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1.
We study the Kolmogorov n-widths and the linear n-widths of weighted Sobolev classes on the unit ball Bd in Lq,μ, where Lq,μ, 1≤ q≤ ∞, denotes the weighted Lq space of functions on Bd with respect to weight . Optimal asymptotic orders of and as n→ ∞ are obtained for all 1≤ p, q≤ ∞ and μ≥0. 相似文献
2.
Let p be a trigonometric polynomial, non-negative on the unit circle . We say that a measure σ on belongs to the polynomial Szegő class, if , σs is singular, and For the associated orthogonal polynomials { n}, we obtain pointwise asymptotics inside the unit disc . Then we show that these asymptotics hold in L2-sense on the unit circle. As a corollary, we get an existence of certain modified wave operators. 相似文献
3.
Let denote the set of real algebraic polynomials of d variables and of total degree at most n. For a compact set KRd set Then the Markov factors on K are defined by (Here, as usual, Sd-1 stands for the Euclidean unit sphere in Rd.) Furthermore, given a smooth curve ΓRd, we denote by DTP the tangential derivative of P along Γ ( T is the unit tangent to Γ). Correspondingly, consider the tangential Markov factor of Γ given by Let . We prove that for every irrational number α>0 there are constants A, B>1 depending only on α such that for every sufficiently large n.Our second result presents some new bounds for Mn( Ωα), where ( d=2, α>1). We show that for every α>1 there exists a constant c>0 depending only on α such that Mn( Ωα) nclogn. 相似文献
4.
Let λ be a positive number, and let be a fixed Riesz-basis sequence, namely, ( xj) is strictly increasing, and the set of functions is a Riesz basis ( i.e., unconditional basis) for L2[− π, π]. Given a function whose Fourier transform is zero almost everywhere outside the interval [− π, π], there is a unique sequence in , depending on λ and f, such that the function is continuous and square integrable on (− ∞, ∞), and satisfies the interpolatory conditions Iλ( f)( xj)= f( xj), . It is shown that Iλ( f)converges to f in , and also uniformly on , as λ→0 +. In addition, the fundamental functions for the univariate interpolation process are defined, and some of their basic properties, including their exponential decay for large argument, are established. It is further shown that the associated interpolation operators are bounded on for every p[1, ∞]. 相似文献
5.
Let K be an arbitrary field of characteristic zero, Pn:= K[ x1,…, xn] be a polynomial algebra, and , for n2. Let σ′Aut K( Pn) be given by It is proved that the algebra of invariants, , is a polynomial algebra in n−1 variables which is generated by quadratic and cubic (free) generators that are given explicitly.Let σAut K( Pn) be given by It is well known that the algebra of invariants, , is finitely generated (theorem of Weitzenböck [R. Weitzenböck, Über die invarianten Gruppen, Acta Math. 58 (1932) 453–494]), has transcendence degree n−1, and that one can give an explicit transcendence basis in which the elements have degrees 1,2,3,…, n−1. However, it is an old open problem to find explicit generators for Fn. We find an explicit vector space basis for the quadratic invariants, and prove that the algebra of invariants is a polynomial algebra over in n−2 variables which is generated by quadratic and cubic (free) generators that are given explicitly.The coefficients of these quadratic and cubic invariants throw light on the ‘unpredictable combinatorics’ of invariants of affine automorphisms and of SL 2-invariants. 相似文献
6.
We prove the relative asymptotic behavior for the ratio of two sequences of multiple orthogonal polynomials with respect to the Nikishin systems of measures. The first Nikishin system is such that for each k, σk has a constant sign on its compact support consisting of an interval , on which almost everywhere, and a discrete set without accumulation points in . If denotes the smallest interval containing , we assume that Δk∩ Δk+1=0/, k=1,…, m−1. The second Nikishin system is a perturbation of the first by means of rational functions rk, k=1,…, m, whose zeros and poles lie in . 相似文献
7.
Let , and for k=0,1,…, denote the orthonormalized Jacobi polynomial of degree k. We discuss the construction of a matrix H so that there exist positive constants c, c1, depending only on H, α, and β such that Specializing to the case of Chebyshev polynomials, , we apply this theory to obtain a construction of an exponentially localized polynomial basis for the corresponding L2 space. 相似文献
| denote the zeros of nth m-orthogonal polynomial for a generalized Jacobi weight This note proves . The gap left over , is filled. 相似文献
9.
We study the uniqueness of limit cycles (periodic solutions that are isolated in the set of periodic solutions) in the scalar ODE
in terms of {
ik}, {
jk}, {
nk}. Our main result characterizes, under some additional hypotheses, the exponents {
ik}, {
jk}, {
nk}, such that for any choice of
the equation has at most one limit cycle. The obtained results have direct application to rigid planar vector fields, thus, planar systems of the form
x′=
y+
xR(
x,
y),
y′=−
x+
yR(
x,
y), where
. Concretely, when the set
has at least three elements (or exactly one) and another technical condition is satisfied, we characterize the exponents {
ik}, {
jk} such that the origin of the rigid system is a center for any choice of
and also when there are no limit cycles surrounding the origin for any choice of
.
相似文献
10.
A simply connected domain
is called a slit disc if
minus a finite number of closed radial slits not reaching the origin. A slit disc is called rational (rationally placed) if the lengths of all its circular arcs between neighboring slits (the arguments of the slits) are rational multiples of 2
π. The conformal mapping
of
onto
,
(0)=0,
′(0)>0, extends to a continuous function on
mapping it onto
. A finite union
E of closed non-intersecting arcs
ek on
is called rational if
for every
k,
νE(
ek) being the harmonic measures of
ek at
∞ for the domain
. A compact
E is rational if and only if there is a rational slit disc
such that
. A compact
E essentially supports a measure with periodic Verblunsky parameters if and only if
for a rationally placed
. For any tuple (
α1,…,
αg+1) of positive numbers with ∑
kαk=1 there is a finite family
of closed non-intersecting arcs
ek on
such that
νE(
ek)=
αk. For any set
and any
>0 there is a rationally placed compact
such that the Lebesgue measure |
EE*| of the symmetric difference
EE* is smaller than
.
