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1.
Let { X( t): t [ a, b]} be a Gaussian process with mean μ L2[ a, b] and continuous covariance K( s, t). When estimating μ under the loss ∫ ab (
(t)−μ(t)) 2 dt the natural estimator X is admissible if K is unknown. If K is known, X is minimax with risk ∫ ab K( t, t) dt and admissible if and only if the three by three matrix whose entries are K( ti, tj) has a determinant which vanishes identically in ti [ a, b], i = 1, 2, 3. 相似文献
2.
The objective of this paper is to establish a complete characterization of tight frames, and particularly of orthonormal wavelets, for an arbitrary dilation factor a>1, that are generated by a family of finitely many functions in L2:= L2(
). This is a generalization of the fundamental work of G. Weiss and his colleagues who considered only integer dilations. As an application, we give an example of tight frames generated by one single L2 function for an arbitrary dilation a>1 that possess “good” time-frequency localization. As another application, we also show that there does not exist an orthonormal wavelet with good time-frequency localization when the dilation factor a>1 is irrational such that aj remains irrational for any positive integer j. This answers a question in Daubechies' Ten Lectures book for almost all irrational dilation factors. Other applications include a generalization of the notion of s-elementary wavelets of Dai and Larson to s-elementary wavelet families with arbitrary dilation factors a>1. Generalization to dual frames is also discussed in this paper. 相似文献
3.
Let
Lf( x)=-\frac1w? i,j ? i( ai,j(·)? jf)( x)+ V( x) f( x){\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)} with the non-negative potential V belonging to reverse H?lder class with respect to the measure ω( x) dx, where ω( x) satisfies the A
2 condition of Muckenhoupt and a
i,j
( x) is a real symmetric matrix satisfying l -1w( x)|x| 2 £ ? ni,j=1ai,j( x)x ix j £ lw( x)|x| 2.{\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. } We obtain some estimates for VaL-a{V^{\alpha}\mathcal{L}^{-\alpha}} on the weighted L
p
spaces and we study the weighted L
p
boundedness of the commutator [ b, Va L-a]{[b, V^{\alpha} \mathcal{L}^{-\alpha}]} when b ? BMOw{b\in BMO_\omega} and 0 < α ≤ 1. 相似文献
4.
Summability of spherical h-harmonic expansions with respect to the weight function ∏ j=1d | xj| 2κj (κ j0) on the unit sphere Sd−1 is studied. The main result characterizes the critical index of summability of the Cesàro ( C,δ) means of the h-harmonic expansion; it is proved that the ( C,δ) means of any continuous function converge uniformly in the norm of C( Sd−1) if and only if δ>( d−2)/2+∑ j=1d κ j−min 1jd κ j. Moreover, it is shown that for each point not on the great circles defined by the intersection of the coordinate planes and Sd−1, the ( C,δ) means of the h-harmonic expansion of a continuous function f converges pointwisely to f if δ>( d−2)/2. Similar results are established for the orthogonal expansions with respect to the weight functions ∏ j=1d | xj| 2κj(1−| x| 2) μ−1/2 on the unit ball Bd and ∏ j=1d xjκj−1/2(1−| x| 1) μ−1/2 on the simplex Td. As a related result, the Cesàro summability of the generalized Gegenbauer expansions associated to the weight function | t| 2μ(1− t2) λ−1/2 on [−1,1] is studied, which is of interest in itself. 相似文献
5.
In this paper we use generalized Fourier-Hermite functionals to obtain a complete orthonormal set in L
2( C
a,b
[0, T]) where C
a,b
[0, T] is a very general function space. We then proceed to give a necessary and sufficient condition that a functional F in L
2( C
a,b
[0, T]) has an integral transform ℱ
γ,β
F also belonging to L
2( C
a,b
[0, T]). 相似文献
6.
