共查询到20条相似文献,搜索用时 682 毫秒
1.
Piotr Niemiec 《Rendiconti del Circolo Matematico di Palermo》2008,57(3):391-399
The aim of the paper is to prove that every f ∈ L
1([0,1]) is of the form f = , where j
n,k
is the characteristic function of the interval [k- 1 / 2
n
, k / 2
n
) and Σ
n=0∞Σ
k=12n
|a
n,k
| is arbitrarily close to ||f|| (Theorem 2). It is also shown that if μ is any probabilistic Borel measure on [0,1], then for any ɛ > 0 there exists a sequence (b
n,k
)
n≧0
k=1,...,2n
of real numbers such that and for each Lipschitz function g: [0,1] → ℝ (Theorem 3).
相似文献
2.
Abdelmajid Siai 《Potential Analysis》2006,24(1):15-45
Let Ω be an open bounded set in ℝN, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|p−2∇u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative
∂/∂νa related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝN, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere
defined in ℝ, with β(0)=γ(0)=0, f∈L1(ℝN), g∈L1(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,
in the sense that, if Tk(r)=max {−k,min (r,k)}, k>0, r∈ℝ, ∇u is the gradient by means of truncation (∇u=DTku on the set {|u|<k}) and
, u measurable; DTk(u)∈Lp(ℝN), k>0}, then
and u satisfies,
for every k>0 and every
.
Mathematics Subject Classifications (2000) 35J65, 35J70, 47J05. 相似文献
3.
We describe an algorithm for large-scale discrete ill-posed problems, called GKB-FP, which combines the Golub-Kahan bidiagonalization
algorithm with Tikhonov regularization in the generated Krylov subspace, with the regularization parameter for the projected
problem being chosen by the fixed-point method by Bazán (Inverse Probl. 24(3), 2008). The fixed-point method selects as regularization
parameter a fixed-point of the function ‖r
λ
‖2/‖f
λ
‖2, where f
λ
is the regularized solution and r
λ
is the corresponding residual. GKB-FP determines the sought fixed-point by computing a finite sequence of fixed-points of
functions ||rl(k)||2/||fl(k)||2\|r_{\lambda}^{(k)}\|_{2}/\|f_{\lambda}^{(k)}\|_{2}, where fl(k)f_{\lambda}^{(k)} approximates f
λ
in a k-dimensional Krylov subspace and rl(k)r_{\lambda}^{(k)} is the corresponding residual. Based on this and provided the sought fixed-point is reached, we prove that the regularized
solutions fl(k)f_{\lambda}^{(k)} remain unchanged and therefore completely insensitive to the number of iterations. This and the performance of the method when applied to
well-known test problems are illustrated numerically. 相似文献
4.
O. M. Fomenko 《Journal of Mathematical Sciences》2006,133(6):1733-1748
Let Sk(Γ) be the space of holomorphic Γ-cusp forms f(z) of even weight k ≥ 12 for Γ = SL(2, ℤ), and let Sk(Γ)+ be the set of all Hecke eigenforms from this space with the first Fourier coefficient af(1) = 1. For f ∈ Sk(Γ)+, consider the Hecke L-function L(s, f). Let
It is proved that for large K,
where ε > 0 is arbitrary. For f ∈ Sk(Γ)+, let L(s, sym
2 f) denote the symmetric square L-function. It is proved that as k → ∞ the frequence
converges to a distribution function G(x) at every point of continuity of the latter, and for the corresponding characteristic
function an explicit expression is obtained. Bibliography: 17 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 221–246. 相似文献
5.
Let f∈C
[−1,1]
″
(r≥1) and Rn(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions Rn(f,α,β,xk)=f(xk),Rn
′(f,α,β,xk)=f′(xk)(k=1,2,…,n), where {xk} are the roots of n-th Jacobi polynomial Pn(α,β,x),α,β>−1 and {x
k
″
} are the roots of (1−x2)Pn″(α,β,x). In this paper, we prove that
holds uniformly on [0,1].
