Mean Convergence of Interpolation at Zeros of Airy Functions

Transcription

1 Mean Convergence of Interpolation at Zeros of Airy Functions D. S. Lubinsky Abstract The classical Erdős-Turán theorem established mean convergence of Lagrange interpolants at zeros of orthogonal polynomials. A non-polynomial extension of this was established by Ian Sloan in 983. Mean convergence of interpolation by entire functions has been investigated by Grozev, Rahman, and Vértesi. In this spirit, we establish an Erdős-Turán theorem for interpolation by entire functions at zeros of the Airy function. Introduction The classical Erdős-Turán theorem involves a weight w on an compact interval, which we take as [,]. We assume that w 0 and is positive on a set of positive measure. Let p n denote the corresponding orthonormal polynomial of degree n 0, so that for m,n 0, p n p m w = δ mn. Let us denote the zeros of p n in [,] by < x nn < x n,n < < x 2n < x n <. Given f : [,] R, let L n [ f] denote the Lagrange interpolation polynomial to f at { x jn } n j=, so that L n[ f] has degree at most n and L n [ f]x jn = f x jn, j n. D. S. Lubinsky Georgia Institute of Technology, School of Mathematics, Atlanta, GA , USA lubinsky@math.gatech.edu

2 2 D. S. Lubinsky Theorem Erdős-Turán Theorem. Let f : [, ] R be continuous. For n, let L n [ f] denote the Lagrange interpolation polynomial to f at the zeros of p n. Then lim f L n [ f] 2 w = 0. n The ramifications of this result continue to be explored to this day. It has been extended in numerous directions: for example, rather than requiring f to be continuous, we can allow it to be Riemann integrable. We may replace w by a positive measure µ, which may have non-compact support. In addition, convergence in L 2 may be replaced, under additional conditions on w, by convergence in L p. There is a very large literature on all of this. See [0,, 2, 3, 22, 23] for references and results. Ian Sloan and his coauthor Will Smith ingeniously used results on mean convergence of Lagrange interpolation in various L p norms, to establish the definitive results on convergence of product integration rules [4, 6, 8, 9, 20, 2]. This is a subject of substantial practical importance, for example in numerical solution of integral equations. One can speculate that it was this interest in product integration that led to Ian Sloan extending the Erdős-Turán theorem to non-polynomial interpolation. Here is an important special case of his general result [7, p. 99]: Theorem 2 Sloan s Erdös-Turán Theorem on Sturm-Liouville Systems. Consider the eigenvalue problem with boundary conditions pxu x+qxu x+[rx+λ]ux = 0 cosαua+sinαu a = 0; cosβub+sinβu b = 0. Assume that p,q,r are continuous and real valued on [a,b], that p > 0 there, while α,β are real. Let {u n } n 0 be the eigenfunctions, ordered so that the corresponding eigenvalues {λ n } are increasing. Given continuous f : [a,b] R, let L n [ f] denote the linear combination of { } n u j j=0 that coincides with f at the n+ zeros of u n+ in the open interval a,b. Let Then wx = px exp x a qt pt dt, x [a,b]. b lim f x L n [ f]x 2 wxdx = 0, n a provided f a = 0 if sinα = 0 and f b = 0 if sinβ = 0. Moreover, there is a constant c independent of n and f such that for all such f,

