SERIES REPRESENTATIONS FOR BEST APPROXIMATING ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

Transcription

1 SERIES REPRESENTATIONS OR BEST APPROXIMATING ENTIRE UNCTIONS O EXPONENTIAL TYPE D. S. LUBINSKY School of Mathematics, Georgia Institute of Technology, Atlanta, GA lubinsky@math.gatech.edu Abstract. Let > not be an even integer. We derive Lagrange type series representations for the entire function of exponential type that minimizes x f (x) Lp(R) amongst all such entire functions f, when p = and p =. This minimum arises as the scaled limit of the L p error of polynomial approximation of x on [ ; ], and is one representation of the L p Bernstein constant.. Introduction Let > be not an even integer. S.N. Bernstein [], [3] established the limit ; = lim n! n E n [x ; L [ ; ]] ; where E n [f; L p [a; b]] = inf k f P k Lp[a;b]: deg (P ) n denotes the error in best L p approximation of a function f on [a; b] by polynomials of degree n. Bernstein s 938 method yielded a formulation of the limit as the error in approximation on the whole real axis by entire functions of exponential type, namely ; = inf k x f (x) k L(R): f is entire of exponential type : Recall here that f is of exponential type A means that for each " >, and for z large enough, f (z) exp (z (A + ")) : Date: September, 5. Research supported by NS grant DMS4446

2 D. S. LUBINSKY Moreover, A is the smallest number with this property. The exact value of ; is not known for any, and the search for it has inspired much research. See [5], [6], [8], [9] for references. There are extensions to spaces other than L. It is known [6], [9] that for p, there exists p; = lim n + n! p En [x ; L p [ ; ]] = inf k x f (x) k Lp(R): f is entire of exponential type : Only for p = and p = is p; known explicitly. Nikolskii [] proved that sin X ; = 8 ( + ) ( ) k (k + ) (for some, while Bernstein observed this remains true for all ) and Raitsin [] proved that sin ; = ( + ) p = ( + ): There is a series representation for the minimizing entire function in L. The author is not certain who rst derived it, but it appears in a recent paper of Ganzburg [6]. An integral representation is given below. In the case p =, M. Ganzburg has informed the author that one can can use ourier transforms, the Paley-Wiener Theorem, and the theory of distributions, to derive a representation. It is the purpose of this paper to derive a series representation for the minimizing entire function in L, using the same method that can be used in L. There is a substantial body of estimates for approximation by entire functions of exponential type, when the approximated function is bounded or has bounded derivatives of some order [], [4], [], [3], [4]. With a view to applications in number theory, there are also explicit representations of the best approximating entire function when p = and g is one of a special class of functions. or example for g (x) = sign (x), the best L entire function was determined by Vaaler [5]. or other special g, it can be determined using the theory of minimal extrapolations [, Chapter 7]. Quite recently Littman [7] has used these ideas to determine a representation for the best L entire function when g (x) = x n +, that is g (x) = x n in [; ) and is on the negative real axis. Then one can deduce from this the extremal entire function for g (x) = x n = x n + x n. or odd integers n, this gives another approach to the L results of this k=

3 BEST ENTIRE UNCTIONS 3 paper. It is not clear that the approach there can be extended to all >. Our approach is based on an integral representation established in [9, Theorem.3, and remarks thereafter]: Theorem. Let p and > ; not an even integer. The unique entire p function Hp; of exponential type satisfying (.) k x H p; (x) k Lp(R) = inf k x f (x) k Lp(R): f is entire of exponential type admits for Re z the representation (.) Hp; (z) = z + sin p; (z) where (.3) p; (z) = and Y = Z! z ; (.4) < x < x < x 3 < ::: with (.5) x 3 ; In the special case p =, ; (z) = cos z: rom this, we shall derive for p = : Theorem. Let > ; not an even integer. Then (.6) H ; (z) = ; (z) fp (z) + z` where ` is the even integer in ( of degree at most `, (.7) P (z) = sin `= X = x s + ds s + z p; (is) ; ; : X = z x `+ x ; x g; ; + ], and P is a polynomial z Z s ; (is) ds: or p =, where ; (z) = cos (z), the representation is more explicit:

