Hilbert Transforms and Orthonormal Expansions for Exponential Weights

Transcription

1 Hilbert Transforms and Orthonormal Expansions for Exponential Weights S. B. Damelin Abstract. We establish the uniform boundedness of the weighted Hilbert transform for a general class of symmetric and nonsymmetric weights on a finite or infinite interval I := (c, d) with c < < d. We then apply these results to study mean and uniform convergence of orthonormal expansions on the line.. Introduction and Statement of Results In this paper we shall study mean and uniform convergence of orthonormal expansions as well as uniform bounds for the weighted Hilbert transform for a large class of exponential decaying weights on an interval I := (c, d) with c < < d. For orthonormal expansions on [, ], there are many well known results for Chebyshev, Jacobi and generalized Jacobi weights starting from Riesz, and we do not review these here. Instead, we refer the reader to [9,2,24,28,32] and the many references cited therein for a detailed and comprehensive account of this vast and interesting subject. Our study involves exponentially decaying weights w on I and functions f which may grow at ± or near the endpoints of I. More precisely, in Theorems.2 and.7 below, we study both mean and uniform convergence of orthonormal expansions for a class of symmetric exponential weights on the line of polynomial decay at infinity, the so called Freud class which will be defined more precisely in Definition. below. Our mean convergence results are motivated by a recent result of Jha and Lubinsky in [2, Theorem.2] and indeed we will show how to improve this latter result. Our uniform convergence result, see Theorem.7, relies on good uniform bounds for the weighted Hilbert transform, see Theorems.6(A- B). We establish these latter bounds for a general class of symmetric and nonsymmetric weights on I. Approximation Theory X: Wavelets, Splines, and Applications Charles K. Chui, Larry L. Schumaker, and Joachim Stöckler (eds.), pp. 9. Copyright oc 2 by Vanderbilt University Press, Nashville, TN. ISBN All rights of reproduction in any form reserved.

2 2 S. B. Damelin. Statement of Results To set the scene for our investigations, let I := (c, d) where c < < d with c and d finite or infinite. (Note that I need not be symmetric about but should contain ). Let w be a nonnegative weight on I with x n w(x) L, n =,,... The idea of this paper arose, partly, from an interesting paper of Geza Freud [8] who studied the dependence of the greatest zero of p n (w 2 ), the nth orthonormal polynomial for w 2, on the corresponding recurrence coefficients. More precisely, given w as above, we recall, see [9,3], that p n := p n (w 2 ) admits the representation p n (x) = γ n x n +, γ n := γ n (w 2 ) > and satisfies I p n (x)p n (x)w 2 (x)dx = δ m,n, m, n. We denote by x j,n := x j,n (w 2 ), j n, the jth simple zero of p n in I, and we order these zeroes as x n,n < x n,n < x 2,n < x,n. If w is even, we write the three term recurrence of p n in the form xp n (x) = (w 2 )p n+ (x) + (w 2 )p n (x), n. Here, p =, p = ( w 2 (x) dx ) /2 and := (w 2 ) = γ n /γ n > are the corresponding recurrence coefficients. Given w as above, we may also form an orthonormal expansion f(x) b j p j (x), b j := j= for any measurable function f : I (, ) for which I I fp j w, j (.) f(x)x j w(x)dx <, j =, 2, 3... (.2) Let S n [.w] := S n [.], n denote the nth partial sums of the orthonormal expansions given by (.2).

3 Hilbert Transforms and Orthonormal Expansions 3 To state our main results, we require some additional notation. To this end, let us agree that henceforth C will denote a positive constant independent of x, y, n, f and j which may take on different values at different times. Moreover, for any two sequences b n and c n of nonzero real numbers, we shall write b n = O(c n ) if there exists a positive constant C, independent of n, such that b n Cc n, n and b n c n if b n = O(c n ) and c n = O(b n ). Henceforth, for functions and sequences of functions, O and will be uniform in x, y, n, f and j. We begin with the definition of a large class of admissible weights, see Definitions. (A-C) below. For clarity of exposition, explicit and easily absorbed examples of admissible weights are presented immediately after Definitions.(A-C). Definition.A. A function g : (, d) (, ) is said to be quasiincreasing if g(x) Cg(y), < x y < d. It is easy to see that any increasing function is quasi-increasing. Similarly, we may define the notion of quasi-decreasing. Definition.B. A weight function w : I (, ) will be called admissible if each of the following conditions below is satisfied: a) Q := log(/w) is continuously differentiable and satisfies Q() = ; b) Q is nondecreasing in I with lim Q(x) = lim Q(x) = ; x c + x d c) The function T (x) := xq (x) Q(x), x is quasi-increasing in (, d) and quasi decreasing in (c, ) with T (x) λ >, x I\; d) Given a sufficiently small B > along with positive ε and δ, log Q (x + δ) ( Q (x ε) ) B C,