3 Interpolation at Airy Zeros 3 b a f x L n [ f]x 2 wxdx C inf c 0,c,...,c n b a f x n j=0 c j u j x 2 wxdx. As a specific example, Sloan considers the Bessel equation. His general theorem [7, p. 02], from which the above result is deduced, involves orthonormal functions, associated reproducing kernels, and interpolation points satisfying two boundedness conditions. In 988, M. R. Akhlaghi [2] extended Sloan s result to convergence in L p for p. Interpolation by trigonometric polynomials is closely related to that by algebraic polynomials, in as much as every even trigonometric polynomial has the form Pcos θ where P is an algebraic polynomial. From trigonometric polynomials, one can pass via scaling limits to entire functions of exponential type, and the latter have a long and gloried history associated with sampling theory. However, to this author s knowledge, the first general result on mean convergence of entire interpolants at equispaced points is due to Rahman and Vértesi [5, Theorem, p. 304]. Define the classic sinc kernel { sinπt St = πt, t 0,, t = 0. Given a function f : R R, and > 0, define the formal Lagrange interpolation series kπ L [ f ;x] = f S x kπ. k= It is easily seen that this series converges uniformly in compact sets if for some p >, we have kπ p f <. k= Theorem 3 Theorem of Rahman and Vértesi. Let f : R R be Riemann integrable over every finite interval and satisfy for some β > p, Then f x C+ x β, x R. lim f x L [ f ;x] p dx = 0. Butzer, Higgins and Stens later showed that this result is equivalent to the classical sampling theorem, and as such is an example of an approximate sampling theorem [3]. Of course there are sampling theorems at nonequally spaced points see for example [7, 25], and in the setting of de Branges spaces, there are more general expansions involving interpolation series. However, as far as this author is aware, there are no analogues of the Rahman-Vértesi theorem in that more general

4 4 D. S. Lubinsky setting. Ganzburg [5] and Littman [9] have explored other aspects of convergence of Lagrange interpolation by entire functions. One setting where mean convergence has been explored, is interpolation at zeros of Bessel functions, notably by Grozev and Rahman [6, Theorem, p. 48]. Let α > and z α z 2k J α z = 2 k 2 k=0 k!γ α + k+ denote the Bessel function of order α. It is often convenient to instead use its entire cousin, G α z = z α J α z. J α has positive zeros and matching negative zeros j α, < j α,2 < j α,3 <, j α, k = j α,k, k, so for f : R C, and > 0, one can define the formal interpolation series jα,k L α, [ f ;x] = f l α,k z, k=,k 0 where for k 0, l α,k z = G α G α z. jα,k z jα,k Theorem 4 Theorem of Rahman and Grozev. Let α 2 and p >, or let < α < 2 and < p < 2 2α+. Let f : R R be Riemann integrable over every finite interval and satisfy for some δ > 0, Then f x C+ x α 2 p δ, x R. lim x α+ 2 f x L [ f ;x] p dx = 0. Note that p = 2 is always included. The proof of this theorem involves a lot of tools: detailed properties of entire functions of exponential type and of Bessel functions, and a converse Marcinkiewicz-Zygmund inequality that is itself of great interest. In this paper, we explore convergence of interpolation at scaled zeros of Airy functions. Recall that the Airy function Ai is given on the real line by [, , p. 447] Aix = cos π 3 t3 + xt dt. 0

5 Interpolation at Airy Zeros 5 { The} Airy function Ai is an entire function of order 3 2, with only real negative zeros a j, where Ai satisfies the differential equation 0 > a > a 2 > a 3 >. Ai z zaiz = 0. The Airy kernel Ai,, much used in random matrix theory, is defined [8] by { AiaAi b Ai aaib Aia,b = a b, a b, Ai a 2 aaia 2, a = b. Observe that l j z = Aiz,a j Aia j,a j = Aiz Ai a j z a j, is the Airy analogue of a fundamental of Lagrange interpolation, satisfying l j a k = δ jk. There is an analogue of sampling series and Lagrange interpolation series involving { l j } : Definition. Let G be the class of all functions g : C C with the following properties: a g is an entire function of order at most 3 2 ; b There exists L > 0 such that for δ 0,π, some C δ > 0, and all z C with argz π δ, gz C δ + z L exp 2 3 z 3 2 ; c ga j 2 j= a j /2 <. In [8, Corollary.3, p. 429], it was shown that each g G admits the expansion gz = j= ga j Aiz,a j Aia j,a j. Moreover, for f, g G, there is the quadrature formula [8, Corollary.4, p. 429] f xgxdx = j= f ga j Aia j,a j. In analogy with the entire interpolants of Grozev-Rahman, we define for f : R R, the formal series

