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1.
Michael Revers 《Monatshefte für Mathematik》2000,40(2):215-221
In 1942, P. Szász published the surprising result that if a function f is of bounded variation on [−1, 1] and continuous at 0 then the sequence of the equidistant Lagrange interpolation polynomials converges at 0 to . In the present note we give a construction of a function continuous on [−1, 1] whose Lagrange polynomials diverge at 0. Moreover, we show that the rate of divergence attains almost the maximal possible rate. 相似文献
2.
On the Zero-Divergence of Equidistant Lagrange Interpolation 总被引:1,自引:0,他引:1
Michael Revers 《Monatshefte für Mathematik》2000,131(3):215-221
In 1942, P. Szász published the surprising result that if a function f is of bounded variation on [−1, 1] and continuous at 0 then the sequence of the equidistant Lagrange interpolation polynomials
converges at 0 to . In the present note we give a construction of a function continuous on [−1, 1] whose Lagrange polynomials diverge at 0.
Moreover, we show that the rate of divergence attains almost the maximal possible rate.
(Received 2 February 2000) 相似文献
3.
Josef Dalík 《Applications of Mathematics》2008,53(6):547-560
We study the problem of Lagrange interpolation of functions of two variables by quadratic polynomials under the condition
that nodes of interpolation are vertices of a triangulation. For an extensive class of triangulations we prove that every
inner vertex belongs to a local six-tuple of vertices which, used as nodes of interpolation, have the following property:
For every smooth function there exists a unique quadratic Lagrange interpolation polynomial and the related local interpolation
error is of optimal order. The existence of such six-tuples of vertices is a precondition for a successful application of
certain post-processing procedures to the finite-element approximations of the solutions of differential problems.
This work was supported by the grant GA ČR 103/05/0292. 相似文献
4.
N. N. Pustovoitov 《Mathematical Notes》1999,65(1):89-98
In the paper order-exact upper bounds for the best approximations of classesH
q
Emphasis>/ω by trigonometric polynomials are obtained. The spectrum of the approximating polynomials lies in sets generated by the level
surfaces of the function ω(t). These sets are a generalization of hyperbolic crosses to the case of an arbitrary function ω(t).
Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 107–117, January, 1999. 相似文献
5.
Recently, Srivastava, Özarslan and Kaanoglu have introduced certain families of three and two variable polynomials, which include Lagrange and Lagrange-Hermite polynomials, and obtained families of two-sided linear generating functions between these families [H.M. Srivastava, M.A. Özarslan, C. Kaanoglu, Some families of generating functions for a certain class of three-variable polynomials, Integr. Transform. Spec. Funct. iFirst (2010) 1-12]. The main object of this investigation is to obtain new two-sided linear generating functions between these families by applying certain hypergeometric transformations. Furthermore, more general families of bilinear, bilateral, multilateral finite series relationships and generating functions are presented for them. 相似文献
6.
7.
Steven G. Krantz Harold R. Parks 《Journal of Mathematical Analysis and Applications》2008,340(2):1263-1270
The classical Lagrange inversion theorem is a concrete, explicit form of the implicit function theorem for real analytic functions. An explicit construction shows that the formula is not true for all merely smooth functions. The authors modify the Lagrange formula by replacing the smooth function by its Maclaurin polynomials. The resulting modified Lagrange series is, in analogy to the Maclaurin polynomials, an approximation to the solution function accurate to o(xN) as x→0. 相似文献
8.
The Padua points are a family of points on the square [−1, 1]2 given by explicit formulas that admits unique Lagrange interpolation by bivariate polynomials. Interpolation polynomials
and cubature formulas based on the Padua points are studied from an ideal theoretic point of view, which leads to the discovery
of a compact formula for the interpolation polynomials. The L
p
convergence of the interpolation polynomials is also studied.
S. De Marchi and M. Vianello were supported by the “ex-60%” funds of the University of Padua and by the INdAM GNCS (Italian
National Group for Scientific Computing). Y. Xu was partially supported by NSF Grant DMS-0604056. 相似文献
9.
V. A. Martirosian S. E. Mkrtchyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2008,43(6):372-376
The paper presents some new results on the possibility of approximation by polynomials with gaps. The approximations are done
in the norm of the space L
p
, 1 ≤ p < + ∞, on the Caratheodory sets in the complex plane. The lacunary versions of some results by Farell—Markushevich, S. Sinanian,
A. L. Shahinian are obtained (Theorems 1, 3, 5). Similar approximations by the real parts of lacunary polynomials are given
(Theorems 2, 4, 6).
Dedicated to the memory of academician S. N. Mergelyan 相似文献
10.
A generating function for nonstandard orthogonal polynomials involving differences: the Meixner case
In this paper we deal with a family of nonstandard polynomials orthogonal with respect to an inner product involving differences.