In Memory of Professor M. T. Cheng
Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang. 相似文献
6.
In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L
2(ℝ
s
). Suppose ψ = (ψ1,..., ψ
r
)
T
and are two compactly supported vectors of functions in the Sobolev space (H
μ(ℝ
s
))
r
for some μ > 0. We provide a characterization for the sequences {ψ
jk
l
: l = 1,...,r, j ε ℤ, k ε ℤ
s
} and to form two Riesz sequences for L
2(ℝ
s
), where ψ
jk
l
= m
j/2ψ
l
(M
j
·−k) and , M is an s × s integer matrix such that lim
n→∞
M
−n
= 0 and m = |detM|. Furthermore, let ϕ = (ϕ1,...,ϕ
r
)
T
and be a pair of compactly supported biorthogonal refinable vectors of functions associated with the refinement masks a, and M, where a and are finitely supported sequences of r × r matrices. We obtain a general principle for characterizing vectors of functions ψν = (ψν1,...,ψνr
)
T
and , ν = 1,..., m − 1 such that two sequences {ψ
jk
νl
: ν = 1,..., m − 1, l = 1,...,r, j ε ℤ, k ε ℤ
s
} and { : ν=1,...,m−1,ℓ=1,...,r, j ∈ ℤ, k ∈ ℤ
s
} form two Riesz multiwavelet bases for L
2(ℝ
s
). The bracket product [f, g] of two vectors of functions f, g in (L
2(ℝ
s
))
r
is an indispensable tool for our characterization.
This work was supported by National Natural Science Foundation of China (Grant Nos. 10771190, 10471123) 相似文献
7.
Ilya A. Krishtal Benjamin D. Robinson Guido L. Weiss Edward N. Wilson 《Journal of Geometric Analysis》2007,17(1):87-96
An orthonormal wavelet system in ℝd, d ∈ ℕ, is a countable collection of functions {ψ
j,k
ℓ
}, j ∈ ℤ, k ∈ ℤd, ℓ = 1,..., L, of the form
that is an orthonormal basis for L2 (ℝd), where a ∈ GLd (ℝ) is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which L
= d = 1, ψ1(x) = ψ(x) = κ[0,1/2)(x) − κ[l/2,1)
(x), a = 2. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate
products Φ(x1, x2, ..., xd) = φ1 (x1)φ2(x2) ... φd(xd) of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to
find “nonseparable” examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems.
For example, if a = (
1-1
1 1
) is the quincunx dilation matrix, it is well-known (see, e.g., [5]) that one can construct nonseparable Haar-type scaling
functions which are characteristic functions of rather complicated fractal-like compact sets. In this work we shall construct
considerably simpler Haar-type wavelets if we use the ideas arising from “composite dilation” wavelets. These were developed
in [7] and involve dilations by matrices that are products of the form ajb, j ∈ ℤ, where a ∈ GLd(ℝ) has some “expanding” property and b belongs to a group of matrices in GLd(ℝ) having |det b| = 1. 相似文献
8.
We obtain new sharp Kolmogorov-type inequalities, in particular the following sharp inequality for 2π-periodic functions x ∈ L
∞
r
(T):
where k, r ∈ N, k < r, r ≥ 3, p ∈ [1, ∞], α = (r – k) / (r – 1 + 1/p), φ
r
is the perfect Euler spline of order r, and ν(x′) is the number of sign changes of x′ on a period.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1642–1649, December, 2008. 相似文献
9.
For every product preserving bundle functor T μ on fibered manifolds, we describe the underlying functor of any order (r, s, q), s ≥ r ≤ q. We define the bundle Kk,lr,s,q YK_{k,l}^{r,s,q} Y of (k, l)-dimensional contact elements of the order (r, s, q) on a fibered manifold Y and we characterize its elements geometrically. Then we study the bundle of general contact elements of type μ. We also determine all natural transformations of Kk,lr,s,q YK_{k,l}^{r,s,q} Y into itself and of T( Kk,lr,s,q Y )T\left( {K_{k,l}^{r,s,q} Y} \right) into itself and we find all natural operators lifting projectable vector fields and horizontal one-forms from Y to Kk,lr,s,q YK_{k,l}^{r,s,q} Y . 相似文献
10.
