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 f²{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.