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1.
Given a specification linear operatorS, we want to test an implementation linear operatorA and determine whether it conforms to the specification operator according to an error criterion. In an earlier paper [3],
we studied a worst case error in which we test whether the error is no more than a given bound ε>0 for all elements in a given
setF, i.e., sup
fεf∥Sf—Af∥≤ε. In this work, we study the average error instead, i. e., ∫
F
∥Sf-Af∥2μ(df)ɛ≤2, where μ is a probability measure onF. We assume that an upper boundK on the norm of the difference ofS andA is given a priori. It turns out that any finite number of tests is in general inconclusive with the average error. Therefore,
as in the worst case, we allow a relaxation parameter α>0 and test for weak conformance with an error bound (1+α)ε. Then a
finite number of tests from an arbitrary orthogonal complete sequence is conclusive. Furthermore, the eigenvectors of the
covariance operatorC
μ of the probability measure μ provide an almost optimal test sequence. This implies that the test set isuniversal; it only depends on the set of valid inputsF and the measure μ, and is independent ofS, A, and the other parameters of the problem. However, the minimal number of tests does depend on all the parameters of the testing
problem, i.e., ε, α,K, and the eigenvalues ofC
μ. In contrast to the worst case setting, it also depends on the dimensiond of the range space ofS andA.
This work was done while consulting at Bell Laboratories, and is partially supported by the National Science Foundation and
the Air Force Office of Scientific Research. 相似文献
2.
In this paper, we introduce the fractional integral operator T of degree α of order m with respect to a dilation A for 0 < α < 1 and . First we establish the Hardy-Littlewood-Sobolev inequalities for T on anisotropic Hardy spaces associated with dilation A, which show that T is bounded from H
p
to H
q
, or from H
p
to L
q
, where 0 < p ≤ 1/(1 + α) and 1/q = 1/p − α. Then we give anisotropic Hardy spaces estimates for a class of multilinear operators formed by fractional integrals
or Calderón-Zygmund singular integrals. Finally, we apply the above results to give the boundedness of the commutators of
T and a BMO function.
Research supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025). 相似文献
3.
Sharp estimates of the point-evaluation functional in weighted Bergman spaces L
p
a
(Ω, dν
α) and for the point-evaluation derivalive functional in Besov spaces B
p
(Ω) are obtained for bounded symmetric domains Ω in ℂ
n
.
Received October 25, 1999, Accepted December 6, 2000 相似文献
4.
A. Rogozhin 《Integral Equations and Operator Theory》2007,57(2):283-301
In this paper we estimate the norm of the Moore-Penrose inverse T(a)+ of a Fredholm Toeplitz operator T(a) with a matrix-valued symbol a∈LN × N∞ defined on the complex unit circle. In particular, we show that in the ”generic case” the strict inequality ||T(a)+|| > ||a−1||∞ holds. Moreover, we discuss the asymptotic behavior of ||T(tra)+|| for
. The results are illustrated by numerical experiments. 相似文献
5.
Oscar F. Bandtlow 《Integral Equations and Operator Theory》2008,61(1):21-43
For a, α > 0 let E(a, α) be the set of all compact operators A on a separable Hilbert space such that s
n
(A) = O(exp(-anα)), where s
n
(A) denotes the n-th singular number of A. We provide upper bounds for the norm of the resolvent (zI − A)−1 of A in terms of a quantity describing the departure from normality of A and the distance of z to the spectrum of A. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in E(a, α).
相似文献
6.
Ferenc Weisz 《Journal of Fourier Analysis and Applications》2000,6(4):389-401
The two-parameter dyadic martingale Hardy spacesH
p are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series
is bounded from Hp to Lp (1/(α+1), 1/(β+1)<p<∞) and is of weak type (H
1
#
, L1), where the Hardy space H
1
#
is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f∈H
1
#
converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on Hp whenever 1/(α+1), 1/(β+1)<p<∞. Thus in case f∈Hp, the (C, α, β) means converge to f in Hp norm. The same results are proved for the conjugate (C, α, β) means, too. 相似文献
7.