6 6 D. S. Lubinsky L [ f ;z] = f j= a j l j z = Note that it samples f only in,0. We prove: f j= a j Aiz,aj Aia j,a j. Theorem 5. Let f : R R be bounded and Riemann integrable in each finite interval, with f x = 0 in [0,. Assume in addition that for some β > 2, and x R, Then f x C+ x β. 2 lim f x L [ f ;x] 2 dx = 0. 3 Observe that the integration is over the whole real line. We expect that there is an analogue of this theorem at least for all p >. However, this seems to require a converse Marcinkiewicz-Zygmund inequality estimating L p norms of appropriate classes of entire functions in terms of their values at Airy zeros. This is not available, so we content ourselves with a weaker result for the related operator L [ f ;z] = f j= This interpolates f at each a 2 j, but not at a 2 j. sup x R a2 j [l2 j z+l 2 j z ]. Theorem 6. a For bounded functions f : R R, and a, L a2 j, [ f ;x] C sup f 4 j where C is independent of and f. b Let 4 5 < p <. Let f : R R be bounded and Riemann integrable in each finite interval, with f x = 0 in [0,.Assume in addition that for some β > p, and x R, we have 2. Then lim f x L [ f ;x] p dx = 0. 5 Note that L [ ] f ; z G, so this also establishes density of that class of functions in a suitable space of functions containing those in Theorem 6. The usual approach to Erdős-Turan theorems is via quadrature formulae and density of polynomials, or entire functions of exponential type, in appropriate spaces. The latter density is not available for G. So in Section 2, we establish convergence for characteristic functions of intervals. We prove Theorems 5 and 6 in Section 3. Throughout C,C,C 2,... denote positive constants independent of n,x,z,t,, and possibly other specified quantities. The same symbol does not necessarily denote the same constant in different occurrences, even when used in the same line.

7 Interpolation at Airy Zeros 7 2 Interpolation of Step Functions We prove: Theorem 7. Let r > 0, and f denote the characteristic function χ [ r,0] of the interval [ r,0]. Then for p > 4 5, and lim L [ f ;x] f x p dx = 0. 6 lim L [ f ;x] f x p dx = 0. 7 This section is organized as follows: we first recall some asymptotics associated with Airy functions. Then we prove some estimates on integrals involving the fundamental polynomials l j. Next we prove the case p = 2 of Theorem 7. Then we estimate a certain sum and finally prove the general case of Theorem 7. Firstly, the following asymptotics and estimates are listed on pages of []: see there. Aix = 2π /2 x /4 exp 2 3 x 3 2 +o, x ; 8 2 Ai x = π /2 x [sin /4 3 x π ] + O x 3 2, x. 9 4 Then as Ai is entire, for x [0,, Aix C+x /4 exp Ai x C+x /4 ; 2 3 x 3 2 and 0 2 Ai x = π /2 x /4 cos + O x x π 4, x. +O Next, the zeros { a j } of Ai satisfy [, p. 450, ,96] a j = [3π 4 j /8] +O 2/3 3π j 2/3 = +o. 2 j 2 x Consequently,

8 8 D. S. Lubinsky In addition, a j+ a j = π +o. 3 a j /2 Ai a j = j π /2 3π 8 4 j /6 +O j 2 4 = j π /2 a j /4 +o. A calculation shows that Ai a j Ai a j = C0 j 5/6 +O j, C 0 = 3 /6 6 2π 2. 5 Define the Scorer function [, p. 448, ] Gix = t 3 sin π 3 + xt dt. 6 0 We shall use an identity for the Hilbert transform of the Airy function [24, p. 7, eqn. 4.4]: π PV Ait dt = Gix. 7 t x Here PV denotes Cauchy principal value integral. We also use [, p. 450, eqn ] 2 Gi x = π /2 x [cos /4 3 x π ] + o, x. 8 4 Finally the Airy kernel Aia,b satisfies [8, p. 432] Thus Aia j,sais,a k ds = δ jk Aia j,a j = δ jk Ai a j 2. Lemma. a As j, l j sl k sds = δ jk Aia j,a j = δ jk Ai a j 2. 9 b Uniformly in j,r with r > a j, 0 r l j tdt = π +o. 20 a j /2 l j tdt = π a j /2 +o+ O a j /4 r 3/4 r. 2 a j

