Sidorov, Sergei

Miskolc Mathematical Notes

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 16 (2015), No. 2, pp. 1191–1197 DOI: 10.18514/MMN.2015.1345ON THE SATURATION EFFECT FOR LINEARSHAPE-PRESERVING APPROXIMATION IN SOBOLEV SPACESS. P. SIDOROVReceived 20 September, 2014Abstract. We show that if a linear finite-dimensional operator defined in Sobolev space pre-serves k-monotonicity then the error of approximation of the operator does not decrease with theincrease of smoothness of approximated functions. In other words, there is a saturation effect forlinear finite-rank operators defined in Sobolev space and preserving k-monotonicity.2010 Mathematics Subject Classification: 41A15; 65D17Keywords: shape-preserving approximation, saturation effect, spline approximation1. INTRODUCTIONIn various applications of computer-aided geometric design it is necessary to ap-proximate functions preserving such properties as monotonicity, convexity, concav-ity, etc. The part of the approximation theory, which is devoted to these kind ofproblems, is known as the theory of shape preserving approximation (see e.g. [3, 9]for a survey).A continuous function f W Œ0;1! R is said to be k-monotone, k  1, on Œ0;1 ifand only if for all choices of k C 1 distinct t0; : : : ; tk in Œ0;1 the inequalityŒt0; : : : ; tkf  0 holds, where Œt0; : : : ; tkf denotes the k-th divided difference off at 0  t0 < t1 < :: : < tk  1. Note that 2-monotone functions are just convexfunctions.Let k denote the set of all k-monotone functions defined on [0,1]. If f is areal-valued and k-times continuously differentiable function defined on Œ0;1, thenf 2k iff f .k/.t/ 0, t 2 Œ0;1.Let W .k/1 Œ0;1 be the Sobolev space of all real-valued, .k   1/-times differenti-able functions whose derivative of order .k 1/ is absolutely continuous and whosederivative of order k is in L1Œ0;1, kf k1 WD ess supx2Œ0;1jf .x/j.It is said that a linear operator L of W .k/1 Œ0;1 into W .k/1 Œ0;1 preservesk-monotonicity, if L.k/k .The author was supported in part by Russian Ministry of Education and Science (project1.1520.2014K).c 2015 Miskolc University Press1192 S. P. SIDOROVRecall that a linear operator mapping a linear normed space into a linear space offinite dimension n is called an operator of finite rank n.LetDi denote the i -th differential operator,Dif .x/D d if .x/=dxi , andD0 D Iis the identity operator. Denote ek.x/D xk , k D 1;2; : : :.Denote B.k/1 WD fW .k/1 Œ0;1 W kDkf k1  1g.Let k 1. The main goal of the paper is to estimate the asymptotic value of relativelinear n-widthinfLn.k/ksupf 2B.k/1kf  Lnf k1;where infimum is taken over all linear continuous operators Ln of W.k/1 Œ0;1 intoW.k/1 Œ0;1 of finite rank n, such that Ln.k/k andDkLnek DDkek . Note thatsome estimates of shape-preserving n-widths have been obtained in papers [4, 7, 8].T. Popoviciu [12] proved in 1937 that Bernstein operatorsBnf .x/ WDnXiD0finC