Multiplicative system

An orthonormal system of functions $\{\phi_n\}$ on $[a,b]$ satisfying the conditions:

1) for any two functions $\phi_k$ and $\phi_l$ the system $\{\phi_n\}$ contains their product $\phi_m(x)=\phi_k(x)\phi_l(x)$;

2) for each function $\phi_k$ the system $\{\phi_n\}$ contains the function $\phi_m(x)=1/\phi_k(x)$.

Examples of multiplicative systems are the exponential system $\{e^{i2\pi nx}\}_{n=-\infty}^\infty$, which is orthogonal on $[0,1]$, and the Walsh system of functions.

References

[1]	S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[2]	H.F. Harmuth, "Transmission of information by orthogonal functions" , Springer (1972)
[3]	R.W. Zeek (ed.) A.E. Showalter (ed.) , Applications of Walsh functions (Proc. Symp. Washington, April 1971) , Univ. Maryland (1971)