4 4 S. B. Damelin for all x close enough to c and d. Definition.B suffices for uniform bounds on the weighted Hilbert transform, see Theorems.6(A-B) below. For Theorems.2,.3,.5 and.7, we need some additional smoothness and regularity assumptions on Q. Definition.C. An admissible weight w will be called strongly admissible if the following additional assumptions on w hold: a) There exists ε (, ) such that for y I\{} ( [ T (y) T y ε ]) ; T (y) b) For every ε >, there exists δ > such that for every x I\{}, x+ δx T (x) Q (s) Q (x) ds ε Q T (x) (x). s x 3/2 x x δx T (x).2 Examples The following are explicit examples of strongly admissible weights. Here and throughout, exp k denotes the kth iterated exponential. a) Symmetric exponential weights on the line of polynomial decay: w α (x) := exp ( x α ), α >, x (, ); (.3) b) Nonsymmetric exponential weights on the line with varying rates of polynomial and faster than polynomial decay: w k,l,α,β (x) := exp( Q k,l,α,β (x)) with Q k,l,α,β (x) := { expl (x α ) exp l (), x [, ), exp k ( x β ) exp k (), x (, ) (.4) where l, k and α, β > ; c) Nonsymmetric exponential weights on (, ) with varying rates of decay near ±: w k,l,α,β (x) := exp( Q k,l,α,β (x)) with Q k,l,α,β (x) := where l, k and α, β >. { expl ( x 2 ) α exp l (), x [, ), exp k ( x 2 ) β exp k (), x (, ), (.5)

5 Hilbert Transforms and Orthonormal Expansions 5.3 Remarks (a) Definitions.(A-C) define a very general class of possibly nonsymmetric exponential weights of minimal smoothness and with varying rates of decrease on (c, ) and [, d). The weights given by (a) are widely called Freud weights and are characterized by one rate of polynomial decay at ±. In this case T in Definition.B(c). Because of their current popularity, we shall henceforth adopt the name Freud weight in what follows. However, we also allow nonsymmetric weights of polynomial and faster than polynomial decay at c and/or d respectively. Notice that our definition allows w to decrease with one rate on [, d), and with another on (c, ). For a detailed perspective on this class of weights and its applications to orthogonal polynomials and various weighted approximation problems of current interest, we refer the reader to [5,6,7,8,9,2,3,5,2,22,23, 24,25,28,29,3] and the many references cited therein. (b) Definitions.B (a-d) involve smoothness and regularity conditions on w which suffice for uniform bounds on the weighted Hilbert transform, see Theorem.6(A-B) below. Note that when w decays as a polynomial, T, whereas otherwise T increases without bound. Definition.B(d) is needed to control the behavior of Q close to c and d when Q grows very quickly. It is trivially satisfied when Q grows as a polynomial. (c) The definition of a strongly admissible weight, see Definition.C, is necessary in proving Theorems.2,.3,.5 and.7. Here, we need bounds on p n and estimates for the spacing of its zeroes, which in turn require additional smoothness and regularity assumptions on Q. Definition.C(b) is a local lip /2 condition on Q, and appeared first in [22]. Notice that we do not require Q to exist everywhere. Definition.C(d) first appeared in [5,5], although it is motivated by a much older growth lemma of E. Borel. We apply it heavily in the proof of Theorem.3 below. We need some additional notation: If w is strongly admissible and even, it is well known, see [22], that the asymptotic behavior of the recurrence coefficients is expressed in terms of the scaled endpoints of the support of the equilibrium measure for w which is one interval. It is also well known, see [22,24,29,3] and the references cited therein, that firstly these scaled endpoints can be explicitly calculated and are given as the positive roots of the equation u = 2 π a u tq (a u t), t 2 and secondly that lim = /2 n a n