inxi .1 x/n ipreserve k-monotonicity of approximated functions, i.e. if f is k-monotone on [0,1],then Bnf is k-monotone on [0,1] as well. However, it is well known [2, 13] that theorder of approximation by Bernstein operators is low. In this paper we present theexample of k-monotonicity preserving operator with the optimal order of approxim-ation n 2 (k  2).2. LEMMASLemma 1. Let L WW .k/1 Œ0;1!W .k/1 Œ0;1 be such that(1) L.k/k;(2) DkLek DDkek .If f 2W .k/1 Œ0;1 satisfies kDkf k1  1, then kDkLf k1  1.Proof. For every f 2W .k/1 Œ0;1 such that kDkf k1  1, we havejDkf .x/j  1D 1kŠDkek.x/for almost all x 2 Œ0;1. Since L.k/  k , for almost all x 2 Œ0;1 the followinginequality holds:jDkLf .x/j  1kŠDkLek.x/:Therefore, we havekDkLf k1  1kŠkDkLekk1:It follows from DkLek DDkek that kDkLf k1  1. LINEAR SHAPE-PRESERVING APPROXIMATION IN SOBOLEV SPACES 1193Lemma 1 says that if L W W .k/1 Œ0;1! W .k/1 Œ0;1 satisfies conditions (i) and (ii)then L preserves the ball B.k/1 , i.e. L.B.k/1 / B.k/1 .Denote j´ WD j=n, j D 0; : : : ;n. Let the linear operator Mk;n W W .k/1 Œ0;1!W.k/1 Œ0;1, k > 2, n 2, be defined byMk;nf .x/Dk 1XlD01lŠxlDlf .0/C 2kŠxkŒ´0;´1;´2Dk 2fC 2n.kC1/ŠxkC1Œ´1;´2;´3Dk 2f   Œ´0;´1;´2Dk 2f  ; (2.1)if x 2 Œ´0;´1IMk;nf .x/Dk 1XlD01lŠ.x  j´ /lDlMk;nf . j´ /C 2kŠ.x  j´ /kŒ j´ ; j´C1; j´C2Dk 2f C 2n.kC1/Š.x  j´ /kC1Œ j´C1; j´C2; j´C3Dk 2f   Œ j´ ; j´C1; j´C2Dk 2f; (2.2)if x 2 . j´ ; j´C1; j D 1;2; : : : ;n 3IMk;nf .x/Dk 1XlD01lŠ.x ´n 2/lDlMk;nf .´n 2/C 2kŠ.x ´n 2/kŒ´n 3;´n 2;´n 1Dk 2f C 2n.kC1/Š.x ´n 2/kC1Œ´n 2;´n 1;´nDk 2f   Œ´n 3;´n 2;´n 1Dk 2f; (2.3)if x 2 .´n 2;´n 1IMk;nf .x/Dk 1XlD01lŠ.x ´n 1/lDlMk;nf .´n 1/C 2kŠ.x ´n 1/kŒ´n 3;´n 2;´n 1Dk 2f C 2n.kC1/Š.x ´n 1/kC1 .Œ´n 2;´n 1;´nDk 2f   Œ´n 3;´n 2;´n 1Dk 1f /; (2.4)if x 2 .´n 1;1:1194 S. P. SIDOROVLemma 2. Let k  2 and let the operator Mk;n W W .k/1 Œ0;1! W .k/1 Œ0;1 bedefined by (2.1), (2.2), (2.3), (2.4). Then(1) Mk;n.k/k;(2) DkMk;nek DDkek;(3) there exists a real c > 0 such that kf  Mk;nf k1  cn 2 for all f 2 B.k/1 .Proof. Note thatDkMk;nf is a piecewise linear function andDkMk;nf is linearon Œ j´ ; j´C1 for each 0 j  n 1. If x 2 . j´ ; j´C1, thenDkMk;nf .x/D 2.j C1 nx/Œ j´ ; j´C1; j´C2Dk 2fC2.nx j /Œ j´C1; j´C2; j´C3Dk 2f; j D 0;1; : : : ;n 3; (2.5)DkMk;nf .x/D 2.n 1 nx/  Œ´n 3;´n 2;´n 1Dk 2fC2.nx nC2/  Œ´n 2;´n 1;´nDk 2f; j D n 2: (2.6)DkMk;nf .x/D 2.n nx/  Œ´n 3;´n 2;´n 1Dk 2fC2.nx nC1/  Œ´n 2;´n 1;´nDk 2f; j D n 1: (2.7)If f 2 k , then it follows from (2.5), (2.6), (2.7) that DkMk;nf  0 on each.xj ;xjC1/, j D 0; : : : ;n 1, i.e. Mk;nf 2k . Therefore, Mk;n.k/k .A direct verification shows that DkMk;nek DDkek .Since f 2 B.k/1 , the .k 2/-th derivative of f can be represented asDk 2f .x/DDk 2f .0/CDk 1f .0/1ŠxCZ 10.x  t /CDkf .t/dt;where kDkf k1  1 and yC D y if y  0, and 0 otherwise.Take an arbitrary x 2 Œ0; 1n (the case x 2 . j´ ; j´C1/, j D 1; : : : ;n  3, can beexamined analogously), thenDk 2Mk;nf .x/ Dk 2f .x/D x2Œ´0;´1;´2Dk 2f C 13nx3Œ´1;´2;´3Dk 2f   Œ´0;´1;´2Dk 2f Z 10.x  t /CDkf .t/dtDZ 10x2 ..