ANZIAM J. 50(2009), 550–561 doi:10.1017/S1446181109000273 ERROR ESTIMATES FOR DOMINICI’S HERMITE FUNCTION ASYMPTOTIC FORMULA AND SOME APPLICATIONS R. KERMAN1, M. L. HUANG ˛ 1 and M. BRANNAN1 (Received 11 October, 2007; revised 24 August, 2009) Abstract The aim of this paper is to find a concrete bound for the error involved when approximating the nth Hermite function (in the oscillating range) by an asymptotic formula due to D. Dominici. This bound is then used to study the accuracy of certain approximations to Hermite expansions and to Fourier transforms. A way of estimating an unknown probability density is proposed. 2000 Mathematics subject classification: primary 33C45; secondary 41A10, 62G07. Keywords and phrases: Hermite function, asymptotic formula, Green’s function, Hermite expansion, Fourier transform, density estimation. 1. Introduction Let n 2 d 2 h .x/ D .−1/nγ ex =2 e−x ; γ D π −1=42−n=2.nW/−1=2; (1.1) n n dxn n be the nth Hermite function, n D 0; 1;:::: Dominici [2] obtained an asymptotic formulap for the Hermite polynomial,p Hn, which yields for hn in the oscillatory range jxj < 2n (or, writing x D 2n sin θ, jθj < π=2), n=2 r p 2n 2 n 1 nπ hn.x/ D hn. 2n sin θ/ ∼ γn cos sin θ C n C θ − : e cos θ 2 2 2 (1.2) As will be seen in Examples 1, 2 and 5 below, this asymptotic formula leads to remarkably accurate results. Our aim in this paper is to estimate the error incurred when hn.x/ is replaced by the right side of (1.2). We go on to apply this new information to the approximation of Hermite expansions and Fourier transforms and then to density estimation. 1Department of Mathematics, Brock University, St. Catharines, Ontario, Canada L2S 3A1; e-mail: [email protected]. c Australian Mathematical Society 2009, Serial-fee code 1446-1811/2009 $16.00 550 [2] Error estimates for Dominici’s Hermite function asymptotic formula and applications 551 In fact, we work with a slightly modified version of the asymptotic function. Our principal result is given in Theorem 1.1. THEOREM 1.1.pLet hn be the nth Hermite function, n D 0; 1;::: and fix A > 0. Then, with x D 2n sin θ, one has, for jxj ≤ A, hn.x/ D dn.x/ C en.x/; where r p 2 n 1 nπ dn.x/ D dn. 2n sin θ/ D cn cos sin θ C n C θ − ; (1.3) cos θ 2 2 2 8 nW >p π −1=42−n=2.nW/−1=2; n even; > <> 20 ..n=2/ C 1/ D cn 2.nW/ (1.4) > π −1=42−n=2.nW/−1=2; n odd; >p p :> 2n C 1=. 2n/ 0 .n C 1=2/ and rn.x/ 1 e .x/ D C O ; (1.5) n n7=4 n9=4 in which jrn.x/j ≤ 2A, n ≥ 5. 0 The constant cn in (1.4) has been chosen to guarantee dn.0/ D hn.0/, and dn.0/ D 0 hn.0/, n D 0; 1;:::: Now, hn satisfies the differential equation d2 L y D 0; L VD C .−x2 C 2n C 1/: n n dx2 Defining fn D −Lndn, we get that en D hn − dn is the unique solution of the initial value problem 0 Ln y D fn; y.0/ D y .0/ D 0: Thus, Z x en.x/ D Gn.x; y/ fn.y/ dy; x 2 R; (1.6) 0 Gn being the Green’s function of the differential operator Ln. In Sections 2–4 we successively estimate j fn.x/j, Gn.x; y/ and, finally, using (1.5), jen.x/j. Section 5 has an application of the latter estimate to Hermite expansions and Section 6 has one to the Fourier transform. In Section 7 we briefly discuss density estimation. p p 2. A uniform bound for f on [− n; nU p n p −1 Setting θ D sin .x= 2n/ in formula (1.3) for dn. 2n sin θ/ gives cn cos(ρ.x// dn.x/ D ; 4 − .2x2=n/1=4 552 R. Kerman, M. L. Huang and M. Brannan [3] in which s 1p 2x2 1 x nπ ρ.x/ D 2nx 4 − C n C sin−1 p − : 4 n 2 2n 2 Next, −1=4n1=4c 2 n p 2 4 2 .Lndn/ .x/ D .2n − x /.