# Bessel system

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

#### References

[1] | S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951) |

**How to Cite This Entry:**

Bessel system.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Bessel_system&oldid=13693