´0  t /C 2.´1  t /CC .´2  t /C/C 13nx3 ..´1  t /C 2.´2  t /CC .´3  t /C/  13nx3 ..´0  t /C 2.´1  t /CC .´2  t /C/  .x  t /CDkfdt;LINEAR SHAPE-PRESERVING APPROXIMATION IN SOBOLEV SPACES 1195and we can obtainjDk 2Mk;nf .x/ Dk 2f .x/j  an 2; x 2 Œ j´ ; j´C1; j D 1; : : : ;n 3;for some a > 0. In the same way it can be shown thatjDk 2Mk;nf .x/ Dk 2f .x/j  n 2; x 2 Œ´n 2;´n 1; and x 2 Œ´n 1;1:Therefore, for every f 2 B.k/1 and any x 2 Œ0;1jDk 2f .x/ Dk 2Mk;nf .x/j  c1n 2: (2.8)If k D 2 then the proposition (3) of Lemma 2 is proved.Let k  3. Since Di .Mk;nf  f /.0/D 0 holds for i D 0; : : : ;k 3, we haveDi .Mk;nf  f /.x/DDi .Mk;nf  f /.0/CZ x0DiC1.Mk;nf  f /.t/dtDZ x0DiC1.Mk;nf  f /.t/dtfor all x 2 Œ0;1. ThenjDi .Mk;nf  f /.x/j  1n2xk 1 i.k 1  i/Š : (2.9)We have used the fact that if f 2 L1Œ0;1 and there exists a real a 2 R such thatjf j  a on [0,1], thenˇˇˇˇZ x0Z tp 10: : :Z t10f .t1/dt1 : : :dtpˇˇˇˇZ x0Z tp 10: : :Z t10jf .t1/jdt1 : : :dtp  axppŠfor every p 2N.The proposition (3) of Lemma 2 follows from (2.9) with i D 0. 3. THE MAIN RESULTTheorem 1. Let k 2N. TheninfLn.k/ksupf 2B.k/1kf  Lnf k1 (n 2; k > 2;n 1; k D 1; (3.1)where infimum is taken over all linear continuous operators Ln of W.k/1 Œ0;1 intoW.k/1 Œ0;1 of finite rank n, such that(1) Ln.k/k;(2) DkLnek DDkek .1196 S. P. SIDOROVProof. We have (by definition)infLn.k/ksupf 2B.k/1kf  Lnf k1 D infXninfLn.k/kLn WW .k/1 Œ0;1!Xnsupf 2B.k/1kf  Lnf k1;(3.2)where the first infimum is taken over all n-dimensional subspaces Xn of W.k/1 Œ0;1,and the second one is taken over all continuous linear operators Ln ofW.k/1 Œ0;1 intoXn satisfying Ln.k/k and DkLnek DDkek .Since f 2B.k/1 , it follows from Lemma 1 thatLnf 2B.k/1 . Then Eq. (3.2) impliesinfLn.k/ksupf 2B.k/1kf  Lnf k1 > infXnsupf 2B.k/1infg2B.k/1 \Xnkf  gk1:It was shown in the paper of Konovalov [6] thatinfXnsupf 2B.k/1infg2B.k/1 \Xnkf  gk1 (n 2; k > 2;n 1; k D 1:To complete the proof it is sufficient to show that there exists a linear finite-dimensional method, that is optimal for n-widths (3.1). If k > 2, then as it followsfrom Lemma 2, the linear operator Mk;n WW .k/1 Œ0;1!W .k/1 Œ0;1 satisfies the con-ditions of Theorem and it is optimal for n-width (3.1). If k D 1, then the piecewiselinear interpolation method with breakpoints j´ , j D 0; : : : ;n, is optimal for n-width(3.1). Theorem 1 shows that the shape-preserving property of operators is negative inthe sense that the error of approximation of such operators does not decrease withthe increase of smoothness of approximated functions. In other words, there is asaturation effect for linear finite-rank operators preserving k-monotonicity. For thefirst time the problem was mentioned by Ronald A. De Vore in his paper [1]. Itis worth noting that non-linear approximation preserving k-monotonicity does nothave this shortcoming [10]. On the other hand, for sequences of linear operatorspreserving k-monotonicity (as well as intersections of cones) there are [5,11] simpleconvergence conditions (Korovkin type results).REFERENCES[1] R. A. De Vore, “Monotone approximation by splines,” SIAM J. Math. Anal., vol. 8, pp. 891–905,1977, doi: 10.1137/0508068.