Bessel system

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A concept in the theory of orthogonal systems. Let and be two complete systems of functions in (i.e. measurable functions that are square-integrable on the segment ), forming a biorthogonal system of functions. The system is said to be a Bessel system if, for any function , the series

is convergent; here, are the coefficients of the expansion

of the function with respect to the system . For a system to be a Bessel system it is necessary and sufficient that it be possible to define a bounded linear operator on the space such that the system defined by the equation () is a complete orthonormal system. If the system is a Bessel system, there exists a constant such that for any


[1] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
How to Cite This Entry:
Bessel system. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by P.I. Lizorkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article