QUADRATURE AND EXTREMAL BANDLIMITED FUNCTIONS

Let A(delta) be the class of functions of exponential type delta > 0. We prove that for integrable F is an element of A(2 pi delta) integral(infinity)(infinity) F(x)dx = delta(-1) Sigma(xi is an element of gamma,r)(1-gamma/pi(xi(2)+gamma(2))+gamma)F(delta(-1)xi), where T-gamma,T-r is the set of zeros of B-gamma,B-r(z) = z sin pi(z+r)-gamma cos pi(z+r). Let a > (2 delta)(-1). It is shown that for any Laguerre-Polya entire function E with E(+/- a) - 0 there exist two integrable functions G(-), G(+) is an element of A(2 pi delta) such that for all real x E(x)[G-(x) - chi([-a,a])(x)] <= 0, E(x)[G+(x) - chi([-a,a])(x)] >= 0. Combining these results we find the minimal value of ||S-T||(1), where S, T is an element of A(2 pi delta) satisfy S(x) <= chi([-a,a])(x) <= T(x) for all real x. We determine extremal functions for which the minimal value is assumed. As an application we give an explicit expression for C(delta,alpha) = inf(g is an element of A2)(delta) sup(x is an element of[-alpha,alpha]) parallel to g parallel to(2)(2)/vertical bar g(x)|(2), where A(2)(delta) is the set of square integrable functions in A(delta). This constant occurs in work of Donoho and Logan regarding reconstruction of bandlimited functions.