9 Interpolation at Airy Zeros 9 Proof. a Now 7 yields l j tdt = Gia j π Ai a j 22 Here using 2, so from 8, 2 3 cos a j 2 + π 3 4 Substituting this and 4 into 22 gives b Using the bound 0, 0 = j + O, j Gia j = π /2 a j /4 j +o. l j tdt = π a j /2 +o. l j t C dt Ai a j 0 t /4 exp 2 3 t 3 2 dt t a j C a j Ai a j C a j 5/4, 23 recall 4. Next, r l j tdt = Ai a j Ai x dx r x+a j 24 = Ai a j x sin /4 2 π /2 3 x π 4 dx r x+a j x 7/4 + O x+a j dx, by 9. Here, r

10 0 D. S. Lubinsky x sin /4 23 x π I = dx r x+a j x 4 3 = r d dx [ cos 2 3 x π 4 x+a j = cos 2 3 r3/2 + π 4 r 3/4 r a j + ] 2 cos dx r 3 x π 4 [ = O r 3/4 r d a j + O r dx = O r 3/4 r a j as is decreasing in [r,. Next, x 3/4 x a j r x 7/4 x+a j dx [ d dx x 3/4 x a j ] x 3/4 x a j dx ] dx, 25 r a j r x 7/4 dx C r 3/4 r a j. Thus, using also 25 in 24, r l j tdt C a j /4 r 3/4 r. a j Together with 20 and 23, this gives the result 2. Lemma 2. Let L, and Then S L x = lim L a L+ L j= Proof. Using 9, and then 4, l j x = L j= Aix,a j Aia j,a j. 26 SL x χ [al+,0]x 2 dx = 0. 27

11 Interpolation at Airy Zeros SL x χ [al+,0]x 2 dx L j= = L j= Ai a j 2 2 L j= 0 a L+ l j xdx+ a L+ π = a j /2 +o { } L π 2 +o+ O j= a j /2 a L+ 3/4 a j /4 al+ + a L+ a j L π = a L+ a j /2 +o+o a L+ 3/4 L a j /4 a L+, a j j= by 4 and 2. Here using 3, L j= Also, from 2, j= 28 L π a j /2 = a j+ a j +o = al+ +o. 29 j= a L+ 3/4 L a j /4 a L+ a j j= C a L+ 7/4 Substituting this and 29 into 28, gives L j= j /6 j L+ L C a L+ 7/4 0 x /6 C a L+ 7/4 L 5/6 logl L+ x dx C a L+ /2 logl. 30 SL x χ [al+,0]x 2 dx = o a L+ +O C a L+ /2 logl = o a L+. Proof of Theorem 7 for p = 2. Given a /r, choose L = L by the inequality Then a L r < a L+. 3

12 2 D. S. Lubinsky By Lemma 2, L [ f ;x] = a j / [ r,0] l j x = L j= l j x. 32 as. Also, as, L [ f ;x] χ [al+,0]x 2 dx = L j= = o a L+ = o, χ[r,0] x χ [al+,0] x 2 dx = χ a [ L+,r] x2 dx = a L+ r a L+ a L l j t χ [al+,0]t 2 dt = O L /3 = o. Then 6 follows. Since L [ f ;x] = L [ f ;x] if L above is even, 7 also follows. The case of odd L is easily handled by estimating separately the single extra term. Next we bound a generalization of S L x: Lemma 3. Let A > 0, 0 β < 5 4 and { } c j be real numbers such that for j, j= c 2 j = c 2 j, 33 and c2 j A + a2 j β. 34 Let Ŝx = j= Then the series converges and for all real x, Here C is independent of x,a, and { c j } j=. c j l j x. Ŝx CA+ x β. 35 Proof. We may assume that A =. We assume first that x,0, as this is the most difficult case. Set a 0 = 0. Choose an even integer j 0 2 such that x [a j0,a j0 2. Let us first deal with central terms: assume that j and j j 0 3. Then l j x = Ai a j Aix A i a j x a j = Ai t Ai a j,