Lattice-universal Orlicz function spacesL
F
α,β[0, 1] with prefixed Boyd indices are constructed. Namely, given 0<α<β<∞ arbitrary there exists Orlicz function spacesL
F
α,β[0, 1] with indices α and β such that every Orlicz function spaceL
G
[0, 1] with indices between α and β is lattice-isomorphic to a sublattice ofL
F
α,β[0, 1]. The existence of classes of universal Orlicz spacesl
Fα,β(I) with uncountable symmetric basis and prefixed indices α and β is also proved in the uncountable discrete case.
Partially supported by BFM2001-1284. 相似文献
8.
We study a class of weighted shifts W
α defined by a recursively generated sequence α ≡ α0, … , α
m−2, (α
m−1, α
m
, α
m+1)∧ and characterize the difference between quadratic hyponormality and positive quadratic hyponormality. We show that a shift
in this class is positively quadratically hyponormal if and only if it is quadratically hyponormal and satisfies a finite
number of conditions. Using this characterization, we give a new proof of [12, Theorem 4.6], that is, for m = 2, W
α is quadratically hyponormal if and only if it is positively quadratically hyponormal. Also, we give some new conditions for
quadratic hyponormality of recursively generated weighted shift W
α (m ≥ 2). Finally, we give an example to show that for m ≥ 3, a quadratically hyponormal recursively generated weighted shift W
α need not be positively quadratically hyponormal. 相似文献
9.
The present contribution deals with the Stokes operator Aq on Lqσ(Ω), 1<q<∞, where Ω is an exterior domain in ℝ2 of class C2. It is proved that Aq admits a bounded H∞-calculus. This implies the existence of bounded imaginary powers of Aq, which has several important applications. – So far this property was only known for exterior domains in ℝn, n≥3. – In particular, this shows that Aq has maximal regularity on Lqσ(Ω). For the proof the resolvent (λ+Aq)−1 has to be analyzed for |λ|→∞ and λ→0. For large λ this is done using an approximate resolvent based on the results of [3], which were obtained by applying the calculus of
pseudodifferential boundary value problems. For small λ we analyze the representation of the resolvent developed in [11] by a potential theoretical method. 相似文献
10.
We introduce the class of operators on Banach spaces having property (H) and study Weyl’s theorems, and related results for operators which satisfy this property. We show that a- Weyl’s theorem holds for every decomposable operator having property (H). We also show that a-Weyl’s theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator Tμ of a group algebra L1(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras. 相似文献
11.
We investigate Besov spaces and their connection with trigonometric polynomial approximation inL
p[−π,π], algebraic polynomial approximation inL
p[−1,1], algebraic polynomial approximation inL
p(S), and entire function of exponential type approximation inL
p(R), and characterizeK-functionals for certain pairs of function spaces including (L
p[−π,π],B
s
a(L
p[−π,π])), (L
p(R),s
a(Lp(R))),
, and
, where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r.
This project is supported by the National Science Foundation of China. 相似文献
12.
Ferenc Weisz 《逼近论及其应用》2000,16(1):52-65
The two-dimensional classical Hardy space Hp(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space Hp(T×T) to Lp(T2) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H 1 # (T×T), L1(T2)), where the Hardy space H 1 # (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H 1 # (T×T)⊃LlogL(T 2) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces Hp(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈Hp(T×T), the (C, α, β) means convergs to f in Hp(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. 相似文献
13.
Ariyadasa Aluthge 《Integral Equations and Operator Theory》2006,55(4):597-600
A bounded linear operator T is clalled p-hyponormal if (T*T)p ≥ (TT)p, 0 < p < 1. It is known that for semi-hyponormal operators (p = 1/2), the spectrum of the operator is equal to the union of the spectra of the general polar symbols of the operator.
In this paper we prove a somewhat weaker result for invertible p-hyponormal operators for 0 < p < 1/2. 相似文献
14.
Ferenc Weisz 《分析论及其应用》2000,16(1):52-65
The two-dimensional classical Hardy space Hp(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded
from the space Hp(T×T) to Lp(T2) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H
1
#
(T×T), L1(T2)), where the Hardy space H
1
#
(T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H
1
#
(T×T)⊃LlogL(T
2) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces
Hp(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈Hp(T×T), the (C, α, β) means convergs to f in Hp(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.
This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt
Foundation. 相似文献
15.