We observe an unknown function of infinitely many variables f = f(t), t = (t1, ..., tn, ... ) ∈, [0, 1]∞, in the Gaussian white noise of level ε > 0. We suppose that in each variable there exists a 1-periodical σ-smooth extension
of the function f(t) to IR
∞. Taking a quantity σ > 0 and a positive sequence a = {ak}, we consider the set
that consists of functions f such that
. We consider the cases ak = kα and ak = exp(λk), α > 0, λ > 0. We would like to estimate a function f ∈
or to test the null hypothesis H0: f = 0 against the alternatives f ∈
, where the set
consists of functions f ∈
such that ∥f∥2 ≥ r. In the estimation problem, we obtain the asymptotics (as ε → 0) of the minimax quadratic risk. In the detection problem,
we study the sharp asymptotics of minimax separation rates f
ɛ
*
that provide distiguishability in the problems. Bibliography: 12 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 91–113. 相似文献
11.
V. A. Kofanov 《Ukrainian Mathematical Journal》2008,60(10):1557-1573
We obtain a new sharp inequality for the local norms of functions x ∈ L
∞, ∞
r
(R), namely,
where φ
r
is the perfect Euler spline, on the segment [a, b] of monotonicity of x for q ≥ 1 and for arbitrary q > 0 in the case where r = 2 or r = 3.
As a corollary, we prove the well-known Ligun inequality for periodic functions x ∈ L
∞
r
, namely,
for q ∈ [0, 1) in the case where r = 2 or r = 3.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1338–1349, October, 2008. 相似文献
12.
In this paper we obtain a general lower bound for the tail distribution of the Fourier spectrum of Boolean functionsf on {1, −1}
N
. Roughly speaking, fixingk∈ℤ+ and assuming thatf is not essentially determined by a bounded number (depending onk) of variables, we have that
. The example of the majority function shows that this result is basically optimal. 相似文献
13.
R. S. Laugesen 《Journal of Fourier Analysis and Applications》2008,14(2):235-266
The affine synthesis operator
is shown to map the coefficient space ℓ
p
(ℤ+×ℤ
d
) surjectively onto L
p
(ℝ
d
), for p∈(0,1]. Here ψ
j,k
(x)=|det a
j
|1/p
ψ(a
j
x−k) for dilation matrices a
j
that expand, and the synthesizer ψ∈L
p
(ℝ
d
) need satisfy only mild restrictions, for example, ψ∈L
1(ℝ
d
) with nonzero integral or else with periodization that is real-valued, nontrivial and bounded below.
An affine atomic decomposition of L
p
follows immediately:
Tools include an analysis operator that is nonlinear on L
p
.
Laugesen’s travel was supported by the NSF under Award DMS–0140481. 相似文献
14.
LetV be ann-dimensional space over an infinite field of characteristic different from 2. Therank ofw ∈ Λ
p
V is the minimal dimension of a subspaceU ⊂V such thatw ∈ Λ
p
U. Extending a well-known result on linear spaces in the Grassmannian, it is shown that ifp≤k<n then the maximal dimension of a subspaceW ⊂ Λ
p
V such that rankw≤k for allω ∈W is
where∈=1 ifk=p orp=2|k,∈=0 otherwise, andm satisfies
.
Supported by The Israel Science Foundation founded by the Academy of Sciences and Humanities. 相似文献
15.
It is well-known that (ℤ+, |) = (ℤ+, GCD, LCM) is a lattice, where | is the usual divisibility relation and GCD and LCM stand for the greatest common divisor
and the least common multiple of positive integers.
