# Bessel Inequality

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## Bessel inequality

[′bes·əl ‚in·ē′kwäl·əd·ē]*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Bessel Inequality

an inequality for the coefficients of the Fourier series for an arbitrary orthonormalized system of functions φ_{k} (*x*) (k = 1, 2,. . .)—that is, a system defined over a certain interval *[a, b]* and satisfying the conditions *(k* ≠;*l)*

If the function *f(x)* is measurable over the interval *[a, b],* and the function *f ^{2}{x)* is integrable over this interval, and is a Fourier series

*of f(x)*for the system φ

_{κ}(

*(x),*then the Bessel inequality

is valid.

The Bessel inequality plays an important role in all investigations which pertain to the theory of orthogonal series. In particular, it shows that the Fourier coefficients of the function *f(x)* tend to zero as *n* → ∞ . For a trigonometric system of functions this inequality was obtained by F. Bessel (1828). If the system of functions φ_{k} is such that for any function *f* the Bessel inequality is converted into an equality, then it is called the Parseval equality.

S. B. STECHKIN