Min Guohua 《分析论及其应用》1992,8(3):28-37
In this paper, the Lp-convergence of Grünwald interpolation Gn(f,x) based on the zeros of Jacobi polynomials J
n
(α,β)
(x)(−1<α,β<1) is considered. Lp-convergence (0<p<2) of Grünwald interpolation Gn(f,x) is proved for p·Max(α,β)<1. Moreover, Lp-convergence (p>0) of Gn(f,x) is obtained for −1<α,β≤0. Therefore, the results of [1] and [3–5] are improved. 相似文献
16.
Bernhard H. Haak Peer Christian Kunstmann 《Integral Equations and Operator Theory》2006,55(4):497-533
We study linear systems, described by operators A, B, C for which the state space X is a Banach space.We suppose that − A generates a bounded analytic semigroup and give conditions for admissibility of B and C corresponding to those in G. Weiss’ conjecture. The crucial assumptions on A are boundedness of an H∞-calculus or suitable square function estimates, allowing to use techniques recently developed by N. Kalton and L. Weis. For observation spaces Y or control spaces U that are not Hilbert spaces we are led to a notion of admissibility extending previous considerations by C. Le Merdy. We also obtain a characterisation of wellposedness for the full system. We give several examples for admissible operators
including point observation and point control. At the end we study a heat equation in X = Lp(Ω), 1 < p < ∞, with boundary observation and control and prove its wellposedness for several function spaces Y and U on the boundary ∂Ω. 相似文献
17.
Nathan S. Feldman 《Integral Equations and Operator Theory》2007,58(2):153-173
A pair of commuting operators, (A,B), on a Hilbert space
is said to be hypercyclic if there exists a vector
such that {A
n
B
k
x : n, k ≥ 0} is dense in
. If f, g ∈H
∞(G) where G is an open set with finitely many components in the complex plane, then we show that the pair (M
*
f
, M
*
g
) of adjoints of multiplcation operators on a Hilbert space of analytic functions on G is hypercyclic if and only if the semigroup they generate contains a hypercyclic operator. However, if G has infinitely many components, then we show that there exists f, g ∈H
∞(G) such that the pair (M
*
f
, M
*
g
) is hypercyclic but the semigroup they generate does not contain a hypercyclic operator. We also consider hypercyclic n-tuples. 相似文献
18.
Ath Kehagias 《Rendiconti del Circolo Matematico di Palermo》2002,51(3):503-526
On a generalized deMorgan lattice (X, ≤, ∨, ∧,′) we introduce a family of join hyperoperations *
p
, parametrized by a parameterp εX. As a result we obtain a family of join spaces (X, *
p
). We show that: for everya,b εX the family {a*pb}
pεX
can be considered as thep-cuts of aL-fuzzy seta*b; in this manner we synthesize aL-fuzzy hyperoperation * which takes pairs fromX toL-fuzzy subsets ofX. We then show that (X, *
p
) is aL-fuzzy hypergroup (in the sense of Corsini) and can be considered as aL-fuzzy join space. Furthermore,a*b is aL-fuzzy interval for alla,b εX. 相似文献
19.
Erwann Delay 《manuscripta mathematica》2007,123(2):147-165
Let (M =]0, ∞[×N, g) be an asymptotically hyperbolic manifold of dimension n + 1 ≥ 3, equipped with a warped product metric. We show that there exist no TT L
2-eigentensors with eigenvalue in the essential spectrum of the Lichnerowicz Laplacian Δ
L
. If (M, g) is the real hyperbolic space, there is no symmetric L
2-eigentensors of Δ
L
. 相似文献
20.
M. Ivette Gomes 《Annals of the Institute of Statistical Mathematics》1984,36(1):71-85
Summary Let {X
n}n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM
n=max (X
1,…,X
n), suitably normalized with attraction coefficients {αn}n≧1(αn>0) and {b
n}n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s
which better approximate the d.f. of(M
n−bn)/an thanG(x), for moderaten. We thus consider differences of the formF
n(anx+bn)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф
α(x)=exp (−x−α), x>0] or a type III [Ψ
α(x)= exp (−(−x)α), x<0] d.f. of largest values, much closer toF
n(anx+bn) than the ultimate itself. 相似文献