Error Estimates for Trigonometric Interpolation. of Periodic Functions in Lip 1. Knut Petras

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1 Error Estimates for Trigonometric Interpolation of Periodic Functions in Lip Knut Petras Abstract. The Peano kernel method is used in order to calculate the best possible constants c, independent of M, f Lip M and x, in the estimate jf (x) intpol[f ](x)j M c sin n x : Here, Lip M denotes the class of all -periodic functions with Lipschitz-constant M, and intpol[f ] is the trigonometric interpolation polynomial of f with respect to the n nodes x =. x Introduction and main results Let f be in Lip M, the set of all -periodic functions with Lipschitzconstant M, and denote by intpol[f] = intpol [f] the trigonometric interpolation polynomial (cf. Zygmund [, pp..]) of f with respect to, = ; ; : : : ; n. Two important error estimates are known in this situation. The rst one is the special case k = and = of Nikolsky's estimate [3] for f (k) Lip M, the n equidistant nodes x = sup jf(x) intpol[f](x)j = M ln n flip M n and the second one is of Gunttner [], sup kf intpol[f]k = M flip M sin n x O(n ); (:) k intpol k (:) n = M ln n n O(n ): Approximation Theory VIII Charles K. Chui and Larry L. Schumaker (eds.), pp. {3. Copyright o c 995 by World Scientic Publishing Co., Inc. All rights of reproduction in any form reserved. ISBN --xxxxxx-x

2 K. Petras The rst one is a local estimate, which takes into account the interpolation property, while the second is a global one. It is now an obvious question, whether there is a local estimate of the form (.), which is globally as good as (.). More precisely, we ask, whether the inequality jf(x) intpol[f](x)j = M n k intpol k sin n x (:3) is valid for all f Lip M. The answer to this question is negative for n >. We prove this by calculating the best possible constant c in the estimate sup jf(x) intpol[f](x)j M c flip M sin n x : (:) Nevertheless, c is only slightly greater than k intpol k. The second purpose of this paper is to show that the Peano kernel method may be applied usefully to interpolation errors. This method has been applied many times in numerical integration but rarely in approximation theory. All the results below could also be proved without using Peano kernels. However, the arguments would be more complicate and generalizations to functions with higher derivatives would be more dicult. The Peano kernel method helps to unify the arguments concerning the estimation of linear functionals. We will give further examples in approximation theory elsewhere. Our main result is the following theorem. It will be proved in Section 3. Theorem where Let n >. For all f Lip M and all x, we have the estimate jf(x) intpol[f](x)j M c n sin c = n b(n)=c (n ) = sin n x ; (:5) : (:6) The asterisk indicates that the last summand has to be halved if n is odd. The so-dened constant c is the least possible constant, such that the estimate (.5) is valid for all f Lip M and all x. Let C = :577::: denote Euler's constant, then, with j" n j 7 n, c = ln n C ln 8 n n " n : (:7)

3 Error Estimates for Interpolation 3 Remark. The theorem may be compared with Gunttners bound (.). We therefore note that Remark. < c ln k intpol k < n n : (:8) For n =, the best possible estimate of the form (.5) is n x ; (:9) jf(x) intpol 3 [f](x)j M 3 k intpol 3 k sin where k intpol 3 k = 5 3. x Peano kernels of interpolation errors for periodic functions Let the -periodic function f have a continuous r th derivative. Then, Z f(z) = Z f(t) dt () r Br z t f (r) (t) dt ; (:) where B r is the r th Bernoulli-monospline. For the error functional L x of a trigonometric interpolation operator, intpol, with respect to arbitrary nodes, we have by Fubini's theorem that where L x [f] = f(x) intpol[f](x) = Z () r K r(l x ; t) = L x B r K r (L x ; t)f (r) (t) dt; (:) t : (:3) The periodicity of f implies that we may add an arbirtrary constant to the Peano kernel. We obtain % r (L x ) := sup jl x [f]j = inf kf c (r) k Z jk r (L x ; t) cj dt; (:) since this number is attained if f is the -periodic function, whose r th derivative is the sign function of K r (L x ; ) c, wherever this term is not zero. The Peano kernel K r (L x ; ) is a periodic spline of degree r, and K r (L x ; ) is a primitive of K r (L x ; ). We now want to count the sign changes of the Peano kernels. We start with the number of sign changes of Peano kernels of high order r. Therefore, we need the following estimate.

4 K. Petras Lemma. Let n (x) = (L x [sin(n )]) (L x [cos(n )]) = : Then, for each r and each x, there exists a ' x [; ) such that K r (L x ; t) ()br=c n (x) (n ) r sin ' x (n )t kl x k (r )(n 3=) r : (:5) Proof: From the boundedness of L x and the uniform convergence of the Fourier series of Br for r, we obtain L x B r t = ()br=c L x [p ( t)] () r r ; (:6) =n where This implies p (x) = n sin x for odd r and cos x for even r. (:7) K r (L x ; t) = " n (x; t) ()br=c (n ) r Lx [s n ] cos(n )t L x [c n ] sin(n )t for odd r, Lx [c n ] cos(n )t L x [s n ] sin(n )t for even r, (:8) where c k (t) = cos kt, s k (t) = sin kt and j" n (x; t)j kl xk =n r < kl x k : (:9) (r )(n 3=) r If x is not a node, L x does not vanish on the whole space of trigonometric polynomials of maximal degree n, so that L x [c n ] = n (x); L x [s n ] = n (x); p n (x) n(x) > : (:) Hence, there exists a ~' x [; ) satisfying cos ~' x = n (x) p n (x) n(x) ; sin ~' x = n (x) p n (x) n(x) : (:) Inserting this in (.8) gives the ' x of the lemma. Lemma. Let a trigonometric interpolation operator involving n nodes in I := [; ) be given. If x is not a node of this operator, the

