ABSTRACT
We remember that a linear combination of the functions of the system of ex ponents e(A) is called a quasi-polynomial in this system.
Let a number a > 0 be fixed. We say that the system e(A) is a system of rightward extension of convergence in Lp (1 < p < oo) if any sequence of quasi polynomials in e(A) converging in Lp(—a, a) also converges in Lp(—a, 6) for any b E (a,oo). Systems of leftward extension of convergence are defined similarly. If e(A) is a system of both rightward and leftward extension of convergence in Lp, we call it a system of extension of convergence to the whole axis in Lp or simply a system of extension of convergence in Lp. In this section we describe two classes of systems of extension of convergence assuming that A is the sequence of all zeros of an entire function
L(z) = J exzt dcr(t), varcr(£) < 0. (1) — a
T h eo rem 1 . Let A be the sequence of all zeros of the entire function (1). I f cr(—a) ^ a (—a + 0), then e(A) is a system of leftward extension of convergence in Lp, 1 < p < oo. I f the relations
a(a) / <r(a - 0), a ( - a ) ^ (r(-a + 0), (2)
hold simultaneously, then e(A) is a system of extension in the class Lp.
