonal system of non-zero elements of $ H $. The right-hand side of Bessel's inequality never contains more than a countable set of non-zero components, whatever the cardinality of the index set $ A $. Bessel's inequality follows from the Bessel identity

$$ \left \| f - \sum _ {i = 1 } ^ { n } x ^ {\alpha _ {i} } \phi _ {\alpha _ {i} } \ \right \| ^ {2} \equiv \ | f | ^ {2} - \sum _ {i = 1 } ^ { n } \lambda _ {\alpha _ {i} } | x ^ {\alpha _ {i} } | ^ {2} , $$

which is valid for any finite system of elements $ \{ {\phi _ {\alpha _ {i} } } : {i = 1 \dots n } \} $. In this formula the $ x ^ {\alpha _ {i} } $ are the Fourier coefficients of the vector $ f $ with respect to the orthogonal system $ \{ \phi _ {\alpha _ {1} } \dots \phi _ {\alpha _ {n} } \} $, i.e.

$$ x ^ {\alpha _ {i} } = \ \frac{1} {\lambda _ {\alpha _ {i} } } (f, \phi _ {\alpha _ {i} } ),\ \ \lambda _ {\alpha _ {i} } = \ ( \phi _ {\alpha _ {i} } , \phi _ {\alpha _ {i} } ). $$

The geometric meaning of Bessel's inequality is that the orthogonal projection of an element $ f $ on the linear span of the elements $ \phi _ \alpha $, $ \alpha \in A $, has a norm which does not exceed the norm of $ f $( i.e. the hypothenuse in a right-angled triangle is not shorter than one of the other sides). For a vector $ f $ to belong to the closed linear span of the vectors $ \phi _ \alpha $, $ \alpha \in A $, it is necessary and sufficient that Bessel's inequality becomes an equality. If this occurs for any $ f \in H $, one says that the Parseval equality holds for the system $ \{ {\phi _ \alpha } : {\alpha \in A } \} $ in $ H $.

For a system $ \{ {\phi _ \alpha } : {\alpha = 1, 2 , . . . } \} $ of linearly independent (not necessarily orthogonal) elements of $ H $ Bessel's identity and Bessel's inequality assume the form

$$ \left \| f - \sum _ {\alpha , \beta = 1 } ^ { n } b _ {n} ^ {\alpha \beta } (f, \phi _ \beta ) \phi _ \alpha \right \| ^ {2\ } \equiv $$

$$ \equiv \ \| f \| ^ {2} - \sum _ {\alpha , \beta = 1 } ^ { n } b _ {n} ^ {\alpha \beta } (f, \phi _ \alpha ) (f, \phi _ \beta ), $$

$$ \| f \| ^ {2} \geq \sum _ {\alpha , \beta = 1 } ^ { n } b _ {n} ^ {\alpha \beta } (f, \phi _ \alpha ) (f, \phi _ \beta ), $$

where $ b _ {n} ^ {\alpha \beta } $ are the elements of the matrix inverse to the Gram matrix (cf. Gram determinant) of the first $ n $ vectors of the initial system.

The inequality was derived by F.W. Bessel in 1828 for the trigonometric system.

References

[1]	L.D. Kudryavtsev, "Mathematical analysis" , 2 , Moscow (1973) (In Russian)

Comments

Usually, the orthogonal system of elements $ \{ \phi _ \alpha \} $ is orthonormalized, i.e. one sets $ \psi _ \alpha = \phi _ \alpha / \| \phi _ \alpha \| $. Bessel's inequality then takes the form

$$ \sum _ {\alpha \in A } | (f, \psi _ \alpha ) | \leq \ \| f \| ^ {2} , $$

which is easier to remember. In this form it is used in approximation theory, Fourier analysis, the theory of orthogonal polynomials, etc.

References

[a1]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5
[a2]	E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff
[a3]	P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 108–126
[a4]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)