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1.
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
2.
P. van der Cruyssen 《BIT Numerical Mathematics》1982,22(4):533-537
Consider the (n+1)st order nonhomogeneous recursionX
k+n+1=b
k
X
k+n
+a
k
(n)
X
k+n-1+...+a
k
(1)
X
k
+X
k
.Leth be a particular solution, andf
(1),...,f
(n),g independent solutions of the associated homogeneous equation. It is supposed thatg dominatesf
(1),...,f
(n) andh. If we want to calculate a solutiony which is dominated byg, but dominatesf
(1),...,f
(n), then forward and backward recursion are numerically unstable. A stable algorithm is derived if we use results constituting a link between Generalised Continued Fractions and Recursion Relations. 相似文献
3.
For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT
α=Σ
j=1
∞
a
j
T
j
, where {a
j
} are the coefficients in the power series expansion (1-t)α=1-Σ
j=1
∞
a
j
t
j
in the open unit disk, which satisfya
j
>0 anda
j
>0 and Σ
j=1
∞
a
j
=1. The operator calculus justifies the notation(I−T)
α
:=I−T
α
(e.g., (I−T
1/2)2=I−T). A vectory∈X is called an, α-fractional coboundary for
T if there is anx∈X such that(I−T)
α
x=y, i.e.,y is a coboundary forT
α
. The fractional Poisson equation forT is the Poisson equation forT
α
. We show that if(I−T)X is not closed, then(I−T)
α
X strictly contains(I−T)X (but has the same closure).
ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ
k=1
∞
T
k
y/k
1-α converges in norm, and conclude that lim
n
‖(1/n
1-α)Σ
k=1
n
T
k
y‖=0 for suchy.
For a Dunford-Schwartz operatorT onL
1 of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T)
α
L
1 for some 0<α<1, then the one-sided Hilbert transform Σ
k=1
∞
T
k
f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T)
α
L
p
with α>1−1/p=1/q, then Σ
k=1
∞
T
k
f/k
1/p
converges a.e., and thus (1/n
1/p
) Σ
k=1
n
T
k
f converges a.e. to zero. Whenf∈(I−T)
1/q
L
p
(the case α=1/q), we prove that (1/n
1/p
(logn)1/q
)Σ
k=1
n
T
k
f converges a.e. to zero. 相似文献
4.
We consider the differential operators Ψ
k
, defined by Ψ1(y) =y and Ψ
k+1(y)=yΨ
k
y+d/dz(Ψ
k
(y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ
k
F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z
2+β
z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ
k
(F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ
k
(f
′/f) =f
(k)/f, we deduce in particular that iff andf
(k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f
′/f :f ∈F} is normal.
The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999,
and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank
Günter Frank for helpful discussions. 相似文献
5.
A criterion of normality based on a single holomorphic function 总被引:1,自引:0,他引:1
Let F be a family of functions holomorphic on a domain D ⊂ ℂ Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k −1, such that h(z) has no common zeros with any f ∈ F. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ f′(z) = h(z); and (b) f′(z) = h(z) ⇒ |f
(k)(z)| ≤ c, where c is a constant. Then F is normal on D. 相似文献
6.
Let function f(z) ≠ 0 be analytic in the unit disk and have sparse nonzero Taylor coefficients. Then the rate of decay of the function f as x → 1 − 0 depends on the rate of sparseness of its nonzero Taylor coefficients. In this paper, we consider the case f(z) = $
\sum\nolimits_{k = 0}^\infty {a_k z^{n_k } }
$
\sum\nolimits_{k = 0}^\infty {a_k z^{n_k } }
, where n
k
≥ A
0(k + 2)
p
logb(k + 2). 相似文献
7.
M. Zippin 《Israel Journal of Mathematics》1981,39(4):349-358
It is proved that there exists a positive function Φ(∈) defined for sufficiently small ∈ 〉 0 and satisfying limt→0 Φ(∈) =0 such that for any integersn 〉>0, ifQ is a projection ofl
1
n
onto ak-dimensional subspaceE with ‖|Q‖|≦1+∈ then there is an integerh〉=k(1−Φ(∈)) and anh-dimensional subspaceF ofE withd(F,l
1
h
) 〈= 1+Φ (∈) whered(X, Y) denotes the Banach-Mazur distance between the Banach spacesX andY. Moreover, there is a projectionP ofl
1
n
ontoF with ‖|P‖| ≦1+Φ(∈).
Author was partially supported by the N.S.F. Grant MCS 79-03042. 相似文献
8.
Let k be a p-adic field of odd residue characteristic and let C be a hyperelliptic (or elliptic) curve defined by the affine equation Y
2=h(X). We discuss the index of C if h(X)=ɛf(X), where ɛ is either a non-square unit or a uniformising element in O
k
and f(X) a monic, irreducible polynomial with integral coefficients. If a root θ of f generates an extension k(θ) with ramification index a power of 2, we completely determine the index of C in terms of data associated to θ. Theorem 3.11 summarizes our results and provides an algorithm to calculate the index for
such curves C.
Received: 14 July 1997 / Revised version: 16 February 1998 相似文献
9.
Explosive solutions of elliptic equations with absorption and nonlinear gradient term 总被引:2,自引:0,他引:2
Marius Ghergu Constantin Niculescu Vicenţiu Rădulescu 《Proceedings Mathematical Sciences》2002,112(3):441-451
Letf be a non-decreasing C1-function such that
andF(t)/f
2
a(t)→ 0 ast → ∞, whereF(t)=∫
0
t
f(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu|
a
=p(x)f(u) in a smooth bounded domain Ω ⊂RN, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded. 相似文献
10.