5 Error Estimates for Interpolation 5 Peano kernel K r (L x ; ) of the error functional L x has exactly n sign changes in I. Proof: For each x and each n, there is a number r = r (x), such that n (x) (n ) r kl x k (r )(n 3=) r : (:) for all r > r. If r > r, there are certain constants c r 6= and c r 6=, as well as functions v r and v r, whose moduli are bounded by =, such that K r (L x ; t) = c r sin ' x (n )t v r (t) (:3) and K r (L x ; t) = c r cos ' x (n )t v r (t) : (:) Hence, K r (L x ; ) has no sign change for j sin ' x (n )t j >. j sin ' x (n)t j, we obtain j cos ' x (n)t j and therefore the monotony of K r (L x ; ). We now count exactly n sign changes of K r (L x ; ) on the interval I. A repeated dierentiation may only increase the number of sign changes. However, the rst Peano kernel is a step function with n jumps in I. This shows the lemma. p 3 If x3 The rst Peano kernel for n equidistant nodes By the shift-invariance, we may choose the nodes x j =, where j = ; ; : : : ; n (for simplicity, we sometimes use the notation x j = j for arbitrary j). Without restriction, we suppose that the evaluation point x is in the interval [; x =]. The interpolation operator may be written in the form where intpol[f](x) = n = p(x) n = n = j f(x ) sin(n )(x x ) sin (x x ) () a f(x ); (3:) p(x) = n sin(n )x and a = a (x) = sin (x x ) (3:)

6 6 K. Petras (cf. Zygmund []). By the preceding investigations, we know that K (L x ; ) is positive on (; x), negative on (x; x ) and has the sign () on (x ; x ) for >. This implies that the subset of [; ), on which the kernel is positive, has a measure between x and. Therefore, the constant c, which minimizes the integral in (.) equals the smallest modulus among all negative values of the Peano kernel. Since the weights a are decreasing for = ; : : : ; n and increasing for n, we obtain c = 8 >< >: jk (L x ; )j Let us rst suppose that n is even. constant c, has the representation K c (L x ; t) jp(x)j = 8 >< >: n = =n for even n and jk (L x ; n 3 n )j for odd n. (3:3) Then, the kernel, modied by the () a for t (x ; x ) \ (x; n ), (3:) () a for t (x ; x ) \ ( n ; x). The respective signs are () in the rst and () in the second case. We obtain % (L x ) jp(x)j x = (x x) =n = (x x) = x n = () a n = =n () a () a () a n =n n = () (x x ) () (x x ) n () a = n =n = () a n =n n = () a () a () (x x ) = We proceed analogously for odd n and nally obtain % (L x ) jp(x)j x = (x b(n)=c b(n)=c = ja (x)j ja (x)j n =b(n3)=c () (x x ): n =b(n3)=c ja (x)j ja (x)j A A =: M (x): (3:5) (3:6)

7 Error Estimates for Interpolation 7 Let g(x) = ( cos x) sin 3 x. In order to show the convexity of M on the interval (; x =), we note that M (x) equals (x x) cos x x sin(x x) sin 3 x x x cos x x sin(x x) sin 3 x x x b(n)=c x x n x x g g A (3:7) = =b(n3)=c b(n)=c x x n x x g g A = =b(n3)=c n x x = () cos sin 3 x x sin 3 (x=) n cos = (x x) sin x x cos x x x cos (x=) sin x x x sin x x sin(x x) : (3:8) Here, denotes the second forward dierence, i.e., f(x k ) = f(x k ) f(x k ) f(x k ). The positivity of the last sum in (3.8) follows from the convexity of the function j cos j sin. The positivity of the other two sums on the right-hand side of (3.8) is proved by setting f(x) = x( cos (x=)) sin x = 3x x cos x sin x (3:9) and using that f () = and f (x) = cos x((tan x) x) for x (; ). The convexity of the function M is shown, such that it attains its maximum at the boundary of [; x =]. We have M () = b(n)=c sin n n (3:) =

8 8 K. Petras and M (x =) = sin n n (3:) With standard techniques (see, e.g., Gunttner []), we obtain the bounds of the theorem for the constants c = M (). It can also be shown that ln 6 88n n C (n ) M (x ) ln(n ) : (3:) We obtain = M () > M (x ) for n > 3. (3:3) For smaller n, we may verify this relation numerically. The theorem is now proved completely, and Remark follows from the lower bound in (3.). Remark may be shown by a simple explicit calculation. References. Gunttner, R., Abschatzungen fur Normen von Interpolationsoperatoren, doctoral thesis, TU Clausthal, 97.. Gunttner, R., Eine optimale Fehlerabschatzung zur trigonometrischen Interpolation, Stud. Sci. Math. Hungar. (975), 3{9. 3. Nikolsky, S. M., An asymptotic estimation of the remainder under approximation by interpolating trogonometric polynomials. C. R. (Doklady) Acad. Sci. URSS (N.S.) 3 (9), {.. Zygmund, A., Trigonometric Series II, Cambridge University Press, London, 959. Knut Petras Institut fur Angewandte Mathematik TU Braunschweig Pockelsstr. 386 Braunschweig, Germany k.petras@tu-bs.de

ON A WEIGHTED INTERPOLATION OF FUNCTIONS WITH CIRCULAR MAJORANT

ON A WEIGHTED INTERPOLATION OF FUNCTIONS WITH CIRCULAR MAJORANT Received: 31 July, 2008 Accepted: 06 February, 2009 Communicated by: SIMON J SMITH Department of Mathematics and Statistics La Trobe University,