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1.
We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ
and expansive matrix M: Λ → Λ if ρ(M
−1) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M
−1. We shall prove further that if the polynomial f(x) = c
0 + c
1
x + ··· + c
k
x
k
∈ Z[x], c
k
= 1 satisfies the condition |c
0| > 2 Σ
i=1
k
|c
i
| then there is a suitable digit set D for which (Z
k
, M, D) is a number system, where M is the companion matrix of f(x).
The research was supported by OTKA-T043657 and Bolyai Fellowship Committee. 相似文献
2.
Bao Yongguang 《分析论及其应用》1995,11(4):15-23
Let ξn −1 < ξn −2 < ξn − 2 < ... < ξ1 be the zeros of the the (n−1)-th Legendre polynomial Pn−1(x) and −1=xn<xn−1<...<x1=1, the zeros of the polynomial
. By the theory of the inverse Pal-Type interpolation, for a function f(x)∈C
[−1,1]
1
, there exists a unique polynomial Rn(x) of degree 2n−2 (if n is even) satisfying conditions Rn(f, ξk) = f (εk) (1 ⩽ k ⩽ n −1); R1
n(f,xk)=f1(xk)(1≤k≤n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation
polynomial {Rn(f, x)} (n is even) and the main result of this paper is that if f∈C
[1,1]
r
, r≥2, n≥r+2, and n is even then |R1
n(f,x)−f1(x)|=0(1)|Wn(x)|h(x)·n3−r·E2n−r−3(f(r)) holds uniformly for all x∈[−1,1], where
. 相似文献
3.
Accuracy of several multidimensional refinable distributions 总被引:3,自引:0,他引:3
Carlos Cabrelli Chritopher Heil Ursula Molter 《Journal of Fourier Analysis and Applications》2000,6(5):483-502
Compactly supported distributions f1,..., fr on ℝd are fefinable if each fi is a finite linear combination of the rescaled and translated distributions fj(Ax−k), where the translates k are taken along a lattice Γ ⊂ ∝d and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x)=Σk∈Λ ck f(Ax−k), where Λ is a finite subset of Γ, the ck are r×r matrices, and f=(f1,...,fr)T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q)<p are exactly reproduced
from linear combinations of translates of f1,...,fr along the lattice Γ. We determine the accuracy p from the matrices ck. Moreover, we determine explicitly the coefficients yα,i(k) such that xα=Σ
i=1
r
Σk∈Γyα,i(k) fi(x+k). These coefficients are multivariate polynomials yα,i(x) of degree |α| evaluated at lattice points k∈Γ. 相似文献
4.
A. K. Varma 《Israel Journal of Mathematics》1972,12(4):337-341
Letx
kn=2θk/n,k=0,1 …n−1 (n odd positive integer). LetR
n(x) be the unique trigonometric polynomial of order 2n satisfying the interpolatory conditions:R
n(xkn)=f(xkn),R
n
(j)(xkn)=0,j=1,2,4,k=0,1…,n−1. We setw
2(t,f) as the second modulus of continuity off(x). Then we prove that |R
n(x)-f(x)|=0(nw2(1/nf)). We also examine the question of lower estimate of ‖R
n-f‖. This generalizes an earlier work of the author. 相似文献
5.
Guizhen LIU 《Frontiers of Mathematics in China》2009,4(2):311-323
Let G be a digraph with vertex set V(G) and arc set E(G) and let g = (g
−, g
+) and ƒ = (ƒ
−, ƒ
+) be pairs of positive integer-valued functions defined on V(G) such that g
−(x) ⩽ ƒ
−(x) and g
+(x) ⩽ ƒ
+(x) for each x ∈ V(G). A (g, ƒ)-factor of G is a spanning subdigraph H of G such that g
−(x) ⩽ id
H
(x) ⩽ ƒ
−(x) and g
+(x) ⩽ od
H
(x) ⩽ ƒ
+(x) for each x ∈ V(H); a (g, ƒ)-factorization of G is a partition of E(G) into arc-disjoint (g, ƒ)-factors. Let
= {F
1, F
2,…, F
m} and H be a factorization and a subdigraph of G, respectively.
is called k-orthogonal to H if each F
i
, 1 ⩽ i ⩽ m, has exactly k arcs in common with H. In this paper it is proved that every (mg+m−1,mƒ−m+1)-digraph has a (g, f)-factorization k-orthogonal to any given subdigraph with km arcs if k ⩽ min{g
−(x), g
+(x)} for any x ∈ V(G) and that every (mg, mf)-digraph has a (g, f)-factorization orthogonal to any given directed m-star if 0 ⩽ g(x) ⩽ f(x) for any x ∈ V(G). The results in this paper are in some sense best possible.
相似文献
6.