6 6 S. B. Damelin and + lim =. n Here, the number a u is, as a real valued function of u, uniquely defined and strictly increasing in (, ) with lim a u = c, u lim a u = d. u We refer the interested reader to [,4,6,3] and the references cited therein for recent and related analogues of the above circle of ideas for weights whose equilibrium measure is supported on more than one interval..4 Mean convergence of orthonormal expansions We are ready to state our first result. Theorem.2. Let I = (, ), w be a symmetric strongly admissible Freud weight, and let < p <. Suppose b, B I satisfy b < /p, B > /p, b B. In addition suppose that if p < 4/3 then (I): a max{b, /p} B n n /6(4/p 3) C B = O(), (II): if p = 4/3 or 4 then b < B, and if p > 4 then a b min{b, /p} n n /6( 4/p) C B = O(). Suppose also that + = + O ( ), n. (.6) n Then there exists an infinite subsequence n = n j of natural numbers such that for j C and for all f satisfying (.2) S n [f]wu b Lp (I) C fwu B Lp (I). (.7) Moreover, if (.7) holds for some infinite subsequence n j and real b and B, then necessarily the following hold: If p < 4/3 then (I): b < /p, B > /p, b B. a max{b, /p} B n n /6(4/p 3) C B = O(),

7 Hilbert Transforms and Orthonormal Expansions 7 (II): If p = 4/3 or 4 then b < B, and if p > 4 then In particular, given δ >, we have for all continuous f with an b min{b, /p} n /6( 4/p) C B = O(). lim (S n[f] f)wu b Lp (I) = (.8) j lim fwu B+δ (x) =. x In [2, Theorem.2], Theorem.2 is established for every n assuming in addition to (.6) the assumption that = [ ( )] + O, n. (.9) a n 2 n 2/3 For the Hermite weight, [2, Theorem.2] essentially appears in earlier papers of Askey and Waigner, see [] and Muckenhoupt, see [26] and [27]. Interesting generalizations of the work of Muckenhoupt in [26] and [27] have also been proved by Mhaskar and Xu in [25]. For a detailed discussion of the conditions (.6) and (.9), we refer the reader to Remark.4 below. In order to establish Theorem.2, we use the following theorem of independent interest. Theorem.3. Let w be strongly admissible and symmetric. If ( ) + = + O, n (.) (nt (a n )) 2/3 then there exists a subsequence n = n j of natural numbers such that = [ ( + O a n 2 The following remark suffices: Remark.4. (nt (a n )) 2/3 )], j. (.) (a) Observe that (.) is a strong asymptotic whereas (.) is a ratio asymptotic. Thus applying the well-known identity a n+ a n nt (a n ),

8 8 S. B. Damelin see [22], it is clear that (.) implies (.) for every n. It is the other direction which is new and nontrivial for strongly admissible weights w. (b) Fortunately, the hypotheses on the recurrence coefficients given by (.6) and (.) are not always vacuous ones. We list some related results on the line, and refer the interested reader to [7,6,2,22,23,28] and the references cited therein for further references and insights. (A) Let w = exp( Q), where Q is an even polynomial of fixed degree. Then (see [6]), = [ ( )] + O a n 2 n 2, n. (B) Let w = W exp( Q), where Q is an even polynomial of fixed degree with nonnegative coefficients and W (x) = x ρ for some real ρ greater than. Then (see [7] and [23]), = [ ( )] + O, n. a n 2 n Note that when ρ, w is not always admissible. (C) Let w = w α be given by (.3). Then (see [2]), = [ + O a n 2 ( )], n. n (D) Let m and w = W w,,2m,2m given by (.4). Then (see [7]), = [ ( )] T (an ) + O, n. a n 2 n As a consequence of Theorem.3, we now state Theorem.5. Let I = (, ), w be a strongly admissible symmetric Freud weight and assume that the recurrence coefficients satisfy (.6). Then there exists N and a infinite set of natural numbers Ω such that { sup x I p n+ (x) p n (x) w(x) x } /4 + n 2/3 a /2 n for all n N and n N, n Ω. a n (.2) The importance of (.2) lies in the fact that for x close to a n, it improves the known bound (see [22]) { sup x I p n (x) w(x) x } /4 + n 2/3 a /2 n (.3) a n

