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1.
It is often observed that interpolation based on translates of radial basis functions or non-radial kernels is numerically
unstable due to exceedingly large condition of the kernel matrix. But if stability is assessed in function space without considering
special bases, this paper proves that kernel-based interpolation is stable. Provided that the data are not too wildly scattered,
the L
2 or L
∞ norms of interpolants can be bounded above by discrete ℓ2 and ℓ ∞ norms of the data. Furthermore, Lagrange basis functions are uniformly bounded and Lebesgue constants grow at most like the
square root of the number of data points. However, this analysis applies only to kernels of limited smoothness. Numerical
examples support our bounds, but also show that the case of infinitely smooth kernels must lead to worse bounds in future
work, while the observed Lebesgue constants for kernels with limited smoothness even seem to be independent of the sample
size and the fill distance. 相似文献
2.
V. V. Makeev 《Journal of Mathematical Sciences》2007,140(4):551-557
Here are three samples of results. (1) Let m be a finite (absolutely) continuous mass distribution in ℝ2, and let ℓ = {ℓ1, ..., ℓ5 ⊂ ℝ2} be a quintuple of rays with common origin such that any two adjacent angles between them make a sum of at most π. Then an
affine image of ℓ subdivides m into five parts with any prescribed ratios.
(2) For each finite continuous mass distribution m in ℝn, there exist n mutually orthogonal hyperplanes any two of which quarter m.
(3) Let m and m′ be two finite continuous mass distributions in ℝRn with common center of symmetry O. Then there exist n hyperplanes through O any two of which quarter both m and m′. Bibliography: 9 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 92–106. 相似文献
3.
Bui Minh Phong 《Acta Mathematica Hungarica》2006,113(1-2):63-71
Summary We prove that if a complex valued completely multiplicative function F and a positive integer ℓ≦5 satisfy the condition F(N) = Uℓ, where Uℓ is the set of all ℓ-th roots of unity, then {F(n+1) F(n) ∣ nε N} = Uℓ. 相似文献
4.
Ricardo E. Carrera 《Mathematica Slovaca》2011,61(3):389-410
W
∞ denotes the category of archimedean ℓ-groups with designated weak unit and complete ℓ-homomorphisms that preserve the weak unit. CmpT
2,∞ denotes the category of compact Hausdorff spaces with continuous skeletal maps. This work introduces the concept of a functorial
polar function on W
∞ and its dual a functorial covering function on CmpT
2,∞. 相似文献
5.
Bernhard H. Haak 《Complex Analysis and Operator Theory》2010,4(2):281-299
In Ho and Russell (SIAM J Control Optim 21(4):614–640, 1983), and Weiss (Syst Control Lett 10(1): 79–82, 1988), a Carleson
measure criterion for admissibility of one-dimensional input elements with respect to diagonal semigroups is given. We extend
their results from the Hilbert space situation (L
2-admissibility on the state space ℓ
2) to the more general situation of L
p
-admissibility on the state space ℓ
q
. For analytic diagonal semigroups we present a new result that does not rely on Laplace transform methods. A comparison of
both criteria leads to a result on L
p
-admissibility for reciprocal systems in the sense of Curtain (Syst Control Lett 49(2):81–89, 2003). 相似文献
6.
Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. We show
that every central unit in the integral loop ring ZL is the product ℓμ0 of an element ℓ ∈ L and a loop ring element μ0 whose support is in the torsion subloop of L and use this result to determine when all central units of ZL are trivial.
Received: 8 October 2004 相似文献
7.
Ronald DeVore Guergana Petrova Przemyslaw Wojtaszczyk 《Constructive Approximation》2011,33(1):125-143
Let f be a continuous function defined on Ω:=[0,1]
N
which depends on only ℓ coordinate variables, f(x1,?,xN)=g(xi1,?,xil)f(x_{1},\ldots,x_{N})=g(x_{i_{1}},\ldots,x_{i_{\ell}}). We assume that we are given m and are allowed to ask for the values of f at m points in Ω. If g is in Lip1 and the coordinates i
1,…,i
ℓ
are known to us, then by asking for the values of f at m=L
ℓ
uniformly spaced points, we could recover f to the accuracy |g|Lip1
L
−1 in the norm of C(Ω). This paper studies whether we can obtain similar results when the coordinates i
1,…,i
ℓ
are not known to us. A prototypical result of this paper is that by asking for C(ℓ)L
ℓ
(log 2
N) adaptively chosen point values of f, we can recover f in the uniform norm to accuracy |g|Lip1
L
−1 when g∈Lip1. Similar results are proven for more general smoothness conditions on g. Results are also proven under the assumption that f can be approximated to some tolerance ε (which is not known) by functions of ℓ variables. 相似文献
8.
