On the Airy Reproducing Kernel, Sampling Series, and Quadrature Formula

We determine the class of entire functions for which the Airy kernel (of random matrix theory) is a reproducing kernel. We deduce an Airy sampling series and quadrature formula. Our results are analogues of well known ones for the Bessel kernel. The need for these arises in investigating universality limits for random matrices at the soft edge of the spectrum. 1. Introduction Universality limits play a central role in random matrix theory [4], [17]. A recent new approach to these [14], [15], [16], [19] involves the reproducing kernel for a suitable space of entire functions . For universality in the bulk, the reproducing kernel is the familiar sine kernel, sin t t . It reproduces PaleyWiener space, the class of entire functions of exponential type at most , that are square integrable along the real axis. More precisely, if g is an entire function of exponential type at most , and g 2 L2 (R), then for all complex z, g (z) = Z 1 1 sin (t z) (t z) g (t) dt: For universality at the hard edge of the spectrum, one obtains instead the Bessel kernel. It is the reproducing kernel for a class of entire functions of exponential type 1, satisfying a weighted integrability condition on the real line [16]. Both the sine and Bessel kernels have also been intensively studied within the context of sampling theory, as well as Lagrange interpolation series, and quadrature formula in approximation theory [6], [9], [10], [11], [12]. In investigating universality limits for random matrices at the soft edge of the spectrum, the authors needed to determine the class of entire functions for which the Airy kernel is a reproducing kernel. An extensive literature search did not turn up the desired results. While Paley-Wiener theorems have been obtained for some related Airy operators [21], and many identities for the Airy kernel are familiar in universality theory [2], [3], [5], [20], they do not readily yield the desired reproducing kernel formula. The associated sampling series, and quadrature formula, also could not be traced. It is Date : January 20, 2009. Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353 1 2 ELI LEVIN1 AND DORON S. LUBINSKY2 the goal of this paper to present these. These will be an essential input to establishing universality at the soft edge of the spectrum for fairly general measures, in a future work. Recall that the Airy kernel Ai ( ; ) of random matrix theory, is de…ned by (1.1) Ai (a; b) = ( Ai(a)Ai0(b) Ai0(a)Ai(b) a b ; a 6= b; Ai0 (a) aAi (a) ; a = b; where Ai is the Airy function, given on the real line by [1, 10.4.32, p. 447], [18, p. 53] (1.2) Ai (x) = 1 Z 1 0 cos 1 3 t + xt dt: The Airy function Ai is an entire function of order 32 , with only real negative zeros fajg, where (1.3) 0 > a1 > a2 > a3 > ::: and [1, 10.4.94, p. 450], [22, pp. 15-16] (1.4) aj = [3 (4j 1) =8] 1 +O 1 j2 = 3 j 2 2=3 (1 + o (1)) : Ai satis…es the di¤erential equation (1.5) Ai00 (z) zAi (z) = 0: We note also that standard estimates for Ai show that Ai (z; ) 2 L2 (R) for each …xed z.: Our main result is as follows: Theorem 1 Let g 2 L2 (R) be the restriction to the real axis of an entire function. The following are equivalent: (I) For all z 2 C; (1.6) g (z) = Z 1 1 Ai (z; s) g (s) ds: (II) All the following are true: (a) g is an entire function of order at most 3 2 . (b) There exists L > 0 with the following property: whenever 0 < < , there exists C such that for jarg (z)j ; (1.7) jg (z)j C (1 + jzj) e 23 z3=2 : (c)