9 Hilbert Transforms and Orthonormal Expansions 9 by a factor of /4 as it should. Under the additional assumption (.9), (.2) is [2, Theorem.] for n. We turn our attention to uniform convergence of orthonormal expansions for strongly admissible symmetric Freud weights on (, ). To this end, we will need bounds on weighted Hilbert transforms. Define formally for continuous f : I (, ) f(t) H[f](x) := lim ε + x t x t ε where the integral above is understood as a Cauchy-Principal valued integral. It is known, see [26,27], that if b < /p, B > /p, b B and < p <, we have H[f]u b Lp ((, )) C fu B Lp ((, )), (.4) provided the right hand side of (.4) is finite. Indeed, relations such as (.4) are essential in studying convergence of orthonormal expansions. This is mainly due to the following identity which follows from the Christoffel-Darboux formula for orthonormal polynomials: S n [f] = {p n H[fp n ] p n H[fp n ]}. (.5) For uniform convergence of orthonormal expansions, it thus seems natural to look for L analogues of (.4), but until recently, even for finite intervals, such analogues have been scarce in the literature. We mention that uniform bounds for weighted Hilbert transforms are also important for the numerical solution and stability of integral equations on finite and infinite intervals, see [] and the references cited therein. To this end, we now state two theorems which hold for any admissible weight on I. We refer the reader to [2,3,4,8,9] and the references cited therein for related results. We recall that for h >, h(f, I)(x) := f(x + h/2) f(x h/2), x ± h/2 I (.6) is the first symmetric difference operator of f. Then we have: Theorem.6A. Let f : I (, ) be measurable, w be admissible, B, ε > and suppose fw( + Q ) B L (I) < and w u (f,i) L (I) u L [, ε]. Then H[fw] L (I) [ C fw( + Q ) B L (I) + ε w u(f, I) L (I) u du ]. (.7)

10 S. B. Damelin Theorem.6B. Let f : I (, ) be differentiable, w be admissible, B > and suppose fw( + Q ) B L (I) < and f w L (I) <. Then H[fw] L (I) C [ fw( + Q ) B L (I) + f w L (I)]. (.8) Note the factor ( + Q ) B on the right-hand side of (.7) and (.8). For Q even and of polynomial growth, (the case T ), we may take it to be essentially u B for some B >. Theorems.6(A-B) improve [9, Theorem.] and [4, Theorem ] in several respects. Firstly they replace the weighting factor of w 2 in [9] by the correct factor w(+ Q ) B. This later observation is crucial in the formulation of Theorem.7 below. Secondly, Theorems.6(A-B) hold simultaneously for a much larger class of possibly nonsymmetric weights with varying rates of decay on (c, ) and [, d). Using Theorems.6(A-B), we are able to announce: Theorem.7. Let I = (, ), w be a strongly admissible symmetric Freud weight, B >, assume (.6) and further that a b min{b,} n n /6 C B = O(), n where C B is if B and log n if B =. Let f : I (, ) be differentiable and suppose that f(t)(t x) has fixed sign in [x ε, x + ε] for some small and positive ε. Suppose finally that fwu B L (I) < and (fw) L [x,x+ε] <. Then there exists an infinite subsequence n = n j of natural numbers such that for all j C S n [f]w (x) C [ fwu B L (I) + (fw) L [x,x+ε]]. (.9) If f is also continuous and for some δ >, then lim x fq wu B+δ (x) = lim (S n[f] f)w (x) =. (.2) j If we assume (.9), then (.2) holds for every n. Theorem.7 provides sufficient conditions for uniform convergence of orthogonal expansions for strongly admissible Freud weights on the line. Its proof will appear in a future paper. The remainder of this paper is devoted to the proofs of Theorem.2, Theorem.3, Theorem.5 and Theorems.6(A-B).

11 Hilbert Transforms and Orthonormal Expansions 2. Proofs of Theorems.2,.3,.5 and.6(a-b) In this section, we prove Theorems.2,.3,.5 and.6(a-b). 2. The Proof of Theorem.3. Let n C, and define m = m(n) = [n /3 (T (a n )) /3 ] where [x] denotes the greatest integer x. It follows using Definition.C (a) and Remark.4(a) that uniformly for r =,..., m and n, T (a n+r ) T (a n ). Armed with this identity, we apply (.) repeatedly and deduce that there exists N = N(m) and D > such that for n N, ( ) D +r A (nt (a n )) 2/3 n, r =, 2,..., m. (2.) Here D > does not depend on n or m so we fix it. Now set D ε = ε(n) = (nt (a n )). 2/3 A careful adaption of the proof of [8, Theorem 6] then shows that π x,n (w) 2( ε) cos m + 2( ε) ( C/m 2 ) ( ) ( 2 Now recall that Thus (2.2) and (2.3) give a n 2 D (nt (a n )) 2/3 ( x,n a n = O n 2/3 T (a n ) 2/3 [ + O ( (nt (a n )) 2/3 ) C. (nt (a n )) 2/3 (2.2) ). (2.3) )], n C. (2.4) Another careful inspection of [8, Theorem 7], reveals that we have x,n lim 2. n max k n α k Thus, we may apply this with (2.3) to obtain max k n A k ( /C a n 2 Choosing an increasing sequence n r with and applying (2.4) (2.6) gives the theorem. (nt (a n )) 2/3 ). (2.5) max A k = r, (2.6) k n r