Vladimir I. Danilov Alexander V. Karzanov Gleb A. Koshevoy 《Journal of Algebraic Combinatorics》2010,32(4):497-531
For a permutation ω∈S
n
, Leclerc and Zelevinsky in Am. Math. Soc. Transl., Ser. 2 181, 85–108 (1998) introduced the concept of an ω-chamber weakly separated collection of subsets of {1,2,…,n} and conjectured that all inclusionwise maximal collections of this sort have the same cardinality ℓ(ω)+n+1, where ℓ(ω) is the length of ω. We answer this conjecture affirmatively and present a generalization and additional results. 相似文献
9.
We are interested in minimizing functionals with ℓ2 data and gradient fitting term and ℓ1 regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by
reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution
turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in
the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s
algorithm to solve the minimization problem with the ℓ1 norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented
efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the ℓ2 gradient fitting term can be used to avoid both edge blurring and staircasing effects.
相似文献
10.
Ferenc Weisz 《数学学报(英文版)》2010,26(9):1627-1640
A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W(hp, e∞) to W(Lp,e∞). This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W(L1, e∞) velong to L1. 相似文献
11.
We study the persistence of the asymptotic stability of delay equations both under linear and nonlinear perturbations. Namely,
we consider nonautonomous linear delay equations v′ = L(t)v
t
with a nonuniform exponential contraction. Our main objective is to establish the persistence of the nonuniform exponential stability of the
zero solution both under nonautonomous linear perturbations, i.e., for the equation v′ = (L(t) + M(t))v
t
, thus discussing the so-called robustness problem, and under a large class of nonlinear perturbations, namely for the equation
v′ = L(t)v
t
+ f(t, v
t
). In addition, we consider general contractions e
−λρ(t) determined by an increasing function ρ that includes the usual exponential behavior with ρ(t) = t as a very special case. We also obtain corresponding results in the case of discrete time. 相似文献
12.
Çiğdem A. Bektaş 《Mathematica Slovaca》2011,61(2):227-234
In this paper we define the sequence space ℓ
M
(Δ
υ
m
, p, q, s) on a seminormed complex linear space, by using a sequence of Orlicz functions. We study some algebraic and topological properties.
We prove some inclusion relations involving ℓ
M
(Δ
υ
m
, p, q, s). spaces 相似文献
13.
For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C
0(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak*-continuous. Given a discrete semigroup S, the convolution algebra ℓ
1(S) also carries a coproduct. In this paper we examine preduals for ℓ
1(S) making both the product and the coproduct weak*-continuous. Under certain conditions on S, we show that ℓ
1(S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple
semigroups and we construct uncountably many such preduals on ℓ
1(S) when S is either ℤ+×ℤ or (ℕ,⋅). 相似文献
14.
Francisco L. Hernández Baltasar Rodriguez-Salinas 《Israel Journal of Mathematics》1998,104(1):191-220
We study the setP
X
of scalarsp such thatL
p
is lattice-isomorphically embedded into a given rearrangement invariant (r.i.) function spaceX[0, 1]. Given 0<α≤β<∞, we construct a family of Orlicz function spacesX=L
F
[0, 1], with Boyd indicesα andβ, whose associated setsP
X
are the closed intervals [γ, β], for everyγ withα≤γ≤β. In particular forα>2, this proves the existence of separable 2-convex r.i. function spaces on [0,1] containing isomorphically scales ofL
p
-spaces for different values ofp. We also show that, in general, the associated setP
X
is not closed. Similar questions in the setting of Banach spaces with uncountable symmetric basis are also considered. Thus,
we construct a family of Orlicz spaces ℓ
F
(I), with symmetric basis and indices fixed in advance, containing ℓ
p
(Γ-subspaces for differentp’s and uncountable Λ⊂I. In contrast with the behavior in the countable case (Lindenstrauss and Tzafriri [L-T1]), we show that the set of scalarsp for which ℓ
p
(Γ) is isomorphic to a subspace of a given Orlicz space ℓ
F
(I) is not in general closed.
