共查询到20条相似文献,搜索用时 46 毫秒
1.
Peter Binev Albert Cohen Wolfgang Dahmen Ronald DeVore 《Constructive Approximation》2007,26(2):127-152
This paper is concerned with estimating the regression function fρ in supervised learning by utilizing piecewise polynomial approximations on adaptively generated partitions. The main point
of interest is algorithms that with high probability are optimal in terms of the least square error achieved for a given number
m of observed data. In a previous paper [1], we have developed for each β > 0 an algorithm for piecewise constant approximation
which is proven to provide such optimal order estimates with probability larger than 1- m-β. In this paper we consider the case of higher-degree polynomials. We show that for general probability measures ρ empirical
least squares minimization will not provide optimal error estimates with high probability. We go further in identifying certain
conditions on the probability measure ρ which will allow optimal estimates with high probability. 相似文献
2.
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
3.
Veerle Ledoux Marnix Van Daele 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,39(4):993-1011
The piecewise perturbation methods (PPM) have proven to be very efficient for the numerical solution of the linear time-independent
Schr?dinger equation. The underlying idea is to replace the potential function piecewisely by simpler approximations and then
to solve the approximating problem. The accuracy is improved by adding some perturbation corrections. Two types of approximating
potentials were considered in the literature, that is piecewise constant and piecewise linear functions, giving rise to the
so-called CP methods (CPM) and LP methods (LPM). Piecewise polynomials of higher degree have not been used since the approximating
problem is not easy to integrate analytically. As suggested by Ixaru (Comput Phys Commun 177:897–907, 2007), this problem can be circumvented using another perturbative approach to construct an expression for the solution of the
approximating problem. In this paper, we show that there is, however, no need to consider PPM based on higher-order polynomials,
since these methods are equivalent to the CPM. Also, LPM is equivalent to CPM, although it was sometimes suggested in the
literature that an LP method is more suited for problems with strongly varying potentials. We advocate that CP schemes can
(and should) be used in all cases, since it forms the most straightforward way of devising PPM and there is no advantage in
considering other piecewise polynomial perturbation methods. 相似文献
4.
We prove that a convex functionf ∈ L
p[−1, 1], 0<p<∞, can be approximated by convex polynomials with an error not exceeding Cω
3
ϕ
(f,1/n)p where ω
3
ϕ
(f,·) is the Ditzian-Totik modulus of smoothness of order three off. We are thus filling the gap between previously known estimates involving ω
3
ϕ
(f,1/n)p, and the impossibility of having such estimates involving ω4. We also give similar estimates for the approximation off by convexC
0 andC
1 piecewise quadratics as well as convexC
2 piecewise cubic polynomials.
Communicated by Dietrich Braess 相似文献
5.
We consider 3-monotone approximation by piecewise polynomials with prescribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L1 approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone approximation by piecewise polynomials with prescribed knots. Such estimates are well known for monotone and convex approximation, and to the contrary, they in general are not valid for higher orders of monotonicity. Also we show that any convex piecewise polynomial can be modified to be, in addition, interpolatory, while still preserving the degree of the uniform approximation. Alternatively, we show that we may smooth the approximating piecewise polynomials to be twice continuously differentiable, while still being 3-monotone and still keeping the same degree of approximation. 相似文献
6.
Best approximations of the classes B
p,θ
r
of periodic functions of many variables in uniform metric
A. S. Romanyuk 《Ukrainian Mathematical Journal》2006,58(10):1582-1596
We obtain estimates exact in order for the best approximations of the classes B
∞,θ
r
of periodic functions of two variables in the metric of L
∞ by trigonometric polynomials whose spectrum belongs to a hyperbolic cross. We also investigate the best approximations of
the classes B
p,θ
r
, 1 ≤ p < ∞, of periodic functions of many variables in the metric of L
∞ by trigonometric polynomials whose spectrum belongs to a graded hyperbolic cross.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1395–1406, October, 2006. 相似文献
7.
《中国科学 数学(英文版)》2017,(8)
This paper proposes a weak Galerkin finite element method to solve incompressible quasi-Newtonian Stokes equations. We use piecewise polynomials of degrees k + 1(k 0) and k for the velocity and pressure in the interior of elements, respectively, and piecewise polynomials of degrees k and k + 1 for the boundary parts of the velocity and pressure, respectively. Wellposedness of the discrete scheme is established. The method yields a globally divergence-free velocity approximation. Optimal priori error estimates are derived for the velocity gradient and pressure approximations. Numerical results are provided to confirm the theoretical results. 相似文献
8.
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
9.
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
10.
A. S. Romanyuk 《Ukrainian Mathematical Journal》2009,61(4):613-626
Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov (B
p,θ
r
) and Nukol’skii (H
p
r
) classes of periodic functions of many variables in the metric of L
q
, 1 ≤ p, q ≤ ∞. We also establish the orders of the best approximations of functions from the same classes in the spaces L
1 and L
∞ by trigonometric polynomials with the corresponding spectrum. 相似文献
11.
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
12.