U. Luther 《Integral Equations and Operator Theory》2006,54(4):541-554
We study integral operators on (−1, 1) with kernels k(x, t) which may have weak singularities in (x, t) with x ∈N1, t ∈N2, or x=t, where N1,N2 are sets of measure zero. It is shown that such operators map weighted L∞–spaces into certain weighted spaces of smooth functions, where the degree of smoothness is the higher the smoother the kernel
k(x, t) as a function in x is. The spaces of smooth function are generalizations of the Ditzian-Totik spaces which are defined in terms of the errors
of best weighted uniform approximation by algebraic polynomials. 相似文献
7.
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
8.
Pedro E. Ferreira 《Annals of the Institute of Statistical Mathematics》1982,34(1):423-431
Summary Let {p(x, θ): θ∈Θ} be a family of densities where θ=(θ1,θ2), being θ1 ∈ Θ1 ak-dimensional parameter of interest, θ2 ∈ Θ2 a nuisance parameter and Θ=Θ1×Θ2. To estimate θ1, vector estimating equations g(x,θ1)=(g1(x,θ1),...,gk(x,θ1))=0 are considered. The standardized form of g(x,θ1) is defined as gs=(Eθ(∂g/∂θ′1))−1g. Then, within the classG
1 of unbiased equations (i.e. satisfying Eθ(g)=0 (θ∈Θ)), an equationg
*=0 is said to be optimum if the covariance matrices ofg
s andg
s
*
are such that
is non-negative definite for allg∈
G
1 and θ∈Θ. Sufficient conditions for optimality are discussed and, in particular, conditions for the optimality of the maximum
conditional likelihood equation are analyzed. Special attention is given to non-regular cases. In addition, measures of the
information about θ1 contained in an estimating equation are presented and a Rao-Blackwell theorem is given.
CIENES 相似文献
9.
Suppose that f(x) = (f
1(x),...,f
r
(x))
T
, x∈R
d
is a vector-valued function satisfying the refinement equation f(x) = ∑Λ
c
κ
f(2x−κ) with finite set Λ of Z
d
and some r×r matricex c
κ. The requirements for f to have accuracy p are given in terms of the symbol function m(ξ).
Supported by NSFC 相似文献
10.
If ψ ∈ L2(R), Λ is a discrete subset of the affine groupA =R
+ ×R, and w: Λ →R
+ is a weight function, then the weighted wavelet system generated by ψ, Λ, and w is
. In this article we define lower and upper weighted densities D
w
−
(Λ) and D
w
+
(Λ) of Λ with respect to the geometry of the affine group, and prove that there exist necessary conditions on a weighted wavelet
system in order that it possesses frame bounds. Specifically, we prove that if W(ψ, Λ, w) possesses an upper frame bound,
then the upper weighted density is finite. Furthermore, for the unweighted case w = 1, we prove that if W(ψ, Λ, 1) possesses
a lower frame bound and D
w
+
(Λ−1) < ∞, then the lower density is strictly positive. We apply these results to oversampled affine systems (which include the
classical affine and the quasi-affine systems as special cases), to co-affine wavelet systems, and to systems consisting only
of dilations, obtaining some new results relating density to the frame properties of these systems. 相似文献
11.
O. V. Lopotko 《Ukrainian Mathematical Journal》2011,63(6):981-992
We obtain an integral representation of even functions of two variables for which the kernel [k
1(x + y) + k
2(x − y)], x, y ∈ R
2, is positive definite. 相似文献
12.
Thierry De Pauw 《Journal of Geometric Analysis》2002,12(1):29-61
A concentrated (ξ, m) almost monotone measure inR
n
is a Radon measure Φ satisfying the two following conditions: (1) Θ
m
(Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R
n
the ratioexp [ξ(r)]r−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim
r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R
n
and a λ-Lipschitz function f from x+W into x+W‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ. 相似文献
13.
Katalin Marton 《Probability Theory and Related Fields》1998,110(3):427-439
Summary. Let X={X
i
}
i
=−∞
∞ be a stationary random process with a countable alphabet and distribution q. Let q
∞(·|x
−
k
0) denote the conditional distribution of X
∞=(X
1,X
2,…,X
n
,…) given the k-length past:
Write d(1,x
1)=0 if 1=x
1, and d(1,x
1)=1 otherwise. We say that the process X admits a joining with finite distance u if for any two past sequences −
k
0=(−
k
+1,…,0) and x
−
k
0=(x
−
k
+1,…,x
0), there is a joining of q
∞(·|−
k
0) and q
∞(·|x
−
k
0), say dist(0
∞,X
0
∞|−
k
0,x
−
k
0), such that
The main result of this paper is the following inequality for processes that admit a joining with finite distance:
Received: 6 May 1996 / In revised form: 29 September 1997 相似文献
14.
MiaoLI QuanZHENG 《数学学报(英文版)》2004,20(5):821-828
Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups. 相似文献
15.