Andrey Shishkov Laurent Véron 《Calculus of Variations and Partial Differential Equations》2008,33(3):343-375
We study the limit behaviour of solutions of with initial data k
δ
0 when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r
β
, β > N(p − 1) − 2, we prove that the limit function u
∞ is an explicit very singular solution, while such a solution does not exist if β ≤ N(p − 1) − 2. If lim
inf
r→ 0
r
2 ln (1/h(r)) > 0, u
∞ has a persistent singularity at (0, t) (t ≥ 0). If , u
∞ has a pointwise singularity localized at (0, 0). 相似文献
11.
M. M. Sheremeta 《Ukrainian Mathematical Journal》1996,48(3):460-466
We prove that, for every sequence (a
k) of complex numbers satisfying the conditions Σ(1/|a
k
|) < ∞ and |a
k+1| − |a
k
| ↗ ∞ (k → ∞), there exists a continuous functionl decreasing to 0 on [0, + ∞] and such thatf(z) = Π(1 −z/|a
k
|) is an entire function of finitel-index. 相似文献
12.
Djemaïa Bensikaddour Sadek Gala Amina Lahmar-Benbernou 《Periodica Mathematica Hungarica》2008,57(1):1-22
The most important result stated in this paper is to show that the solutions of the Poisson equation −Δu = f, where f ∈ (Ḣ1(ℝ
d
) → (Ḣ−1(ℝ
d
)) is a complex-valued distribution on ℝ
d
, satisfy the regularity property D
k
u ∈ (Ḣ1 → Ḣ−1) for all k, |k| = 2. The regularity of this equation is well studied by Maz’ya and Verbitsky [12] in the case where f belongs to the class of positive Borel measures.
相似文献
13.
P. Quittner W. Reichel 《Calculus of Variations and Partial Differential Equations》2008,32(4):429-452
Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂ν
u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L
∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s|
p
) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that
p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) = s
p
then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of
∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential
equation is of the form h(x, u) with h satisfying suitable growth conditions. 相似文献
14.
Yehoram Gordon 《Israel Journal of Mathematics》1966,4(3):177-188
GivenF(z),f
1(z), ..,f
n(z) defined on a finite point setE, and givenB — the set of generalised polynomials Σ
k
=1/n
a
kfk(z) — the definition of a juxtapolynomial is extended in the following manner: for a fixedλ(0<λ≦1),f(z) εB is called a generalizedλ-weak juxtapolynomial toF(z) onE if and only if there exists nog(z) εB for whichg(z)=F(z) wheneverf(z)=F(z) and |g(z)−F(z) |<λ|f(z)−F(z)| wheneverf(z)≠F(z). The properties of suchf(z) are investigated with particular attention given to the real case.
This note is an extension of a part of the author’s M.Sc. Thesis under the supervision of Prof. B. Grünbaum to whom the author
wishes to express his sincerest appreciation. The author also wishes to thank Dr. J. Lindenstrauss for his valuable remarks
in the preparation of this paper. 相似文献
15.
Huiling Li Peter Y. H. Pang Mingxin Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,89(1):295-311
We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation
of the type −Δ
p
u = a(x)u
m
−b(x)f(u) with p > 1 and 0 < m < p−1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to
quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p = 2 to the quasilinear case with p > 1. 相似文献
16.
The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ |f
(k)(z)| < |h(z)|; (b) f
(k)(z) ≠ h(z). Then F is normal on D. 相似文献
17.
S. Staněk 《Ukrainian Mathematical Journal》2008,60(2):277-298
We present existence principles for the nonlocal boundary-value problem (φ(u(p−1)))′=g(t,u,...,u(p−1), αk(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α
k: C
p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)n(φ(u(2n−)))′=f(t,u,...,u(2n−1)), u(2k)(0)=0, αku(2k)(T)+bku(2k=1)(T)=0, 0≤k≤n−1, is given.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008. 相似文献
18.
带非局部源的退化半线性抛物方程的解的爆破性质 总被引:1,自引:0,他引:1
This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation
u
t
− (x
a
u
x
)
x
=∫
0
a
f(u)dx in (0,a) × (0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under
appropriate hypotheses, the global existence and blow-up in finite time of positve solutions are obtained. It is also proved
that the blow-up set is almost the whole domain. This differs from the local case. Furthermore, the blow-up rate is precisely
determined for the special case: f(u)=u
p
, p>1. 相似文献
19.
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
20.
K. Kubilius 《Lithuanian Mathematical Journal》1999,39(3):251-261
The uniform distance between the solution of a nonlinear equation driven by a functionh with boundedp-variation and its Milstein-type approximation is estimated by δ
n
v γ
p
(n) v γ
p
2
(n), where δ
n
=max(t
k
−t
k−1
) is the maximum step size of the approximation on the interval [0,T], γ
p
(n)=max υ
p
1/p
(h;[t
k-1,t
k
]), 1 <p < 2, and υ
p
(h;[t
k-1,t
k
]) is thep-variation of the functionh on [t
k-1,t
k]. In particular, ifh is a Lipschitz function of order α, then the uniform distance has the bound δ
n
α
for δn <1.
Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius; Vilnius Technical University, Saulétekio 11, 2054 Vilnius,
Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 317–330, July–September, 1999. 相似文献