The number $
d = \prod\nolimits_{k = 1}^r {p_k^{d^{(k)} } }
$
d = \prod\nolimits_{k = 1}^r {p_k^{d^{(k)} } }
is said to be an exponential divisor or an e-divisor of $
n = \prod\nolimits_{k = 1}^r {p_k^{n^{(k)} } }
$
n = \prod\nolimits_{k = 1}^r {p_k^{n^{(k)} } }
(n > 1), written as d |
e
n, if d
(k) for all prime divisors p
k
of n. It is easy to see that (ℤ+\{1}, |
e
is a poset under the exponential divisibility relation but not a lattice, since the greatest common exponential divisor (GCED)
and the least common exponential multiple (LCEM) do not always exist. 相似文献
16.
R. E. Maiboroda 《Ukrainian Mathematical Journal》1998,50(7):1067-1079
For a process X(t)=Σ
j=1
M
g
j
(t)ξ
j
(), where gj(t) are nonrandom given functions, is a stationary vector-valued Gaussian process, Eξk(t) = 0, and Eξk(0) Eξl(τ) = r
kl(τ), we construct an estimate for the functions r
kl(τ) on the basis of observations X(t), t ∈ [0, T]. We establish conditions for the asymptotic normality of as T → ∞. We consider the problem of the optimal choice of parameters of the estimate depending on observations.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 937–947, July, 1998. 相似文献
17.
This paper exploits and extends results of Edmonds, Cunningham, Cruse and McDiarmid on matroid intersections. Letr
1 andr
2 be rank functions of two matroids defined on the same setE. For everyS ⊂E, letr
12(S) be the largest cardinality of a subset ofS independent in both matroids, 0≦k≦r
12(E)−1. It is shown that, ifc is nonnegative and integral, there is ay: 2
E
→Z
+ which maximizes
and
, subject toy≧0, ∀j∈E,
. 相似文献
18.
LetH be any complex inner product space with inner product <·,·>. We say thatf: ℂ→ℂ is Hermitian positive definite onH if the matrix
is Hermitian positive definite for all choice ofz
1,…,z
n inH for alln. It is strictly Hermitian positive definite if the matrix (*) is also non-singular for any choice of distinctz
1,…,z
n inH. In this article, we prove that if dimH≥3, thenf is Hermitian positive definite onH if and only if
whereb
k,l
≥0 for allk, l in ℤ, and the series converges for allz in ℂ. We also prove thatf of the form (**) is strictly Hermitian positive definite on anyH if and only if the setJ={(k,l):b
k,l
>0} is such that (0,0)∈J, and every arithmetic sequence in ℤ intersects the values {k−l: (k, l)∈J} an infinite number of times. 相似文献
(1) |
(1) |
19.
T.-A. Tanaka 《Archiv der Mathematik》2002,78(3):202-209
Transcendence of the number ?k=0¥ ark \sum_{k=0}^\infty \alpha^{r_k} , where a \alpha is an algebraic number with 0 < | a | \mid\alpha\mid > 1 and {rk}k\geqq0 \{r_k\}_{k\geqq0} is a sequence of positive integers such that limk?¥ rk+1/rk = d ? \mathbbN \{1} \lim_{k\to\infty}\, r_{k+1}/r_k = d \in \mathbb{N}\, \backslash \{1\} , is proved by Mahler's method. This result implies the transcendence of the number ?k=0¥ akdk \sum_{k=0}^\infty \alpha^{kd^k} . 相似文献
20.
Tao Feng 《Designs, Codes and Cryptography》2012,62(3):253-258
In 1976, Helleseth conjectured that two binary m-sequences of length 2
m
− 1 can not have a three-valued crosscorrelation function when m is a power of 2. We show that this conjecture is true when −1 is a correlation value. In other words, if C1,k{{\mathcal{C}}_{1,k}} is the cyclic code of length 2
m
− 1 with two zeros α, α
k
, where α is a primitive element of
\mathbbF2m{{\mathbb{F}}_{2^m}} and gcd(k, 2
m
− 1) = 1, then its dual C1,k^{{\mathcal{C}}_{1,k}^{\perp}} can not have three weights when m is a power of 2. 相似文献