For all integers m3 and all natural numbers a1, a2,…, am−1, let n= R( a1, a2,…, am−1) represent the least integer such that for every 2-coloring of the set {1,2,…, n} there exists a monochromatic solution to | Let t=min{a1,a2,…,am−1} and b=a1+a2++am−1−t. In this paper it is shown that whenever t=2, R(a1,a2,…,am−1)=2b2+9b+8.
It is also shown that for all values of t, R(a1,a2,…,am−1)tb2+(2t2+1)b+t3.
相似文献
7.
A new generalized Radon transform
R
α, β
on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform,
the generalized biaxially symmetric potential operator Δ
α, β
, and the Jacobi polynomials
Pk(b, a)(
t)P_{k}^{(\beta,\,\alpha)}(t). The transform
R
α, β
and its dual
Ra, b*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for
R
α, β
for functions in
La, bp(\mathbb
R2+)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform
R
α, β
is used to represent the solutions of the partial differential equations
Lu:=?
j=1majD
a, bju=
fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients
a
j
and the Cauchy problem for the generalized wave equation associated with the operator Δ
α, β
. Another application is that, by an invariant property of
R
α, β
, a new product formula for the Jacobi polynomials of the type
Pk(b, a)(
s)
C2ka+b+1(
t)=
còò
Pk(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.
相似文献
8.
We discuss the construction of finite difference schemes for the two-point nonlinear boundary value problem:
y
(2n)+
f(
x,y)=0,
y
(2j)(
a)=
A
2j
,
y
(2j)(
b)=
B
2j
,
j=0(1)
n–1,
n2. In the case of linear differential equations, these finite difference schemes lead to (2
n+1)-diagonal linear systems. We consider in detail methods of orders two, four and six for two-point boundary value problems involving a fourth order differential equation; convergence of these methods is established and illustrated by numerical examples.
相似文献
9.
If
L1 and
L2 are linear equations, then the disjunctive Rado number of the set {
L1,
L2} is the least integer
n, provided that it exists, such that for every 2-coloring of the set {1,2,…,
n} there exists a monochromatic solution to either
L1 or
L2. If such an integer
n does not exist, then the disjunctive Rado number is infinite. In this paper, it is shown that for all integers and
b1, the disjunctive Rado number for the equations
x1+
a=
x2 and
x1+
b=
x2 is
a+
b+1-gcd(
a,
b) if is odd and the disjunctive Rado number for these equations is infinite otherwise. It is also shown that for all integers
a>1 and
b>1, the disjunctive Rado number for the equations
ax1=
x2 and
bx1=
x2 is
cs+t-1 if there exist natural numbers
c,
s, and
t such that
a=
cs and
b=
ct and
s+
t is an odd integer and
c is the largest such integer, and the disjunctive Rado number for these equations is infinite otherwise.
相似文献
10.
Orthogonal expansions in product Jacobi polynomials with respect to the weight function
Wα, β(
x)=∏
dj=1 (1−
xj)
αj (1+
xj)
βj on [−1, 1]
d are studied. For
αj,
βj>−1 and
αj+
βj−1, the Cesàro (
C,
δ) means of the product Jacobi expansion converge in the norm of
Lp(
Wα, β, [−1, 1]
d), 1
p<∞, and
C([−1, 1]
d) if
Moreover, for
αj,
βj−1/2, the (
C,
δ) means define a positive linear operator if and only if
δ∑
di=1 (
αi+
βi)+3
d−1.
相似文献
11.
Let
ga(
t) and
gb(
t) be two positive, strictly convex and continuously differentiable functions on an interval (
a,
b) (−∞
a <
b ∞), and let {
Ln} be a sequence of linear positive operators, each with domain containing 1,
t,
ga(
t), and
gb(
t). If
Ln(ƒ;
x) converges to ƒ(
x) uniformly on a compact subset of (
a,
b) for the test functions ƒ(
t) = 1,
t,
ga(
t),
gb(
t), then so does every ƒ ε
C(
a,
b) satisfying ƒ(
t) =
O(
ga(
t)) (
t →
a+) and ƒ(
t) =
O(
gb(
t)) (
t →
b−). We estimate the convergence rate of
Lnƒ in terms of the rates for the test functions and the moduli of continuity of ƒ and ƒ′.