We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume
that for any submodular function f: ?→R on a distributive lattice ?⊆2
V
with ?,V∈? and f(?)=0 and for any vector x∈R
V
where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|2) times, the descent method finds a sequence of subsets Z
1,Z
2,···,Z
k
of V such that f(Z
1)>f(Z
2)>···>f(Z
k
)=min{f(Y) | Y∈?}, where k is O(|V|2). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient
membership algorithms.
Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001 相似文献
16.
Wojciech Bartoszek 《Israel Journal of Mathematics》1995,90(1-3):115-123
LetK be a compact group of linear operators of thed-dimensional spaceR
d
andG
K,d
denote the semidirect productK byR
d
. It is shown that if an adapted probability measureμ onG
K,d
is not scattered (i.e. for some compactF we havex
0 ∈ R
d
(gF)>0), then there exists a nonzero vectorx
0 ∈R
d
such thatk
1(x
0)=k
2(x
0) holds for all (k
1,x
1) and (k
2,x
2) belonging to the topological supportS(μ) of the measureμ. As a result we obtain that every adapted and strictly aperiodic probability measure on the group of all rigid motions of
thed-dimensional Euclidian space is scattered.
I thank the Foundation for Research Development for financial support. 相似文献
17.
R. Nair 《Israel Journal of Mathematics》2009,171(1):197-219
We consider a system of “generalised linear forms” defined at a point x = (x
(i, j)) in a subset of R
d
by
for k ≥ 1. Here d = d
1 + ⋯ + d
l
and for each pair of integers (i, j) ∈ D, where D = {(i, j): 1 ≤ i ≤ l, 1 ≤ j ≤ d
i
} the sequence of functions (g
(i, j), k
(x))
k=1∞ are differentiable on an interval X
ij
contained in R. We study the distribution of the sequence on the l-torus defined by the fractional parts X
k
(x) = ({ L
1(x)(k)}, ..., {L
l
(x)(k)}) ∈ T
l
, for typical x in the Cartesian product . More precisely, let R = I
1 × ⋯ × I
l
be a rectangle in T
l
and for each N ≥ 1 define a pair correlation function
and a discrepancy , where the supremum is over all rectangles in T
l
and χ
R
is the characteristic function of the set R. We give conditions on (g
(i, j), k
(x))
k=1∞ to ensure that given ε > 0, for almost every x ∈ T
l
we have Δ
N
(x) = o(N(log N)
l+∈). Under related conditions on(g
(i, j), k
(x))
k =1∞ we calculate for appropriate β ∈ (0, 1) the Hausdorff dimension of the set {x : lim sup
N→∞
N
β Δ
N
(x > 0)}. Our results complement those of Rudnick and Sarnak and Berkes, Philipp, and Tichy in one dimension and M. Pollicott
and the author in higher dimensions. 相似文献
18.
Consider the retarded difference equationx
n
−x
n−1
=F(−f(x
n
)+g(x
n−k
)), wherek is a positive integer,F,f,g:R→R are continuous,F andf are increasing onR, anduF(u)>0 for allu≠0. We show that whenf(y)≥g(y) (resp. f(y)≤g(y)) fory∈R, every solution of (*) tends to either a constant or −∞ (resp. ∞) asn→∞. Furthermore, iff(y)≡g(y) fory∈R, then every solution of (*) tends to a constant asn→∞.
Project supported by NNSF (19601016) of China and NSF (97-37-42) of Hunan 相似文献
19.
Vincenzo De Filippis 《Proceedings Mathematical Sciences》2010,120(3):285-297
Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, H and G non-zero generalized derivations of R. Suppose that there exists an integer n ≥ 1 such that (H(u)u − uG(u))
n
= 0, for all u ∈ L, then one of the following holds: (1) there exists c ∈ U such that H(x) = xc, G(x) = cx; (2) R satisfies the standard identity s
4 and char (R) = 2; (3) R satisfies s
4 and there exist a, b, c ∈ U, such that H(x) = ax+xc, G(x) = cx+xb and (a − b)
n
= 0. 相似文献
20.
Raphael Yuster 《Order》2003,20(2):121-133
Let TT
k
denote the transitive tournament on k vertices. Let TT(h,k) denote the graph obtained from TT
k
by replacing each vertex with an independent set of size h≥1. The following result is proved: Let c
2=1/2, c
3=5/6 and c
k
=1−2−k−log k
for k≥4. For every ∈>0 there exists N=N(∈,h,k) such that for every undirected graph G with n>N vertices and with δ(G)≥c
k
n, every orientation of G contains vertex disjoint copies of TT(h,k) that cover all but at most ∈n vertices. In the cases k=2 and k=3 the result is asymptotically tight. For k≥4, c
k
cannot be improved to less than 1−2−0.5k(1+o(1)).
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献