This type of inner product is the so-called Δ-Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond
to the Pascal distribution (the orthogonal polynomials associated to this distribution are known as Meixner polynomials).
The aim of this work is to obtain a generating function for the Δ-Meixner–Sobolev orthogonal polynomials and, by using a limit process, recover a generating function for Laguerre–Sobolev
orthogonal polynomials. 相似文献
11.
A. V. Andrianov 《Mathematical Notes》1999,66(3):261-270
The paper deals with approximations of functions with bounded mixed difference by Haar polynomials. Approximations of the
following two different types are studied: the classical best approximation by Haar polynomials with indices lying in hyperbolic
crosses and the bestm-term approximation by Haar polynomials.
Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 323–335, September, 1999. 相似文献
12.
Leonid S. Frank 《Annali di Matematica Pura ed Applicata》1979,122(1):315-363
Summary Difference approximations for differential singular perturbations with small parameter ɛ are considered. We point out ellipticity
and coerciveness conditions which arenecessary andsufficient for a two-sided a priori estimate to hold for the solution of difference singular perturbation uniformly with respect to
the ratio of both small parameters: the original one ɛ and the meshsize h.
Entrata in Redazione il 27 maggio 1978. 相似文献
13.
N. L. Patsko 《Mathematical Notes》1995,58(2):867-876
We consider approximations in the spaceL
p[0,a] to differentiable functions whoselth derivative belongs toL
p[0,a]. The function to be approximated is extended to the entire axis by Lagrange interpolation polynomials, and spline approximation with equally spaced nodes on the entire axis is then applied. This procedure results in a good approximation to the original function.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 281–294, August, 1995.The author is grateful to Yu. N. Subbotin for posing the problem and for his attention to the work, as well as to N. I. Chernykh for useful remarks. 相似文献
14.
Alexey Kuznetsov 《Journal of Mathematical Analysis and Applications》2021,493(2):124575
We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [5]. 相似文献
15.
K. Yu. Fedorovskii 《Mathematical Notes》1996,59(4):435-439
We study approximations of functions byn-analytic polynomials in the uniform norm on closed rectifiable Jordan curves in the complex plane. It is shown that, in contrast
to the case of uniform approximations by complex polynomials, there are no topological criteria for the existence of such
approximations. We obtain a criterion for the existence ofn-analytic polynomial approximations in terms of analytic properties of these curves.
Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 603–609, April, 1996.
The author is extremely grateful to A. G. Vitushkin and P. V. Paramonov for the statement of the problem and their attention
to the work.
The work was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00225. 相似文献
16.
Lagrange基函数的复矩阵有理插值及连分式插值 总被引:1,自引:0,他引:1
顾传青 《高等学校计算数学学报》1998,20(4):306-314
1引言 矩阵有理插值问题与系统线性理论中的模型简化问题和部分实现问题有着紧密的联系~[1][2],在矩阵外推方法中也常常涉及线性或有理矩阵插值问题~[3]。按照文~[1]的阐述。目前已经研究的矩阵有理插值问题包括矩阵幂级数和Newton-Pade逼近。Hade逼近,联立Pade逼近,M-Pade逼近,多点Pade逼近等。显然,上述各种形式的矩阵Pade逼上梁山近是矩 相似文献
17.
Diego Dominici 《Central European Journal of Mathematics》2007,5(2):280-304
We analyze the Charlier polynomials C
n
(χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving
some special functions. We give numerical examples showing the accuracy of our formulas.
相似文献
18.
A new approach to the a posteriori analysis of distributed optimal control problems is presented. The approach is based on
functional type a posteriori estimates that provide computable and guaranteed bounds of errors for any conforming approximations
of a boundary value problem. Computable two-sided a posteriori estimates for the cost functional and estimates for approximations
of the state and control functions are derived. Numerical results illustrate the efficiency of the approach. Bibliography:
35 titles.
__________
Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 3–14 相似文献
19.
We show that for a broad class of interpolatory matrices on [-1,1] the sequence of polynomials induced by Hermite—Fejér interpolation to f(z)=z diverges everywhere in the complex plane outside the interval of interpolation [-1,1] . This result is in striking contrast to the behavior of the Lagrange interpolating polynomials.
June 15, 1998. Date accepted: January 26, 1999. 相似文献
20.
A discrete Fourier analysis on the fundamental domain Ω
d
of the d-dimensional lattice of type A
d
is studied, where Ω2 is the regular hexagon and Ω3 is the rhombic dodecahedron, and analogous results on d-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis
in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange
interpolation on the simplex is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation
is shown to be in the order of (log n)
d
. The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already
appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature
formulas, which provides only the second known example of such formulas. 相似文献