相似文献
12.
We propose a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling
function φ in the multivariate setting. For simplicity, we start with a standard dilation matrix 2
I2×2 in the bivariate setting and show how to construct compactly supported functions ψ
1,. . .,ψ
n with
n>3 such that {2
kψ
j(2
kx−ℓ,2
ky−
m),
k,ℓ,
m∈
Z,
j=1,. . .,
n} is an orthonormal basis for
L2(ℝ
2). Here,
n is dependent on the size of the support of φ. With parallel processes in modern computer, it is possible to use these orthonormal
wavelets for applications. Furthermore, the constructive method can be extended to construct compactly supported multi-wavelets
for any given compactly supported orthonormal multi-scaling vector. Finally, we mention that the constructions can be generalized
to the multivariate setting.
Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday
Mathematics subject classifications (2000) 42C15, 42C30.
相似文献
13.
In this paper we study the existence of periodic solutions of the fourth-order equations
uiv −
pu″ −
a(
x)
u +
b(
x)
u3 = 0 and
uiv −
pu″ +
a(
x)
u −
b(
x)
u3 = 0, where
p is a positive constant, and
a(
x) and
b(
x) are continuous positive 2
L-periodic functions. The boundary value problems (
P1) and (
P2) for these equations are considered respectively with the boundary conditions
u(0) =
u(
L) =
u″(0) =
u″(
L) = 0. Existence of nontrivial solutions for (
P1) is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for (
P2) is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation
uiv +
pu″ +
a(
x)
u −
b(
x)
u2 −
c(
x)
u3 = 0, where
p is a constant, and
a(
x),
b(
x), and
c(
x) are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentration-compactness arguments are used.
相似文献
14.
We prove that there does not exist an orthonormal basis {
b
n
} for
L
2(
R) such that the sequences {
μ(
b
n
)},
{m([^(
bn)])}\{\mu(\widehat{b_{n}})\}
, and
{D(
bn)D([^(
bn)])}\{\Delta(b_{n})\Delta(\widehat{b_{n}})\}
are bounded. A higher dimensional version of this result that involves generalized dispersions is also obtained. The main
tool is a time-frequency localization inequality for orthonormal sequences in
L
2(
R
d
). On the other hand, for
d>1 we construct a basis {
b
n
} for
L
2(
R
d
) such that the sequences {
μ(
b
n
)},
{m([^(
bn)])}\{\mu(\widehat{b_{n}})\}
, and
{D(
bn)D([^(
bn)])}\{\Delta(b_{n})\Delta(\widehat{b_{n}})\}
are bounded.
相似文献
15.
Let
m and
M be symmetric means in two and three variables, respectively. We say that
M is type 1 invariant with respect to
m if
M(
m(
a,
c),
m(
a,
b),
m(
b,
c))≡
M(
a,
b,
c). If
m is strict and isotone, then we show that there exists a unique
M which is type 1 invariant with respect to
m. In particular, we discuss the invariant logarithmic mean
L3, which is type 1 invariant with respect to
L(
a,
b)=(
b−
a)/(log
b−log
a). We say that
M is type 2 invariant with respect to
m if
M(
a,
b,
m(
a,
b))≡
m(
a,
b). We also prove existence and uniqueness results for type 2 invariance, given the mean
M(
a,
b,
c). The arithmetic, geometric, and harmonic means in two and three variables satisfy both type 1 and type 2 invariance. There are means
m and
M such that
M is type 2 invariant with respect to
m, but not type 1 invariant with respect to
m (for example, the Lehmer means).
L3 is type 1 invariant with respect to
L, but not type 2 invariant with respect to
L.
相似文献
16.