−4x C 2.4n C 1/x C n/; 2n − x27=4 so p j j D fn.x/ .Lndn/ . 2n sin θ/ cn q ≤ 16 sin6 θ − 40 sin4 θ C 47 sin2 θ C 2 8n cos11=2 θ p cn ≤ 2−3=4 35 ; (2.1) n p when jθj < π=2 or jxj < n. 3. The Green’s function of Ln As observed in Arfken [1, Pages 637–638], two linearly independent solutions of Ln y D 0 are n 1 2 −x2=2 φ n.x/ D F − I I x e 1 1 1 2 2 and .n − 1/ 3 2 −x2=2 φ n.x/ D F − I I x xe ; 2 1 1 2 2 where the confluent hypergeometric function of the first kind 1 k X .a/k x F .aI bI x/ VD ; a; b; x 2 R; b 6D 0; −1;::: 1 1 .b/ kW kD0 k and 0(λ C k/ (λ)k D : 0(λ) D 0 D D 0 D Indeed, φ1n.0/ 1, φ1n.0/ 0 and φ2n.0/ 1, φ2n.0/ 1, which means the Green’s function of Ln is Gn.x; y/ D φ1n.y)φ2n.x/ − φ1n.x)φ2n.y/: Slater [5, Page 68] gives the following asymptotic formula of Tricomi for the confluent hypergeometric function: p .1−b/=2 x=2 1 1 F1.aI bI x/ D 0.b/.kx/ e Jb−1 2 kx 1 C O p ; k in which a 2 R, b > 0, x ≥ 0 and k D b=2 − a. As usual, Jν denotes the νth order Bessel function of the first kind. [4] Error estimates for Dominici’s Hermite function asymptotic formula and applications 553 As 2 1=2 2 1=2 J− .y/ D cos y and J .y/ D sin y; 1=2 πy 1=2 πy the principal term in Gn.x; y/ will be p p p sin 2n C 1x p sin 2n C 1y cos 2n C 1y p − cos 2n C 1x p n C n C p 2 1 2 1 sin 2n C 1.x − y/ D p I 2n C 1 more precisely, for jxj; jyj ≤ A, A > 0 being fixed, p sin 2n C 1.x − y/ 1 Gn.x; y/ D p 1 C O p : (3.1) 2n C 1 n 4. The proof of Theorem 1.1 According to (1.6) and (3.1), for jxj ≤ A, A > 0 being fixed, rn.x/ 1 en.x/ D 1 C O p ; n7=4 n where p Z x C − 7=4 sin 2n 1.x y/ rn.x/ VD n p fn.y/ dy: 0 2n C 1 In view of (2.1), again, if jxj ≤ A, 7=4 p p n −3=4 cn −5=4 1=4 jrn.x/j ≤ p 2 35 A < 2 35n cn A: 2n C 1 n However, Stirling’s formula in the form p θ 0.x/ D x x−1=2e−x−1 2π exp ; 0 < θ < 1; 12x (see Whittaker and Watson [7, Page 253]) yields r 2 1 1 cn ≤ exp : π 24.n C 1/ n1=4 Hence, for n ≥ 5, − r 2 3=4 35 1 1 2A 1 jen.x/j ≤ exp A 1 C O p ≤ 1 C O p : 2 n7=4 π 144 n n7=4 n 554 R. Kerman, M. L. Huang and M. Brannan [5] 5. Hermite series As is well known, any square-integrable function f on R can be represented in terms of its Hermite series. That is, 1 1 X Z f D h f; hni hn with h f; hni D f .x/hn.x/ dx; (5.1) nD0 −∞ the convergence in (5.1) being both almost everywhere and in the mean of order two. For the purpose of computation we will replace the square-integrable function f by f A VD f χ.−A;A/, A > 0. The assumption is that f has its essentially compact support − R j j2 contained in . A; A/. This can be gauged by how small jx|≥A f .x/ dx is. Next, we observe that one can compute hn and h f; hni quickly and accurately, using, say, the formula (1.1) for hn, only when n ≤ N, with N somewhat less than 100. Fortunately, when n > N; dn and h f; dni are readily calculated and they approximate hn and h f; hni extremely well. We will use the root-mean-square norm Z A 1=2 1 2 M2.gI A/ D jg.x/j dx 2A −A to measure how well N 1 X X h f A; hni hn C h f A; dni dn nD0 nDNC1 approximates f . Applying Theorem 1.1 we obtain the following theorem. THEOREM 5.1. Suppose f is square-integrable on R and set f A VD f χ.−A;A/ for a 2 chosen A > 0. Given N 2 ZC,N A, consider the approximation N 1 X X DN f VD h f A; hni hn C h f A; dni dn (5.2) nD0 nDNC1 to the Hermite series of f A.