用指数型整函数的最佳限制逼近
Best Restriction Approximation by Entire Functions of Exponential Type
我们研究了由仅有实零点的代数多项式导出的微分算子确定的广义Sobolev类利用指数型整函数作为逼近工具的最佳限制逼近问题.利用Fourier变换和周期化等方法,得到在L2(R)范数下的广义Sobolev光滑函数类的相对平均宽度和最佳限制逼近的精确常数,以及当0是这个代数多项式的一个至多2重的零点时,得到最佳限制逼近在L1(R)范数和一致范数下的广义Sobolev类的精确到阶的结果.
We studied the best restriction approximation problems using entire functions of exponential type as the approximation tools on some generalized Sobolev classes of smooth functions defined by the differential operator induced by an algebraic polynomial with only real zeros. By the methods of Fourier transform and periodization, etc, we obtained the exact constants of the average relative widths and the best restriction approximation on the generalized Sobolev classes in the L2(R) norm, and obtained the asymptotic results of the best restriction approximation on the generalized Sobolev classes in the L1(R) norm and the uniform norm for the case that the polynomial has a zero of multiplicity at most 2 at the point 0.
最佳限制逼近 / 相对平均宽度 / 指数型整函数 {{custom_keyword}} /
restriction approximation / relative width / entire function {{custom_keyword}} /
[1] Chen D. R., Average Kolmogorov n-widths and optimal recovery of Sobolev classes in Lp(R). Chinese Ann. Math., Ser. B, 1992, 13(4):396-405.
[2] Korneǐchuk N. P., Ligun A. A., Doronin V. G., Approximation with Constraints, "Naukova Dumka", Kiev, 1982.
[3] Kolmogorov A., Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse, Ann. of Math. (2), 1936, 37(1):107-110.
[4] Konovalov V. N., Estimates of diameters of Kolmogorov type for classes of differentiable periodic functions, Mat. Zametki, 1984, 35(3):369-380.
[5] Konovalov V. N., Approximation of Sobolev classes by their finite-dimensional sections, Mat. Zametki, 2002, 72(3):370-382.
[6] Lorentz G. G., Golitschek Manfred V., Makovoz Y., Constructive Approximation, Springer-Verlag, Berlin, 1996.
[7] Liu Y. P., Xu G. Q., Widths and average widths of sobolev classes, Acta Mathematica Scientia, 2003, 23(2):178-184.
[8] Liu Y. P., Xiao W. W., Relative average widths of Sobolev spaces in L2((Rd)), Anal. Math., 2008, 34(1):71-82.
[9] Magaril-Ilyaev G. G., Tikhomirov V. M., Average dimension and ν-widths of classes of functions on the whole line, J. Complexity, 1992, 8(1):64-71.
[10] Mitrinovi? D. S., Pe?ari? J. E., Fink Arlington M., Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Academic Pub, Dordrecht, 1991.
[11] Nikolskiǐ S. M., Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag, New York, 1975.
[12] Pinkus A., n-widths in Approximation Theory, Springer-Verlag, Berlin, 1985.
[13] Poularikas A. D., editor, Transforms and Applications Handbook, CRC Press, Boca Raton, FL, Third Edition, 2010.
[14] Sun Y. S., Fang G. S., Approximation theory of functions, Vol II, Beijing Normal University Pubisher, Beijing, 1990.
[15] Stein E. M., Functions of exponential type, Ann. of Math., 1957, 65(2):582-592.
[16] Sun Y. S., Approximation Theory of Functions, Vol. I, Beijing Normal University Pubisher, Beijing, 1990.
[17] Stein E. M., Weiss Guido, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N. J., 1971.
[18] Tikhomirov V. M., Approximation characteristics of smooth functions of several variables, Theory of Cubature Formulas and Numerical Mathematics (Proc. Conf., Novosibirsk, 1978) (Russian), pages 183-188, 254. "Nauka" Sibirsk. Otdel., Novosibirsk, 1980.
[19] Timan A. F., Theory of Approximation of Functions of a Real Variable, Courier Dover Publications, New York, 1963.
[20] Yang W., Relative widths of differentiable function classes and convolution classes with 2π period in one variable case and hexagonal period in 2-dimensional case, PhD thesis, Beijing Normal University, Beijing, 2009.
[21] Yang W., Liu Y. P., Relative widths of the class of differentiable functions and the periodic convolution class in L1 metric, Beijing Shifan Daxue Xuebao, 2008, 44(6):573-576.
国家自然科学基金资助项目(11401451,11471043)
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