12 2 S. B. Damelin 2.2 Proof of Theorems.2 and.5. We sketch the important ideas of the proof. The remaining technical details are very similar to [2, Theorem.], and so we refer to reader to that paper for these. Step : We reduce the proof of Theorem.5 to one important case. (a) By symmetry we may assume that x. (b) It suffices to prove (.2) for n sufficiently large. (c) Using infinite finite inequalities it suffices to prove (.2) for ( x a n C ). n 2/3 (d) For x a n/2, so (.2) follows from (.3). x + n 2/3 Step 2: Without loss of generality we may thus assume that a n ( a n/2 x a n C ) n 2/3 for some C > which will be chosen later. Let us define for y τ n (y) := n A 2 (αk+ 2 αk)p 2 2 k(y). n k= Then the Dombrowski-Fricke identity, see [7], gives p n+ (y) p n (y) w(x) { 2τ n+ (y)w 2 (y) } /2 + { 2τn (y)w 2 (y) } /2, (2.7) for y 2min {, + }. Applying Theorem.3, we deduce that there exists D > and an infinite set of natural numbers Ω such that (2.7) holds for all y a n ( Dn 2/3), n Ω.

13 Hilbert Transforms and Orthonormal Expansions 3 Choose C = D in the above and assume, as we may, that (2.7) holds for x and n Ω. Step 3: Estimation of τ n (x) for n Ω: We write τ n (x) := A 2 n [n/4] + k= n k=[n/4]+ (α2 k+ α 2 k)p 2 k(x) Now applying (.6), we have that = τ n, (x) + τ n,2 (x). τ n,2 (x)w 2 (x) C n w2 (x)λ n (x), where λ n are the Cotes numbers for w 2. But for this range of x, it is well known that λ n (x) w 2 (x) n x a n + n 2/3. a n Thus, τ n,2 (x)w 2 (x) Ca n x + n 2/3. a n Moreover, using infinite-finite range inequalities yields τ n, (x)w 2 (x) = O (exp( Cn)). Combining the above estimates for both τ n, and τ n,2 yields Theorem.5. Theorem.2 then follows using [2, Theorem.2], Theorem.5 and Theorem.3 setting n = n j. The crucial point in the proofs of both Theorem.2 and Theorem.5 is the removal of the strong asymptotic (.9), and this is achieved by means of Theorem The Proof of Theorems.6(A-B). We begin with Theorem.6(A). We may assume that x [, d). The other case is similar. We consider several subcases. We suppose first that d =, c =, and that x is sufficiently close to d. More precisely, let us choose a constant D > so close to d so that for D < x < d Q (x + ε) < ε.

14 4 S. B. Damelin (We will in practice always take ε small since clearly if Theorem.6(A-B) holds for such ε, then it holds for larger ε). Fix x and define A(x) = A ε (x) := 2Q (x + ε). Note Q (y) = Q (y) for y close enough to d, and Q (y) = Q (y) for y close enough to c. This follows from the definition of T and Definitions.(B-C). Let us write H[fw](x) = ( We first estimate I (x). Here so Similarly, I 2 (x) = I (x) 2 t >2x t >2x + + 2x 2x = I (x) + I 2 (x) + I 3 (x). t x t x t t /2 = t /2 f(t) w(t) t C fw( + Q ) B L (I) C fw( + Q ) B L (I). 2x f(t)w(t) t x C fw( + Q ) B L (I) C fw( + Q ) B L (I). t 2 ) f(t)w(t) t x fw( + Q ) B L (I) 3x x du u t ( + Q (t) ) B 2x x t (2.8) We proceed with I 3 (x). Note that by choice of A(x) and using [22, Lemma 3.2a], w(y) w(x) uniformly for every y I with x y 2A(x). (The latter lemma implies that Q(s) Q(r), s/r.) We then split I 3 (x) as follows: Write for β := β(ε) >, 2x f(t)w(t) I 3 (x) = t x ( x/β x ε = + x/β x A(x) x+a(x) ) 2x x ε x A(x) x+a(x) = I 3 (x) + I 32 (x) + I 33 (x) + I 34 (x) + I 35 (x). f(t)w(t) t x