Supported in part by DGICYT grant PB 94-0243. 相似文献
15.
Francisco L. Hernández Baltasar Rodríguez-Salinas 《Israel Journal of Mathematics》1995,90(1-3):167-188
Given 0<α≤p≤β<∞, we construct Orlicz function spacesL
F
[0, 1] with Boyd indicesα andβ such thatL
p
is lattice isomorphic to a sublattice ofL
F
[0, 1]. Forp>2 this shows the existence of (non-trivial) separable r.i. spaces on [0, 1] containing an isomorphic copy ofL
p
. The discrete case of Orlicz spaces ℓ
F
(I) containing an isomorphic copy of ℓ
p
(Γ) for uncountable sets Γ ⊂I is also considered.
Supported in part by DGICYT, grant PB91-0377. 相似文献
16.
David Soudry 《Israel Journal of Mathematics》2000,120(1):511-561
In this paper we prove the full multiplicativity (in both variables) of gamma factors for generic representations of SO2ℓ+1 × GL
n
. These gamma factors are initially defined as proportionality factors of local functional equations, derived from a corresponding
global theory of certain Rankin-Selberg integrals which interpolate standardL-functions for SO2ℓ+1 × GL
n
. 相似文献
17.
We establish a strong regularity property for the distributions of the random sums Σ±λ
n
, known as “infinite Bernoulli convolutions”: For a.e. λ ∃ (1/2, 1) and any fixed ℓ, the conditional distribution of (w
n+1...,w
n+ℓ) given the sum Σ
n=0
∞
w
n
λ
n
, tends to the uniform distribution on {±1}ℓ asn → ∞. More precise results, where ℓ grows linearly inn, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system
of entropyh hasK-partitions of any prescribed conditional entropy in [0,h]. This answers a question of Rokhlin and Sinai from the 1960’s, for the case of Bernoulli systems.
The authors were partially supported by NSF grants DMS-9729992 (E. L.), DMS-9803597 (Y. P.) and DMS-0070538 (W. S.). 相似文献
18.
This paper considers metrics valued in abelian ℓ-groups and their induced topologies. In addition to a metric into an ℓ-group, one needs a filter in the positive cone to determine which balls are neighborhoods of their center. As a key special
case, we discuss a topology on a lattice ordered abelian group from the metric d
G
and the positive filter consisting of the weak units of G; in the case of
\mathbb Rn{\mathbb R^{n}} , this is the Euclidean topology. We also show that there are many Nachbin convex topologies on an ℓ-group which are not induced by any positive filter of the ℓ-group. 相似文献
19.
Sándor Baran 《Lithuanian Mathematical Journal》2011,51(2):122-140
Asymptotic properties of the variances of the spatial autoregressive model X
k,ℓ
= αX
k−1,ℓ
+ βX
k,ℓ−1 + γX
k−1,ℓ−1 + ε
k,ℓ
are investigated in the unit root case, that is, where the parameters are on the boundary of the domain of stability that
forms a tetrahedron in [−1, 1]3. The limit of the variance of n
−ϱ
X
[ns],[nt] is determined, where ϱ = 1/4 on the interior of the faces of the domain of stability, ϱ = 1/2 on the edges, and ϱ = 1 on the vertices. 相似文献
20.
This paper deals with a reducible sℓ(2,C) action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau conjecture:
Suppose that sℓ(2,C) acts on the formal power series ring via (1.1). Then I(f) = (ℓ
i1) ⊕ (ℓ
i2) ⊕... ⊕ (ℓ
is
) modulo some one dimensional sℓ(2,C) representations where (ℓ
i
) is an irreducible sℓ(2,C) representation of ℓ
i
dimension and {ℓ
i1
ℓ
i2,...,ℓ
is
} ⊆ {ℓ
1
, ℓ
2...,ℓ
r
}. Unlike classical invariant theory which deals only with irreducible action and 1-dimensional representations, we treat
the reducible action and higher dimensional representations successively. 相似文献