[2] M. Floater, “On the convergence of derivatives of Bernstein approximation,” Journal of Approx-imation Theory, vol. 134, no. 1, pp. 130 – 135, 2005, doi: 10.1016/j.jat.2004.02.009.[3] S. G. Gal, Shape-Preserving Approximation by Real and Complex Polynomials. Springer, 2008.[4] J. Gilewicz, V. N. Konovalov, and D. Leviatan, “Widths and shape-preserving widths of Sobolev-type classes of s-monotone functions,” J. Approx. Theory, vol. 140, no. 2, pp. 101–126, 2006, doi:10.1016/j.jat.2005.11.016.LINEAR SHAPE-PRESERVING APPROXIMATION IN SOBOLEV SPACES 1197[5] H. H. Gonska, “Quantitative Korovkin type theorems on simultaneous approximation,” Mathem-atische Zeitschrift, vol. 186, no. 3, pp. 419–433, 1984, doi: 10.1007/BF01174895.[6] V. Konovalov, “Estimates of Kolmogorov type widths for classes of differentiable functions,” Mat.Notes, vol. 35, no. 3, pp. 193–199, 1984, doi: 10.1007/BF01139916.[7] V. Konovalov and D. Leviatan, “Shape preserving widths of Sobolev-type classes of k-monotonefunctions on a finite interval,” Israel Journal of Mathematics, vol. 133, pp. 239–268, 2003, doi:10.1007/BF02773069.[8] V. Konovalov and D. Leviatan, “Shape-preserving widths of weighted Sobolev-type classes of pos-itive, monotone, and convex functions on a finite interval,” Constructive Approximation, vol. 19,pp. 23–58, 2003, doi: 10.1007/s00365-001-0027-3.[9] K. A. Kopotun, D. Leviatan, A. Prymak, and I. A. Shevchuk, “Uniform and pointwise shapepreserving approximation by algebraic polynomials,” Surveys in Approximation Theory, vol. 6,pp. 24–74, 2011.[10] K. Kopotun and A. Shadrin, “On k-monotone approximation by free knot splines,” SIAM J. Math.Anal., vol. 34, pp. 901–924, 2003, doi: 10.1137/S0036141002358514.[11] F. J. Mun˜oz-Delgado, V. Ramı´rez-Gonza´lez, and D. Ca´rdenas-Morales, “Qualitative Korovkin-type results on conservative approximation,” J. Approx. Theory, vol. 94, pp. 144–159, 1998, doi:10.1006/jath.1998.3182.[12] T. Popoviciu, About the Best Polynomial Approximation of Continuous Functions. MathematicalMonography. Sect. Mat. Univ. Cluj, 1937, (In Romanian), fasc. III.[13] E. Voronovskaya, “Determination de la forme asymptotique de l’approximation des fonctions parles polynomes de M. Bernstein,” Dokl. Akad. Nauk SSSR, vol. A, pp. 79–85, 1932.Author’s addressS. P. SidorovDepartment of Mechanics and Mathematics, Saratov State University, Saratov, Russian FederationE-mail address: sidorovsp@info.sgu.ru

On saturation effect for linear shape-preserving approximation in Sobolev spaces

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