Veerle Ledoux Marnix Van Daele 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,62(6):993-1011
The piecewise perturbation methods (PPM) have proven to be very efficient for the numerical solution of the linear time-independent Schrödinger equation. The underlying idea is to replace the potential function piecewisely by simpler approximations and then to solve the approximating problem. The accuracy is improved by adding some perturbation corrections. Two types of approximating potentials were considered in the literature, that is piecewise constant and piecewise linear functions, giving rise to the so-called CP methods (CPM) and LP methods (LPM). Piecewise polynomials of higher degree have not been used since the approximating problem is not easy to integrate analytically. As suggested by Ixaru (Comput Phys Commun 177:897–907, 2007), this problem can be circumvented using another perturbative approach to construct an expression for the solution of the approximating problem. In this paper, we show that there is, however, no need to consider PPM based on higher-order polynomials, since these methods are equivalent to the CPM. Also, LPM is equivalent to CPM, although it was sometimes suggested in the literature that an LP method is more suited for problems with strongly varying potentials. We advocate that CP schemes can (and should) be used in all cases, since it forms the most straightforward way of devising PPM and there is no advantage in considering other piecewise polynomial perturbation methods. 相似文献
13.
We prove theorems that characterize the classes of functions whose best approximations by algebraic polynomials tend to zero
with given order. We construct approximations of solutions of operator-differential equations by polynomials in the inverse
operator.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1506–1516, November, 1998. 相似文献
14.
Xu Shusheng 《分析论及其应用》1989,5(1):33-45
We denote En(f) and E
k
n
(f) the best uniform approximations to a continuous function f defined on [a,b] by the sets of algebraic polynomials of degree
≤n and algebraic polynomials of degree ≤n with the coefficients of xk (k≤n) being zero. In this paper, in cases of r<k and r≥k while [a, b]=[−1,1] (or r<k,k≤r<2k and r>2k while [a,b]=[0, 1]),
we separately discuss the condtions for r-times continuously differentiable function f which enables
. 相似文献
15.
We establish that, for p ∈ [2, ∞), q = 1 or p = ∞, q ∈ [ 1, 2], the classes W
prof functions of many variables defined by restrictions on the L
p-norms of mixed derivatives of order r = (r
1, r
2, ..., r
m) are better approximated in the L
q-metric by periodic generalized splines than by generalized trigonometric polynomials. In these cases, the best approximations
of the Sobolev classes of functions of one variable by trigonometric polynomials and by periodic splines coincide.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1011–1020, August, 1998. 相似文献
16.
A. I. Stepanets 《Ukrainian Mathematical Journal》2005,57(4):640-665
We determine exact values of the best n-term approximations with restrictions on polynomials used for the approximation of λ, q-ellipsoids in the spaces S
ϕ
p,μ
.
__________
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 533–553, April, 2005. 相似文献
17.
In this paper, we present rational approximations based on Fourier series representation. For periodic piecewise analytic functions, the well-known Gibbs phenomenon hampers the convergence of the standard Fourier method. Here, for a given set of the Fourier coefficients from a periodic piecewise analytic function, we define Fourier-Padé-Galerkin and Fourier-Padé collocation methods by expressing the coefficients for the rational approximations using the Fourier data. We show that those methods converge exponentially in the smooth region and successfully reduce the Gibbs oscillations as the degrees of the denominators and the numerators of the Padé approximants increase.
Numerical results are demonstrated in several examples. The collocation method is applied as a postprocessing step to the standard pseudospectral simulations for the one-dimensional inviscid Burgers' equation and the two-dimensional incompressible inviscid Boussinesq convection flow.
18.
A. Leibman 《Israel Journal of Mathematics》2012,188(1):131-176
Generalized polynomials are functions obtained from conventional polynomials by applying the operations of taking the integer
part, addition, and multiplication. We construct a system of “basic” generalized polynomials with the property that any bounded
generalized polynomial is representable as a piecewise polynomial function of these basic ones. Such a representation is unique
up to a function vanishing almost everywhere, which solves the problem of determining whether two generalized polynomials
are equal a.e. The basic generalized polynomials are jointly equidistributed, thus we also obtain an effective algorithm of
finding the limiting distribution of values of one or several generalized polynomials. 相似文献
19.
O. F. Kalaida 《Journal of Mathematical Sciences》1995,75(4):1807-1811
Rational analogues of Taylor and Fourier polynomials and polynomials of the Lagrange type are constructed and investigated.
These analogues are shown to approximate a function with a remainder term of the same order as in the case of the aforementioned
polynomials. Conditions are established under which a polynomial of the Lagrange type and its rational analogue are two-sided
approximations of a function on a segment and their derivatives are two-sided approximations of the derivative of the function
at collocation nodes. Bibliography: 2 titles.
Translated fromObchuslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 32–38. 相似文献
20.
A generalized polynomial is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication,
and taking the integer part; a generalized polynomial mapping is a vector-valued mapping whose coordinates are generalized polynomials. We show that any bounded generalized polynomial
mapping u: Z
d
→ R
l
has a representation u(n) = f(ϕ(n)x), n ∈ Z
d
, where f is a piecewise polynomial function on a compact nilmanifold X, x ∈ X, and ϕ is an ergodic Z
d
-action by translations on X. This fact is used to show that the sequence u(n), n ∈ Z
d
, is well distributed on a piecewise polynomial surface (with respect to the Borel measure on that is the image of the Lebesgue measure under the piecewise polynomial function defining ). As corollaries we also obtain a von Neumann-type ergodic theorem along generalized polynomials and a result on Diophantine
approximations extending the work of van der Corput and of Furstenberg–Weiss. 相似文献