15 Hilbert Transforms and Orthonormal Expansions 5 Then, Similarly, Next, Similarly, I 3 (x) C fw( + Q ) B L (I) C fw( + Q ) B L (I) C fw( + Q ) B L (I). x/β x/β ( + Q (t)) B x t x t I 32 (x) C fw( + Q ) B logx L (I) ( + Q (x/β)) B C fw( + Q ) B L (I). I 33 (x) C fw( + Q ) B L (I) I 35 (x) = x A(x) x ε C fw( + Q ) B log Q (x + ε) L (I) ( + Q (x ε)) B C fw( + Q ) B L (I). 2x x+a(x) f(t)w(t) t x C fw( + Q ) B L (I)( + Q (x + A(x))) B x ( + Q (t)) B x t A(x) u du C fw( + Q ) B L (I)( + Q (x + A(x))) B (log Q (x + ε) + log x) C fw( + Q ) B L (I). Finally we consider I 34 (x). We write I 34 (x) = x+a(x) x A(x) x+a(x) x A(x) f(t) f(t)w(t) t x w(t) w(x) t x = I 34 (x) + I 342 (x). + w(x) x+a(x) x A(x) f(t) f(x) t x

16 6 S. B. Damelin We begin by making the substitution t = u/2 + x into I 342 (x). Then we have 2A(x) I 342 (x) Cw(x) f(x + u/2) f(x u/2) u du Also I 34 (x) = C C x+a(x) x A(x) 2A(x) ε C fw( + Q ) B L (I) u (f, I)w L (I) u du u(f, I)w L (I) u du. w(t) w(x) f(t) t x x+a(x) x A(x) w (t) w (η) CA(x)w(x A(x))w (x + A(x))Q (x + A(x)) fw( + Q ) B L (I) C fw( + Q ) B L (I). We observe that if x < D, then the estimates for I (x) go through as before as we only needed the fact that D > to ensure that the integral converged for t much larger then x. I 2 (x) follows without change except that we use the boundedness of Q rather than its sign. For I 3 (x), x is bounded and w, so the proof is easier than before. If < x <, then we write ( f(t)w(t) x x+ ) w(t)f(t) H[f; w](x) = t x = + + x x+ t x. For the first two integrals, t is bounded away from x, and for the third we proceed as above, but the proof is easier since x is bounded and w. Suppose now that d and c are finite. Let us first suppose that x is close to d. Then choose D > such that D x < d, and write ( H[f; w](x) = + c d ) w(t)f(t) t x. For the first integral, t x is bounded away from, so bounding this term gives us the required estimate. For the second integral, split as in I 3 (x). (Note that here we choose ε at the start small enough so that x ± ε I). A very similar and easier argument works if also γ x D for some fixed and positive γ, for here w. If for this γ, x < γ, then split H[f; w](x) = d c f(t)w(t) t x = ( x γ + c x+γ x γ + d x+γ ) w(t)f(t) t x.