相似文献
11.
We consider a system of heat equations
ut=
Δu and
vt=
Δv in
Ω×(0,
T) completely coupled by nonlinear boundary conditions
We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on
∂Ω with
for
p,
q>0, 0≤
α<1 and 0≤
β<
p.
相似文献
12.
Let
LN+1 be a linear differential operator of order
N+1 with constant coefficients and real eigenvalues
λ1,…,
λN+1, let
E(
ΛN+1) be the space of all
C∞-solutions of
LN+1 on the real line. We show that for
N2 and
n=2,…,
N, there is a recurrence relation from suitable subspaces
to
involving real-analytic functions, and with
if and only if contiguous eigenvalues are equally spaced.
相似文献
14.
A sequence
of continuous linear operators
is said to be hypercyclic if there exists a vector
, called hypercyclic for
, such that {
Tnx:
n0} is dense. A continuous linear operator, acting on some suitable function space, is PDE-preserving for a given set of convolution operators, when it map every kernel set for these operators invariantly. We establish hypercyclic sequences of PDE-preserving operators on
, and study closed infinite-dimensional subspaces of, except for zero, hypercyclic vectors for these sequences.
相似文献
15.
In this paper we investigate the
L2-solutions of vector refinement equations with exponentially decaying masks and a general dilation matrix. A vector refinement equation with a general dilation matrix and exponentially decaying masks is of the form
where the vector of functions
φ=(
φ1,…,
φr)
T is in
is an exponentially decaying sequence of
r×
r matrices called refinement mask and
M is an
s×
s integer matrix such that lim
n→∞M-n=0. Associated with the mask
a and dilation matrix
M is a linear operator
Qa on
given by
The iterative scheme
is called vector subdivision scheme or vector cascade algorithm. The purpose of this paper is to provide a necessary and sufficient condition to guarantee the sequence
to converge in
L2-norm. As an application, we also characterize biorthogonal multiple refinable functions, which extends some main results in [B. Han, R.Q. Jia, Characterization of Riesz bases of wavelets generated from multiresolution analysis, Appl. Comput. Harmon. Anal., to appear] and [R.Q. Jia, Convergence of vector subdivision schemes and construction of biorthogonal multiple wavelets, Advances in Wavelet (Hong Kong, 1997), Springer, Singapore, 1998, pp. 199–227] to the general setting.
相似文献
16.
Let
be an orthonormal Jacobi polynomial of degree
k. We will establish the following inequality:
where
δ-1<
δ1 are appropriate approximations to the extreme zeros of
. As a corollary we confirm, even in a stronger form, T. Erdélyi, A.P. Magnus and P. Nevai conjecture [T. Erdélyi, A.P. Magnus, P. Nevai, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994) 602–614] by proving that
in the region
.
相似文献
17.
Consider a parametric statistical model,
P(d
x|
θ), and an improper prior distribution,
ν(d
θ), that together yield a (proper) formal posterior distribution,
Q(d
θ|
x). The prior is called
strongly admissible if the generalized Bayes estimator of every bounded function of
θ is admissible under squared error loss. Eaton [M.L. Eaton, A statistical diptych: Admissible inferences-recurrence of symmetric Markov chains, Annals of Statistics 20 (1992) 1147–1179] used the Blyth–Stein Lemma to develop a sufficient condition, call it
, for strong admissibility of
ν. Our main result says that, under mild regularity conditions, if
ν satisfies
and
g(
θ) is a bounded, non-negative function, then the
perturbed prior distribution g(
θ)
ν(d
θ) also satisfies
and is therefore strongly admissible. Our proof has three basic components: (i) Eaton's [M.L. Eaton, A statistical diptych: Admissible inferences-recurrence of symmetric Markov chains, Annals of Statistics 20 (1992) 1147–1179] result that the condition
is equivalent to the
local recurrence of the Markov chain whose transition function is
R(d
θ|
η)=∫
Q(d
θ|
x)
P(d
x|
η); (ii) a new result for general state space Markov chains giving conditions under which local recurrence is equivalent to recurrence; and (iii) a new generalization of Hobert and Robert's [J.P. Hobert, C.P. Robert, Eaton's Markov chain, its conjugate partner and
-admissibility, Annals of Statistics 27 (1999) 361–373] result that says Eaton's Markov chain is recurrent if and only if the chain with transition function
is recurrent. One important application of our results involves the construction of strongly admissible prior distributions for estimation problems with restricted parameter spaces.
相似文献
18.
Let
hp, 1<
p<∞, be the best ℓ
p-approximation of the element
from a proper affine subspace
K of
,
hK, and let
denote the strict uniform approximation of
h from
K. We prove that there are a vector
and a real number
a, 0
a1, such that
for all
p>1, where
with
γp=
o(
ap/
p).
相似文献
20.
Let
and let
wρ(
x)|
x|
ρexp(-
Q(
x)), where
and
is an even function. In this paper we consider the properties of the orthonormal polynomials with respect to the weight
, obtaining bounds on the orthonormal polynomials and spacing on their zeros. Moreover, we estimate
An(
x) and
Bn(
x) defined in Section 4, which are used in representing the derivative of the orthonormal polynomials with respect to the weight
.
相似文献