Let
Λ(
λj)
∞j=0 be a sequence of distinct real numbers. The span of {
xλ0,
xλ1, …,
xλn} over
is denoted by
Mn(
Λ)span{
xλ0,
xλ1, …,
xλn}. Elements of
Mn(
Λ) are called Müntz polynomials. The principal result of this paper is the following Markov-type inequality for products of Müntz polynomials. T
2.1.
LetΛ(
λj)
∞j=0andΓ(
γj)
∞j=0be increasing sequences of nonnegative real numbers. LetThen18(
n+
m+1)(λ
n+γ
m).
In particular ,
Under some necessary extra assumptions, an analog of the above Markov-type inequality is extended to the cases when the factor
x is dropped, and when the interval [0, 1] is replaced by [
a,
b](0, ∞).
相似文献
17.
Let
TR be a time-scale, with
a=inf
T,
b=sup
T. We consider the nonlinear boundary value problem
where
λR+:=[0,∞), and satisfies the conditions
We prove a strong maximum principle for the linear operator defined by the left-hand side of (1), and use this to show that for every solution (
λ,
u) of (1)–(2),
u is positive on
T a,
b . In addition, we show that there exists
λmax>0 (possibly
λmax=∞), such that, if 0
λ<
λmax then (1)–(2) has a unique solution
u(
λ), while if
λλmax then (1)–(2) has no solution. The value of
λmax is characterised as the principal eigenvalue of an associated weighted eigenvalue problem (in this regard, we prove a general existence result for such eigenvalues for problems with general, nonnegative weights).
相似文献
18.
Let
= {
Ut: t > 0} be a strongly continuous one-parameter group of operators on a Banach space
X and
Q be any subset of a set
(
X) of all probability measures on
X. By
(
Q;
) we denote the class of all limit measures of {
Utn(μ
1 * μ
2*…*μ
n)*δ
xn}, where {μ
n}
Q, {
xn}
X and measures
Utnμ
j (
j=1, 2,…,
n;
N=1, 2,…) form an infinitesimal triangular array. We define classes
Lm(
) as follows:
L0(
)=
(
(
X);
),
Lm(
)=
(
Lm−1(
);
) for
m=1, 2,… and
L∞(
)=
m=0∞Lm(
). These classes are analogous to those defined earlier by Urbanik on the real line. Probability distributions from
Lm(
),
m=0, 1, 2,…, ∞, are described in terms of their characteristic functionals and their generalized Poisson exponents and Gaussian covariance operators.
相似文献
19.
Suppose
L is a second order elliptic differential operator in
d and let
α>1. Baras and Pierre have proved in 1984 that
Γ is removable for
Lu=
uα if and only if its Bessel capacity Cap
2, α′(
Γ)=0. We extend this result to a general equation
Lu=
Ψ(
u) where
Ψ(
u) is an increasing convex function subject to
Δ2 and
2 conditions. Namely, we prove that
Γ is removable for
Lu=
Ψ(
u) if and only if its Orlicz capacity is zero, that is, the integral ∫
B dx Ψ(∫
Γ |
x−
y|
2−d ν(
dy)) is equal to 0 or ∞ for every measure
ν concentrated on
Γ, where
B stands for any ball containing
Γ.
相似文献
20.
By using Krasnoselskii's fixed point theorem and upper and lower solutions method, we find some sets of positive values λ determining that there exist positive
T-periodic solutions to the higher-dimensional functional difference equations of the form where
A(
n)=diag[
a1(
n),
a2(
n),…,
am(
n)],
h(
n)=diag[
h1(
n),
h2(
n),…,
hm(
n)],
aj,
hj :
Z→
R+, τ :
Z→
Z are
T -periodic,
j=1,2,…,
m,
T1, λ>0,
x :
Z→
Rm,
f :
R+m→
R+m, where
R+m={(
x1,…,
xm)
TRm,
xj0,
j=1,2,…,
m},
R+={
xR,
x>0}.
相似文献