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1.
A polynomial of degree n in z1 and n1 in z isdefined by an interpolation projection from the space A(Np) of functions f analytic in the circular annulusp1 < <p and continuous on itsboundaries = p1, p. The points ofinterpolation are chosen to be spaced at equal angles aroundthe two boundaries, with arguments on the inner boundary midwaybetween those on the outer boundary. By calculating the Lebesgueconstants numerically, is found to be close to a minimax approximation for all p 1and all degrees n in the range 1 n 15. In the limiting casesp = 1 and, it is proved that is asymptotic to 21 log n. More specifically and , where nis the Lebesgueconstant of Gronwall for equally spaced interpolation on a circleby a polynomial of degree n. It is also demonstrated that is not in general monotonic in p, and that is not everywhere differentiable in p. 相似文献
2.
A polynomial of degree n in z1 and n 1 in z isdefined by an interpolation projection from the space A(N) of functions f analytic in thecircular annulus 1<|z| and continuous on its boundaries|z|=1, . The points of interpolation are chosen to coincidewith the n roots of zn=n and the n roots of zn=n.We prove Mason's conjecture that the corresponding Lebesguefunction attains its maximal value on the inner circle. We alsoestimate the bound of the Lebesgue constant . It is proved that the following estimate for theoperator norm holds:
where n, is the Lebesgue constant of Gronwall for equally spacedinterpolation on a circle by a polynomial of degree n. 相似文献
3.
《Journal of Algebra》1999,211(2):562-577
LetRbe a Krull ring with quotient fieldKanda1,…,aninR. If and only if theaiare pairwise incongruent mod every height 1 prime ideal of infinite index inRdoes there exist for all valuesb1,…,bninRan interpolating integer-valued polynomial, i.e., anf ∈ K[x] withf(ai) = biandf(R) ⊆ R.IfSis an infinite subring of a discrete valuation ringRvwith quotient fieldKanda1,…,aninSare pairwise incongruent mod allMkv ∩ Sof infinite index inS, we also determine the minimald(depending on the distribution of theaiamong residue classes of the idealsMkv ∩ S) such that for allb1,…,bn ∈ Rvthere exists a polynomialf ∈ K[x] of degree at mostdwithf(ai) = biandf(S) ⊆ Rv. 相似文献
4.
Lacunary Interpolation by Antiperiodic Trigonometric Polynomials 总被引:17,自引:0,他引:17
The problem of lacunary trigonometric interpolation is investigated. Does a trigonometric polynomial T exist which satisfies T(x
k) = a
k, D
m
T(x
k) = b
k, 0 k n – 1, where x
k = k/n is a nodal set, a
k and b
k are prescribed complex numbers,
and m N. Results obtained by several authors for the periodic case are extended to the antiperiodic case. In particular solvability is established when n as well as m are even. In this case a periodic solution does not exist. 相似文献
5.
The paper obtains error estimates for approximation by radial basis functions on the sphere. The approximations are generated
by interpolation at scattered points on the sphere. The estimate is given in terms of the appropriate power of the fill distance
for the interpolation points, in a similar manner to the estimates for interpolation in Euclidean space. A fundamental ingredient
of our work is an estimate for the Lebesgue constant associated with certain interpolation processes by spherical harmonics.
These interpolation processes take place in ``spherical caps' whose size is controlled by the fill distance, and the important
aim is to keep the relevant Lebesgue constant bounded. This result seems to us to be of independent interest.
March 27, 1997. Dates revised: March 19, 1998; August 5, 1999. Date accepted: December 15, 1999. 相似文献
6.
Robert Schaback 《Constructive Approximation》2005,21(3):293-317
In many cases, multivariate interpolation by smooth radial basis
functions converges toward polynomial interpolants, when the
basis functions are scaled to become flat.
In particular, examples show and this paper proves that
interpolation by scaled Gaussians
converges toward the de Boor/Ron least polynomial interpolant.
To arrive at this result, a few new tools are necessary.
The link between radial basis functions
and multivariate polynomials is provided by
radial polynomials ||x-y||22l\|x-y\|_2^{2\ell} that already occur in the seminal paper by C.A. Micchelli
of 1986. We study the polynomial
spaces spanned by linear combinations of
shifts of radial polynomials and introduce the notion
of a discrete moment basis to define
a new well-posed multivariate polynomial interpolation process
which is of minimal degree and also least and degree-reducing
in the sense of de Boor and Ron.
With these tools at hand, we generalize
the de Boor/Ron interpolation process and show that it
occurs as the limit of interpolation by Gaussian
radial basis functions. As a byproduct, we get
a stable method for preconditioning the matrices arising
with interpolation by smooth radial basis functions. 相似文献
7.
