Bernstein-type constants for approximation of \(| x |^\alpha\) by partial Fourier-Legendre and Fourier-Chebyshev sums. (English) Zbl 1528.41013
Summary: In this paper, we study the approximation of \(f_\alpha (x) = | x |^\alpha\), \(\alpha > 0\) in \(L_\infty [-1, 1]\) by its Fourier-Legendre partial sum \(S_n^{(\alpha)} (x)\). We derive the upper and lower bounds of the approximation error in the \(L^\infty\)-norm that are valid uniformly for all \(n \geq n_0\) for some \(n_0 \geq 1\). Such an optimal \(L^\infty\)-estimate requires a judicious summation rule that can recover the lost half order if one uses a naive summation. Consequently, we can obtain the explicit Bernstein-type constant
\[
B_\infty^{(\alpha)} := \lim_{n \to \infty} n^\alpha \| f_\alpha - S_n^{(\alpha)} \|_{L^\infty} = \frac{2 \mathit{\Gamma} (\alpha)}{\pi} \Big| \sin \frac{\alpha \pi}{2} \Big|.
\]
Interestingly, using a similar argument, we can show that the Fourier-Chebyshev sum has the same Bernstein-type constant \(B_\infty^{(\alpha)}\) as the Legendre case.
MSC:
| 41A10 | Approximation by polynomials |
| 41A25 | Rate of convergence, degree of approximation |
| 41A50 | Best approximation, Chebyshev systems |
Keywords:
Bernstein constants; Legendre polynomial expansion; Chebyshev polynomial expansion; lower and upper bounds; optimal estimatesReferences:
| [1] | Alzer, H., On some inequalities for the Gamma and Psi functions, Math. Comp., 66, 217, 373-389 (1997) · Zbl 0854.33001 |
| [2] | Babuška, I.; Guo, B. Q., Direct and inverse approximation theorems for the \(p\)-version of the finite element method in the framework of weighted Besov spaces. I. Approximability of functions in the weighted Besov spaces, SIAM J. Numer. Anal., 39, 5, 1512-1538 (2001) · Zbl 1008.65078 |
| [3] | Babuška, I.; Hakula, H., Pointwise error estimate of the Legendre expansion: the known and unknown features, Comput. Methods Appl. Mech. Engrg., 345, 748-773 (2019) · Zbl 1440.65027 |
| [4] | Bernstein, S. N., Sur la meilleure approximation de \(| x |\) par des polynomes de degre donnes, Acta Math., 37, 1-57 (1913) · JFM 44.0475.01 |
| [5] | Bernstein, S. N., Sur la meilleure approximation de \(| x |^p\) par des polynomes de degres tres eleves, Bull. Acad. Sci. USSR, Ser. Math., 2, 181-190 (1938) · JFM 65.1198.01 |
| [6] | Buhmann, M. D.; Mache, D. H., Advanced Problems in Constructive Approximation (2002), Birkhäuser Verlag: Birkhäuser Verlag Switzerland |
| [7] | Bustoz, J.; Ismail, M. E.H., On Gamma function inequalities, Math. Comp., 47, 176, 659-667 (1986) · Zbl 0607.33002 |
| [8] | Carpenter, A. J.; Varga, R. S., Some numerical results on best uniform polynomial approximation of \(x^\alpha\) on \([ 0 , 1 ]\), (Methods of Approximation Theory in Complex Analysis and Mathematical Physics. Methods of Approximation Theory in Complex Analysis and Mathematical Physics, Leningrad, 1991. Methods of Approximation Theory in Complex Analysis and Mathematical Physics. Methods of Approximation Theory in Complex Analysis and Mathematical Physics, Leningrad, 1991, Lecture Notes in Math., vol. 1550 (1993), Springer: Springer Berlin), 192-222 · Zbl 0784.65009 |
| [9] | Ganzburg, M. I., The Bernstein constant and polynomial interpolation at the Chebyshev nodes, J. Approx. Theory, 119, 2, 193-213 (2002) · Zbl 1035.41015 |
| [10] | Ganzburg, M. I., Polynomial interpolation and asymptotic representations for zeta functions, Diss. Math., 496, 117 (2013) · Zbl 1296.11108 |
| [11] | M.I. Ganzburg, D.S. Lubinsky, Best approximating entire functions to \(| x |^\alpha\) in \(L_2\), in: Complex Analysis and Dynamical Systems III, in: Contemp. Math. Amer. Math. Soc., vol. 455, Providence, RI, 2008, pp. 93-107. · Zbl 1153.41310 |
| [12] | Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series and Products (2015), Academic Press: Academic Press New York · Zbl 1300.65001 |