17 Hilbert Transforms and Orthonormal Expansions 7 Then the estimate goes through very much as above. Finally, suppose that d is finite and c =. (The other case is similar). Then choose A >, and write ( Ax ) d w(t)f(t) H[f; w](x) = + + Ax t x. This proves Theorem.6(A). Theorem.6(B) follows by noting that in the proof of Theorem.6(A), for a given x, we may take ε small enough so that w(x ± u/2) w(x) for all u ε. This easily yields the result. Acknowledgments The author is is indebted to Larry Schumaker and an an anonymous referee who helped to improve this paper in many ways. References. Askey, R., and S. Wainger, Mean convergence of expansions in Laguerre and Hermite series, J. Math 87, De Bonis, M. C, B. Della Vecchia, and G. Mastroianni, Approximation of the Hilbert transform on the real semiaxis using Laguerre zeros, preprint. 3. De Bonis, M. C, and G. Mastroianni, Some simple quadrature rules to evaluate the Hilbert transform on the real line, preprint. 4. De Bonis, M. C, B. Della Vecchia, and G. Mastroianni, Approximation of the Hilbert transform on the real line using Hermite zeroes, Mathematics of Computation, to appear. 5. Damelin, S. B., Converse and smoothness theorems for Erdös weights in L p, ( < p ), J. Approx. Theory 93 (998), Damelin, S. B., On the distribution of interpolation arrays for exponential weights, Electronic Transactions of Numerical Analysis, to appear. 7. Damelin, S. B., Asymptotics of recurrence coefficients for exponential weights, Magnus s method revisited, manuscript. 8. Damelin, S. B., and K. Diethelm, Interpolatory product quadratures for Cauchy principal value integrals with Freud weights, Numer. Math 83 (999), Damelin, S. B., and K. Diethelm, Boundness and numerical approximation of the weighted Hilbert transform on the real line, Numerical Functional Analysis and Optimization 22(,2), (2), Damelin, S.B., and K. Diethelm, The numerical approximation and stability of certain integral equations on the line, manuscript.

18 8 S. B. Damelin. Damelin, S.B., P. Dragnev, and A. B. J. Kuijlaars, The support of the equilibrium measure for a class of external fields on a finite interval, Pacific J. Math 99(2) (2), Damelin, S. B., H. S. Jung, and K. H. Kwon, Mean convergence of Hermite Fejèr and Hermite interpolation of higher order for Freud type weights, J. Approx. Theory 3 (2), Damelin, S. B., H. S. Jung, and K. H. Kwon, On mean convergence of Lagrange and extended Lagrange interpolation for exponential weights: Methods, recent and new results, manuscript. 4. Damelin, S. B., and A. B. J. Kuijlaars, The support of the extremal measure for monomial external fields on [, ], Trans. Amer. Math. Soc. 35 (999), Damelin, S. B., and D. S. Lubinsky, Jackson theorems for Erdös weights in L p ( < p ), J. Approx. Theory 94 (998), Deift, P., T. Kriecherbauer, K. McLaughlin, S. Venakides, and X. Zhou, Strong asymptotics of orthonormal polynomials with respect to exponential weights, Commun. Pure Appl. Math. 52 (999), Dombrowski, J. M., and G. H Fricke, The absolute continuity of phase operators, Trans. Amer. Math. Soc, 23, Freud, G., On the greatest zero of orthonormal polynomials, J. Approx. Theory 46 (986) Freud, G., Orthonormal Polynomials, Akadémiai Kiadó, Budapest, Jha, S. W., and D. S. Lubinsky, Necessary and sufficient conditions for mean convergence of orthonormal expansions for Freud weights, Constr. Approx. (995), Kriecherbauer, T., and K. McLaughlin, Strong asymptotics of polynomials orthonormal with respect to Freud weights, International Mathematics Research Notices 6 (999), Levin, A. L., and D. S. Lubinsky, Orthonormal Polynomials for Exponential Weights, Springer Verlag Magnus, A. P., On Freud s equations for exponential weights, J. Approx. Theory 46 (986), Mhaskar, H. N., Introduction to the Theory of Weighted Polynomial Approximation, World Scientific, Singapore, Mhaskar, H. N., and Y. Xu, Mean convergence of expansions in Freud type orthogonal polynomials, SIAM J. Math. Anal 22, Muckenhoupt, B., Mean convergence of Hermite and Laguerre series I, Trans. Amer. Math. Soc 47,

19 Hilbert Transforms and Orthonormal Expansions Muckenhoupt, B., Mean convergence of Hermite and Laguerre series II, Trans. Amer. Math. Soc 47, Nevai, P., Geza Freud, orthogonal polynomials and Christoffel functions (a case study), J. Approx. Theory 48 (986), Rakhmanov, E. A., On asymptotic properties of polynomials orthogonal on the real axis, Mat. Sb. 9 (982), English transl: Math. USSR-Sb. 47 (984), Stahl, H., and V. Totik, General Orthogonal Polynomials, Cambridge University Press, Cambridge, Totik, V., Weighted Approximation with Varying Weight, Lect. Notes in Math. Vol. 569, Springer-Verlag, Berlin, Xu, Y., Mean convergence of generalised Jacobi series and interpolating polynomials, J. Approx Theory 72, Steven B. Damelin Department of Mathematics and Computer Science Georgia Southern University Post Box 893 Statesboro GA USA damelin@gasou.edu