4 4 D. S. LUBINSKY Theorem.3 Let > ; not an even integer. Then (.8) H ; (z) = (cos z) fp (z) + z` X = ( ) `+ z g; where ` is the even integer in ( of degree at most `, (.9) P (z) = sin `= X = ; + ], and P is a polynomial z Z s cosh s ds: Remarks (a) or an odd integer, the case p = may be deduced from results of Littman [7]. However, for all, the result of Theorem. is known, and appears in a paper of Ganzburg [6]. The author is not sure where it rst appeared. (b) There are several ways to prove Theorems. and.3. The simplest is based on the integral representation (.) and the residue theorem. Another approach is to take scaled limit of polynomials of best L p approximation on [ ; ]. (c) When =, the representation simpli es to H ; (z) = cos (z) f + z X = ( ) z g: (d) Let >. We note that because of homogeneity of x, the entire function H of type that best approximates x in L p (R) is ust (.) H (z) = H p; (z) ; This follows as H (z) is of exponential type whenever H is of exponential type.. (e) One may also derive a representation for H; based on the observation that the best L polynomial approximation of x is ust a partial sum of its orthonormal expansion in Legendre polynomials. or example, for < <, the representation is H ; (z) = Z s J (x; s) ds;

5 BEST ENTIRE UNCTIONS 5 where J (x; s) is a kernel familiar in universality theory, J (x; s) = sin (x + s) sin (x s) + x + s x s x sin x cos s s sin s cos x = : x s There is an analogue for all >, and this provides a di erent representation to the ourier transform one that may be derived from Raitsin s work. We prove the results in Section. Throughout, C; C ; C,... denote positive constants independent of variables x; z; s; t and indices ; k; m; n; :::. The same symbol may denote di erent constants in different occurrences.. Proofs We shall concentrate on the proof for p = and then indicate the di erences for p = : We begin by summarizing results from [9]. Lemma. Let >, not an even integer. Then (a) There exist alternation points y and (.) y H ; y = with = y < x < y < x < y < ::: = ( ) + Here is the least integer exceeding. (b) or ; (.) ( ) ; y Cy : (c) There exist C and C such that for Im z ; x H ; (x) L (R) : (.3) ; (z) C cos z z C : Proof (a) This is part of Theorem. in [9]. (b) In [9], we transformed approximation over the whole real line to the non-negative axis [; ), involving a factor x =p in the L p norm. In

6 6 D. S. LUBINSKY the case of the L norm, this factor is, and we considered an entire function H ;= such that x = H ;= (x) = inf L[;) x = f (x) ; L[;) where the inf is over all entire functions f such that f (x ) is of exponential type. In Section of [9], we showed that z H; (z) = H ;= : H ;= admits an integral representation like that in (.), involving a function ;= and also has alternation points fy g. The relationship between these quantities and the corresponding ones for H; was established in [9, Section ], and is z ; (z) = ;= ; (.4) y = p y : It was also shown in [9, Theorem.] that ( ) ;= (y ) Cy : Then the identities (.4) give the result. (c) In [9, Theorem 7.(d)], it was shown that log ;= (z) log cos p z log z c + log + z + C; where c is the closest zero of cos p or ;= () to z. Now the substitution z! z easily yields (.3). Proof of Theorem. Assume that P is given by (.7) and H; by (.). We start with the identity `= X = P (z) = H ; (z) z ; (z) z = s s z + s + ( sin )`=+ z ` s s z + s ; which follows from the formula for a nite geometric sum. We multiply sin by s = ; (is), integrate over (; ), and use (.) and (.7). This gives for Re (z) > ; Z + ( )`=+ z` (.5) s `+ (z + s ) ; (is) ds:

7 BEST ENTIRE UNCTIONS 7 We now express the integral in the last right-hand side as a contour integral over the imaginary axis, assuming that Re (z) >. Since we see that (.6) i `+ + ( i) `+ = sin ( )`=+ ; = = i sin Z Z L ( )`=+ Z (is) `+ + ( is) `+ z (is) ; (is) ds t `+ (z t ) ; (t) dt; s `+ (z + s ) ; (is) ds where L is the directed contour along the imaginary axis from i to i. We next re-express this as an in nite series. To start, let R = yn for some large n, and consider a rectangular positively oriented contour in the right-half plane determined as follows: let L R be a vertical line segment along the imaginary axis from ir to ir. Let M R denote a vertical line segment from R ir to R+iR. inally, let H R denote the two horizontal line segments with Im (z) = R oining L R and M R. Note that L R has the opposite orientation to L. If R is large enough so that this contour encloses z, the integrand in (.6) has simple poles at z and at x ; n. The residue theorem gives (.7) = Z Z Z + + i L r M R H R z `+ nx z; (z) + = z t `+ (z t ) ; (t) dt x `+ x ; x We next show that the integral over M R [H R! as R = y n!. irstly, since ; has only real zeros, we see that for t = R+is M R ; ; (R + is) ; (R) CR ; by Lemma.(b). Then t i ZMR `+ (z t ) ; (t) dt C R `+ R z Z R CR `! ; R : ds ; (R + is)