| [13] | Gui, W.; Babuška, I., The \(h , p\) and \(h-p\) versions of the finite element method in \(1\) dimension. I. The error analysis of the \(p\)-version, Numer. Math., 49, 6, 577-612 (1986) · Zbl 0614.65088 |
| [14] | Liu, W. J.; Wang, L.-L., Asymptotics of the generalized Gegenbauer functions of fractional degree, J. Approx. Theory, 253, Article 105378 pp. (2020) · Zbl 1435.30123 |
| [15] | Liu, W. J.; Wang, L.-L.; Li, H. Y., Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces, Math. Comp., 88, 320, 2857-2895 (2019) · Zbl 1428.41020 |
| [16] | Liu, W. J.; Wang, L.-L.; Wu, B. Y., Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation, Adv. Comput. Math., 47, 79 (2021) · Zbl 1496.41003 |
| [17] | Lubinsky, D. S., Series representations for best approximating entire functions of exponential type, (Wavelets and Splines: Athens 2005, Mod. Methods Math (2005), Nashboro Press: Nashboro Press Brentwood, TN), 356-364 · Zbl 1098.41027 |
| [18] | Lubinsky, D. S., On the Bernstein constants of polynomial approximation, Constr. Approx., 25, 3, 303-366 (2007) · Zbl 1118.41003 |
| [19] | Nikolskii, S. M., On the best mean approximation by polynomials of the functions \(| x - c |^s\), Izv. Akad. Nauk SSSR, 11, 139-180 (1947) · Zbl 0029.02801 |
| [20] | Olver, F. W.J.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., NIST Handbook of Mathematical Functions (2010), Cambridge University Press: Cambridge University Press New York · Zbl 1198.00002 |
| [21] | Pachón, R.; Trefethen, L. N., Barycentric-Remez algorithms for best polynomial approximation in the Chebfun system, BIT, 49, 4, 721-741 (2009) · Zbl 1179.65012 |
| [22] | Pinkus, A., Weierstrass and approximation theory, J. Approx. Theory, 107, 1, 1-66 (2000) · Zbl 0968.41001 |
| [23] | Raitsin, R. A., On the best approximation in the mean by polynomials and entire functions of finite degree offunctions having an algebraic singularity, Izv. Vyssh. Uchebn. Zaved. Mat., 13, 4, 59-61 (1969), (Russian) |
| [24] | Revers, M., On the asymptotics of polynomial interpolation to \(| x |^\alpha\) at the Chebyshev nodes, J. Approx. Theory, 165, 70-82 (2013) · Zbl 1264.41005 |
| [25] | Revers, M., Towards best approximations for \(| x |^\alpha \), (Abell, M.; Iacob, E.; Stokolos, A.; Taylor, S.; Tikhonov, S.; Zhu, J., Topics in Classical and Modern Analysis—In Memory of Yingkang Hu (2019), Springer: Springer Switzerland), 303-313 |
| [26] | Revers, M., Asymptotics of polynomial interpolation and the Bernstein constants, Results Math., 76, 2, 25 (2021), Paper No. 100 · Zbl 1468.41005 |
| [27] | Shen, J.; Tang, T.; Wang, L.-L., Spectral Methods: Algorithms, Analysis and Applications (2011), Springer-Verlag: Springer-Verlag New York · Zbl 1227.65117 |
| [28] | Totik, V., Approximation on compact subsets of \(r\), (Buhmann, M.; Mache, D., Advanced Problems in Constructive Approximation (2002), Springer-Verlag: Springer-Verlag Berlin Heidelberg), 263-274 · Zbl 1044.41005 |
| [29] | Trefethen, L. N., Is Gauss quadrature better than Clenshaw-Curtis?, SIAM Rev., 50, 1, 67-87 (2008) · Zbl 1141.65018 |
| [30] | Varga, R. S., (Scientific Computation on Mathematical Problems and Conjectures. Scientific Computation on Mathematical Problems and Conjectures, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 60 (1990), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA) · Zbl 0703.65004 |
| [31] | Varga, R. S.; Carpenter, A. J., On the Bernstein conjecture in approximation theory, Constr. Approx., 1, 4, 333-348 (1985) · Zbl 0648.41013 |
| [32] | Wang, H. Y., A new and sharper bound for Legendre expansion of differentiable functions, Appl. Math. Lett., 85, 95-102 (2018) · Zbl 1401.41008 |
| [33] | Xiang, S. H., Convergence rates on spectral orthogonal projection approximation for functions of algebraic and logarithmatic regularities, SIAM J. Numer. Anal., 59, 3, 1374-1398 (2021) · Zbl 1465.41003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