13 Interpolation at Airy Zeros 3 for some t between x and a j, so t + / a j. Using, 4, and the continuity of Ai, we see that Thus as x + a j0, and 34 holds, l j x + t /4 C C. a j /4 c j l j x C+ x β. 36 j: j j 0 3 Again, we emphasize that C is independent of L and x and { } c j. We turn to the estimation of j0 S 4 x = + c j l j x. j= j= j 0 +3 Recall from 4 that Ai a j has sign j. Then S x = Aix j0 4/2 k= + k= j 0 /2+2 c 2k [ ] Ai a 2k x a 2k Ai a 2k x a 2k = AixΣ + Σ 2, 37 where Σ = = j0 4/2 k= j0 4/2 k= + + k= j 0 /2+2 k= j 0 /2+2 c 2k x a 2k x a 2k Ai a 2k c 2k a 2k a 2k Ai a 2k x a 2k x a 2k 38 and Σ 2 = j0 4/2 k= + k= j 0 /2+2 c 2k Ai a 2k Ai. a 2k x a 2k Then if I denotes the set of integers k with either k j 0 4/2 or k j 0 /2+ x a 2, we see that 2k x a 2k, and a 2k a 2k are bounded above and below by positive a 2k 2 a 2k constants independent of k,x, so using 4 and 34, so

14 4 D. S. Lubinsky Σ C C k I a 2k 2 a 2k a 2k /4+β x a 2k 2 [ a,\[ a j0 3, a j0 + ] C [ [ a a a j0,\ j0 3 t /4+β x t 2 dt = ] a j0 5/4+β a j0, a j 0 + s /4+β 2 ds x a j0 a j0 s C a j0 β 3/4, β > 3/4 a j0 5/4+β log a j0, β = 3/4 + a j0 x a, β < 3/4 j0 + a j0 3 x a j0 + C a j0 /4+β + a j0 /2 a j0 /4+β C+ x /4 β, 39 recall that β < 4 5, and that x a j0 3 a j0 a j0 2 Ca j0 /2, by 3. Next, using 3 5 and 34, Σ 2 C k I C C k I a 2k β Ai a 2k Ai a 2k x a 2k k 5/6 a 2k a 2k 2 a 2k 5/4+β x a 2k t 5/4+β x t dt [ a,\[ a j0 3, a j0 + ] C [ ] [ a a a j0,\ j0 3 a j0, a j 0 + a j0 a j0 5/4+β C a j0 5/4+β C a j0 s 5/4+β ds x a j0 s /4+β + log x a j0 3 a j0 a j0 + log x a j0 + a j0 a j0 a j0 + a j0 5/4 β log j0 C+ x /4 β, recall β < 5 4. Substituting this and 39 into 37 gives S x C Aix + x /4 β C+ x β, in view of 0. This and 36 gives 35. Finally, the case where x 0 is easier. We deduce: Proof of Theorem 7 for the general case. Recall that f = χ [ r,0]. Assume first p > 2. The previous lemma with all c j = and β = 0 shows that

15 Interpolation at Airy Zeros 5 L [ f ;x] f x sup L [ f ;x] + C, x R where C is independent of. We can then apply the case p = 2 of Theorem 7: lim sup L [ f ;x] f x p dx C p 2 limsup L [ f ;x] f x 2 dx = 0. Next, if 4 5 < p < 2, and s 2r, Hölder s inequality gives s lim sup L [ f ;x] f x p dx s s limsup Next, for x s > 2r, and all, we have Thus L [ f ;x] f x L [ f ;x] f x 2 dx s a j [ r,0] C Aix x C x 5/4 p Aix Ai a j x a j a j [ r,0] a j /4 j /6 j Cr 3/2 C x 5/4 r 5/4 = Cr 5/4 x 5/4. 2 2s p 2 = lim sup L [ f ;x] f x p dx C x 5p/4 dx Cs 5p/4 0 x s x s as s, recall p > 5/4. Together with 40, this gives 6. Of course 7 also follows as L [ f ;x] differs from L [ f ;x] in at most one term, which can easily be estimated. 3 Proof of Theorems 5 and 6 Proof of Theorem 5. Suppose first that f is bounded and Riemann integrable, and supported in r,0], some r > 0. Let ε > 0. Then we can find a piecewise constant step function g also compactly supported in r,0] such that both

