Bessel system

A concept in the theory of orthogonal systems. Let $\left\{{\psi_n}\right\}$ and $\left\{{g_n}\right\}$ be two complete systems of functions in $L_2 \! \left({a, b}\right) = L_2$ (i.e. measurable functions that are square-integrable on the segment $a, b$), forming a biorthogonal system of functions. The system $\left\{{\psi_n}\right\}$ is said to be a Bessel system if, for any function $f \in L_2$, the series $$ \sum_{n \, = \, 1}^{\infty} c_{n}^{2} $$

is convergent; here, $c_n = \left({f, g_n}\right)$ are the coefficients of the expansion $$ f \sim \sum_{n \, = \, 1}^{\infty} c_n \psi_n $$

of the function $f$ with respect to the system $\left\{{\psi_n}\right\}$. For a system $\left\{{\psi_n}\right\}$ to be a Bessel system it is necessary and sufficient that it be possible to define a bounded linear operator $A$ on the space $L_2$ such that the system $\left\{{\phi_n}\right\}$ defined by the equation $A \psi_n = \phi_n$ ($n = 1, 2, \dots$) is a complete orthonormal system. If the system $\left\{{\psi_n}\right\}$ is a Bessel system, there exists a constant $M$ such that for any $f \in L_2$ $$ \sum_{n \, = \, 1}^{\infty} \left({f, g_n}\right)^2 \leq M \left\|{f}\right\|_{L_2}^2. $$

References

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