8 8 D. S. LUBINSKY as ` <, recall our choice of `. Next by Lemma.(c), and as cos (s ir) sinh (R) ; t i ZHR `+ (z t ) ; (t) dt C R `+ Z R ds R z ; (s ir) CR C 3 = sinh (R)! ; R!. Thus, letting R = yn! in (.7), and recalling the opposite orientation of L R and L gives Z t `+ i L (z t ) cos (t) dt = z `+ z; (z) + X x `+ : z x ; x Combining this, (.5) and (.6), gives P (z) = H ; (z) ; (z) z + z+ z ; (z) z` = X = z x `+ x : ; x So we obtain (.6) for Re (z) >. Its validity follows for all complex z, by analytic continuation. Proof of Theorem.3 Here we follow exactly the same steps, except that the calculations are easier because ; (z) = cos z; and so the zeros x are x = ; ; while ; x = ( ) : or the dimensions of the contour above, we choose R = n, and in estimating the integral over M R, use cosh (R + is) sinh (s) as well as cosh (R + is) ; s C; some C independent of R = n. Then i (z t ) cos (t) dt C R `+ R z ZMR t `+ Z R R ds cos (R + is) Z CR ` ds max ; sinh (s)! ;

9 R! as ` <. BEST ENTIRE UNCTIONS 9 References [] N.I. Achieser, Theory of Approximation, (translated by C.J. Hyman), Dover, New York, 99. [] S.N. Bernstein, Sur la Meilleure Approximation de x par des Polynomes de Degre Donnes, Acta Math., 37(93), -57. [3] S.N. Bernstein, Sur la Meilleure Approximation de x p par des polynomes de degres tres eleves, Bull. Acad. Sc. USSR, Ser. Math., (938), 8-9. [4] D.P. Dryanov, M.A. Qazi, and Q.I. Rahman, Entire unctions of Exponential Type in Approximation Theory, (in) Constructive Theory of unctions, (ed. B. Boanov), DARBA, So a, 3, pp [5] M. I. Ganzburg, Limit Theorems and Best Constants of Approximation Theory (in), Handbook on Analytic Computational Methods in Applied Mathematics, (G. Anastassiou, ed.), CRC Press, Boca Raton, l.. [6] M. Ganzburg, The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes, J. Approx. Theory, 9(), [7]. Littman, Entire Approximations to the Truncated Powers, to appear in Constructive Approximation. [8] D.S. Lubinsky, Towards a Representation of the Bernstein Constant, (in) Advances in Constructive Approximation, (M. Neamtu, E.B. Sa, eds.), Nashboro Press, Nashville, 4, pp [9] D.S. Lubinsky, On the Bernstein Constants of Polynomial Approximation, to appear in Constructive Approximation. [] S.M. Nikolskii, On the Best Mean Approximation by Polynomials of the unctions x c s, Izvestia Akad. Nauk SSSR, (947), 39-8 (in Russian). [] R. A. Raitsin, On the Best Approximation in the Mean by Polynomials and Entire unctions of inite Degree of unctions having an Algebraic Singularity, Izv. Vysch. Uchebn. Zaved. Mat., 3(969), [] H.S. Shapiro, Topics in Approximation Theory, Lecture Notes in Mathematics, Vol. 87, Springer, New York, 97. [3] A.. Timan, Theory of Approximation of unctions of a Real Variable (translated by J. Berry),, Dover, New York, 994. [4] R.M. Trigub, E.S. Belinsky, ourier Analysis and Approximation of unctions, Kluwer, Dordrecht, 4. [5] J. Vaaler, Some Extremal unctions in ourier Analysis, Bulletin of the American Mathematical Society, (985), 83-6.