16 6 D. S. Lubinsky g f in R and f g 2 < ε 2. This follows directly from the theory of Riemann sums and the boundedness of f. Theorem 7 implies that for any such step function g, /2 lim gx L [g;x] dx 2 = 0. Then using also the orthonormality relation 9, lim sup f x L [ f ;x] 2 dx a j /2 /2 f x gx dx 2 + limsup /2 + limsup ε + 0+limsup = ε + limsup L [g f ;x] 2 dx = ε +C limsup j= a j r,0 a j r,0 gx L [g;x] 2 dx a 2 j f g l j x dx f g 2 a j Ai a j 2 /2 a j a j 0 /2 = ε +C f g xdx 2 Cε. r /2 f g 2 a j /2 /2 a Here C is independent of ε,g and f, and we have used 3, 4, that max j j 0 as, and the theory of Riemann sums. So we have the result for such compactly supported f. Now assume that f is supported in, r and for some β > 2, 2 holds. Then using 9 again,

17 Interpolation at Airy Zeros 7 f x L [ f ;x] 2 dx /2 f xdx 2 + r /2 a j, r /2 C + x dx 2β +C r Cr 2 β +C Cr 2 β +C a j, r where C is independent of r and. So a j f 2 a j Ai a j 2 a j, r /2 2β a j a j /2 2β a j a j /2 /2 +2β t dt 2β Cr /2 β, r /2 lim sup f x L [ f ;x] dx 2 Cr /2 β. This can be made arbitrarily small for large enough r. Together with the case above, this easily implies the result. Proof of Theorem 6. a From Lemma 3 with β = 0, a2 j l2 sup f j x+l 2 j x a2 j. C sup f j x R j= Here C is independent of f,. b For p = 2, the exact same proof as of Theorem 5 gives lim L [ f ;x] f x2 dx = 0. If p > 2, we can use the boundedness of the operators, to obtain lim sup L [ f ;x] f x p dx limsup C f p 2 L R limsup sup L [ f ;x] f x x R p 2 L [ f ;x] f x 2 dx = 0. Now let 4 5 < p < 2. Let s > 0. Hölder s inequality gives L [ f ;x] f x 2 dx

18 8 D. S. Lubinsky s lim sup L [ f ;x] f x p dx s s p limsup L [ f ;x] f 2 x 2 dx 2s p 2 = 0, 4 s by the case p = 2. Next, our bound 2 on f and Lemma 3 show that for all x, L [ f ;x] f x C+ x β. Then x s L [ f ;x] f x p dx C + x pβ dx Cs pβ 0 as s, s as β > p. Acknowledgements Research supported by NSF grant DMS References. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, M. R. Akhlaghi, On L p -Convergence of Nonpolynomial Interpolation, J. Approx. Theory, 55988, P. L. Butzer, J. R. Higgins, and R. L. Stens, Classical and Approximate Sampling Theorems: Studies in the L p R and the uniform norm, J. Approx. Theory, , P. Erdös and P. Turán, On interpolation, I. Quadrature and Mean Convergence in the Lagrange Interpolation, Ann. of Math , M. Ganzburg, Polynomial interpolation and asymptotic representations for zeta functions, Dissertationes Math. Rozprawy Mat., , 7 pp. 6. G. R. Grozev, Q. I. Rahman, Lagrange Interpolation in the Zeros of Bessel functions by Entire Functions of Exponential Type and Mean Convergence, Methods and Applications of Analysis, 3996, J. R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, Oxford University Press, Oxford, Eli Levin, D. S. Lubinsky, On The Airy Reproducing Kernel, Sampling Series, and Quadrature Formula, Integral Equations and Operator Theory, , F. Littman, Interpolation and Approximation by Entire Functions, in Approximation Theory XI, Nashboro Press, Brentwood 2008, pp D. S. Lubinsky, A Taste of Erdős on Interpolation, in Paul Erdős and his Mathematics, L. Bolyai Society Math. Studies II, Bolyai Society, Budapest, 2000, pp G. Mastroianni, G. V. Milovanovic, Interpolation Processes Basic Theory and Applications, Springer Monographs in Mathematics XIV, Springer, Berlin, P. Nevai, Lagrange Interpolation at Zeros of Orthogonal Polynomials, in Approximation Theory II, eds. G. G. Lorentz et al., Academic Press, New York, 976, pp

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