Xu S. S. 《Journal of Approximation Theory》1993,74(3)
In 1990, A. Horwitz proved a theorem about interpolation by restricted range polynomials and asked some ralted questions. This paper gives aflirmative answers to Horwitz′ questions and generalizes his theorem. 相似文献
8.
利用K泛函的定义首次研究了在Besov空间中,一类三角插值多项式的逼近和饱和问题,确定了逼近的饱和类与饱和阶. 相似文献
9.
We obtain asymptotic equalities for the upper bounds of approximations by interpolation trigonometric polynomials on classes of convolutions of periodic functions admitting a regular extension to a fixed strip of the complex plane. 相似文献
10.
Take a unit square and turn it into an annulus by
cutting a parallel square hole in it. We prove that if the
square hole has edge length $1 - 1/ \sqrt2 \approx 0.29$ and
a finite number of strips cover the annulus, then after
appropriate rearrangement the same strips can cover the unit
square as well. 相似文献
11.
We consider certain modified interpolation polynomials for functions from the space L
p[0, 2], 1 p . An estimate for the rate of approximation of an original function f by these polynomials in terms of its modulus of continuity is obtained. We establish that these polynomials converge almost everywhere to f. 相似文献
12.
M. M. Rizk 《Ukrainian Mathematical Journal》2001,53(1):155-165
We obtain explicit expansions of the fundamental Hermite interpolation polynomials in terms of Chebyshev polynomials in the case where the nodes considered are either zeros of the (n + 1)th-degree Chebyshev polynomial or extremum points of the nth-degree Chebyshev polynomial. 相似文献
13.
In this paper, an interpolation polynomial operator Fn (f; l, x ) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function. f(x)∈ C^b[-1,1] (0≤b≤1) Fn(f; l,x) converges to f(x) uniformly, where l is an odd number. 相似文献
14.
15.
Foundations of Computational Mathematics - Sparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For... 相似文献
16.
A. S. Serdyuk 《Ukrainian Mathematical Journal》2000,52(7):1141-1146
We establish an asymptotically exact estimate for the error of approximation of 2-periodic functions of high smoothness by interpolation trigonometric polynomials in the metric of L
1. 相似文献
17.
This note presents a commutant lifting theorem (CLT) of Agler type for the annulus \({\mathbb A}\) . Here the relevant set of test functions are the minimal inner functions on \({\mathbb A}\) —those analytic functions on \({\mathbb A}\) which are unimodular on the boundary and have exactly two zeros in \({\mathbb A}\) —and the model space is determined by a distinguished member of the Sarason family of kernels over \({\mathbb A}\) . The ideas and constructions borrow freely from the CLT of Ball et al. (Indiana Univ Math J 48(2):653–675, 1999) and Archer (Unitary dilations of commuting contractions. PhD thesis, University of Newcastle, 2004) for the polydisc, and Ambrozie and Eschmeier (A commutant lifting theorem on analytic polyhedra. Topological algebras, their applications, and related topics, 83108, Banach Center Publications, vol 67. Polish Academy of Sciences, Warsaw, 2005) for the ball in \({\mathbb C^n}\) , as well as generalizations of the de Branges–Rovnyak construction like found in Agler (On the representation of certain holomorphic functions defined on a polydisc. Topics in operator theory: Ernst D. Hellinger memorial volume, operator theory: advances and applications, vol 48. Birkhäuser, Basel, pp 47–66, 1990) and Ambrozie et al. (J Oper Theory 47(2):287–302, 2002). It offers a template for extending the result in McCullough and Sultanic (Complex Anal Oper Theory 1(4):581–620, 2007) to infinitely many test functions. Among the needed new ingredients is the formulation of the factorization implicit in the statement of the results in Ball et al. (Indiana Univ Math J 48(2):653–675, 1999) and Archer (Unitary dilations of commuting contractions. PhD thesis, University of Newcastle, 2004) and McCullough and Sultanic (Complex Anal Oper Theory 1(4):581–620, 2007) in terms of certain functional Hilbert spaces of Hilbert space valued functions. 相似文献
18.
基于Jacobi多项式零点的Grünwald插值算子 总被引:1,自引:0,他引:1
本文考虑基于一般Jacobi多项式J_n~(α,β)(x)(—1<α,β<1)零点的Grnwald插值多项式G_n(f,x);主要证明了G_n(f,x)在(—1,1)内几乎一致收敛于连续函数f(x),并给出了点态逼近估计;拓广和完善了文献[1,2]的结果。 相似文献
19.
本文考虑了函数f∈L_P[0,2π],1≤p<∞的特定的修正插值多项式,并给出了插值多项式对函数f的逼近速度的估计.本文的估计改进了Metelichenko最近的结果. 相似文献
20.
We obtain asymptotic equalities for the upper bounds of approximations of periodic infinitely differentiable functions by interpolation trigonometric polynomials in the metric of L
1 on the classes of convolutions. 相似文献