This paper investigates the relative Kolmogorov $n$-widths of $2\pi$-periodic smooth classes in $\widetilde{L}_{q}$. We estimate the relative widths of $\widetilde{W}^{r} H_{p}^{\omega}$ and its generalized class $K_{p}H^{\omega}(P_{r})$, where $K_{p}H^{\omega}(P_{r})$ is defined by a self-conjugate differential operator $P_{r}(D)$ induced by $P_{r}(t):= t^{\sigma} \Pi_{j=1}^{l}(t^{2}- t_{j}^{2}),~t_{j} > 0,~j=1, 2 ,\cdots, l,~l \geq 1,~\sigma \geq 1,~r=2l+\sigma .$ Also, the modulus of continuity of the $r$-th derivative, or $r$-th self-conjugate differential, does not exceed a given modulus of continuity $\omega$. Then we obtain the asymptotic results, especially for the case $p=\infty , 1\leq q \leq \infty$.
Yongping LIU
,
Man LU
. APPROXIMATION PROBLEMS ON THE SMOOTHNESS CLASSES*[J]. Acta mathematica scientia, Series B, 2024
, 44(5)
: 1721
-1734
.
DOI: 10.1007/s10473-024-0505-4
[1] Kolmogoroff A. Uber die beste Annaherung von Funktionen einer gegebenen Funktionenklasse. Annals of Mathematics, 1936, 37: 107-110
[2] Tikhomirov B M. Widths of sets in functional spaces and the theory of best approximations. Uspekhi Matematicheskikh Nauk, 1960, 15(3): 81-120
[3] Sun Yongsheng.Approximation Theory of Functions Ⅰ(in Chinese). Beijing: Beijing Normal University Press, 1989
[4] Sun Yongsheng, Fang Gensun.Approximation Theory of Functions Ⅱ(in Chinese). Beijing: Beijing Normal University Press, 1990
[5] Pinkus A.$N$-widths in Approximation Theory. Berlin: Springer Science & Business Media, 2012
[6] Lorentz G G, von Golitschek M, Makovoz Y. Constructive Approximation: Advanced Problems. Berlin: Springer, 1996
[7] Liu Y P, Xu G Q. Widths and average widths of Sobolev classes. Acta Mathematica Scientia, 2003, 23B(2): 178-184
[8] Pinkus A. $N$-widths of Sobolev spaces in $L_p$. Constructive Approximation, 1985, 1: 15-62
[9] Konovalov V N. Estimates of Kolmogorov-type widths for classes of differentiable periodic functions. Matematicheskie Zametki, 1984, 35(3): 369-380
[10] Konovalov V N, Leviatan D. Shape preserving widths of Sobolev-type classes of $s$-monotone functions on a finite interval. Israel Journal of Mathematics, 2003, 133: 239-268
[11] Cohen A, DeVore R. Kolmogorov widths under holomorphic mappings. IMA Journal of Numerical Analysis, 2016, 36(1): 1-12
[12] Kashin B S, Temlyakov V N. A remark on compressed sensing. Mathematical Notes, 2007, 82: 748-755
[13] Mojgani R, Balajewicz M, Hassanzadeh P. Kolmogorov $n$-width and Lagrangian physics-informed neural networks: a causality-conforming manifold for convection-dominated PDEs. Computer Methods in Applied Mechanics and Engineering, 2023, 404: 115810
[14] Babenko V F. On best uniform approximations by splines in the presence of restrictions on their derivatives. Matematicheskie Zametki, 1991, 50(6): 24-30
[15] Babenko V F. Approximation in the mean under restriction on the derivatives of the approximating functions// Questions in Analysis and Approximations, Kiev: Akad Nauk Ukrain SSR Inst Mat, 1989: 9-18
[16] Konovalov V N. Approximation of Sobolev classes by their finite-dimensional sections. Mathematical Notes, 2002, 72(3): 337-349
[17] Konovalov V N. Approximation of Sobolev classes by their sections of finite dimension. Ukrainian Mathematical Journal, 2002, 54(5): 795-805
[18] Makovoz Y. On $n$-widths of certain functional classes defined by linear differential operators. Proceedings of the American Mathematical Society, 1983, 89(1): 109-112
[19] Korneichuk N P. Inequalities for differentiable periodic functions and best approximation of one class of functions by another. Mathematics of the Ussr Izvestiya, 1972, 6(2): 417-428
[20] Yang Lianhong, Liu Yongping. Relative widths of smooth functions determined by linear differential operator. Journal of Mathematical Analysis and Applications, 2009, 351(2): 734-746
[21] Yang Wei.Relative Widths of Differentiable Function Classes and Convolution Classes with $2\pi$ Periodic in one Variable Case and Hexagonal Periodic in 2-Dimensional Case [D]. Beijing: Beijing Normal University, 2009
[22] Liu Yongping, Xu Guiqiao, Zhang Jie. Best restricted approximation of smooth function classes. Trudy Inst Mat I Mekh UrO RAN, 2018, 24(4): 283-294
[23] Ditzian Z, Tikhonov S. Ul'yanov and Nikol'skii-type inequalities. Journal of Approximation Theory, 2005, 133(1): 100-133
[24] Korneichuk N P.Extremal Problems of Approximation Theory. Moscow: Nauka, 1976
[25] Ismagilov R S. On $n$-dimensional diameters of compacts in a Hilbert space. Funktsional Anal i Prilozhen, 1968, 2(2): 32-39
[26] Tikhomirov V M. Some remarks on relative diameters. Banach Center Publications, 1989, 1(22): 471-474
