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2.
For bounded Lipschitz domains D in it is known that if 1< p<∞, then for all β[0, β0), where β0= p−1>0, there is a constant c<∞ with for all . We show that if D is merely assumed to be a bounded domain in that satisfies a Whitney cube-counting condition with exponent λ and has plump complement, then the same inequality holds with β0 now taken to be . Further, we extend the known results (see [H. Brezis, M. Marcus, Hardy's inequalities revisited, Dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997–1998) 217–237; M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, A geometrical version of Hardy's inequality, J. Funct. Anal. 189 (2002) 537–548; J. Tidblom, A geometrical version of Hardy's inequality for W1,p( Ω), Proc. Amer. Math. Soc. 132 (2004) 2265–2271]) concerning the improved Hardy inequality c= c( n, p), by showing that the class of domains for which the inequality holds is larger than that of all bounded convex domains. 相似文献
3.
We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium N*=1/( a+∑ i=0mbi) of the following differential equation with piecewise constant arguments: where r( t) is a nonnegative continuous function on [0,+∞), r( t)0, ∑ i=0mbi>0, bi0, i=0,1,2,…, m, and a+∑ i=0mbi>0. These new conditions depend on a, b0 and ∑ i=1mbi, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case m=0 and r( t)≡ r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms: where r( t) is a nonnegative continuous function on [0,+∞), r( t)0, 1− ax− g( x, x,…, x)=0 has a unique solution x*>0 and g( x0, x1,…, xm) C1[(0,+∞)×(0,+∞)××(0,+∞)]. 相似文献
4.
The best possible constant An in an inequality of Markov type , where · [0, ∞) denotes the sup-norm on the half real line [0, ∞) and pn is an arbitrary polynomial of degree at most n, is determined in terms of the weighted Chebyshev polynomials associated with the Laguerre weight e−x on [0, ∞). 相似文献
5.
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
6.
We characterize the set of functions which can be approximated by polynomials with the following norm
for a big class of weights
w0,
w1, …,
wk 相似文献
7.
Let f:
be a continuous, 2π-periodic function and for each
n ε
let
tn(
f; ·) denote the trigonometric polynomial of degree
n interpolating f in the points 2
kπ/(2
n + 1) (
k = 0, ±1, …, ±
n). It was shown by J. Marcinkiewicz that lim
n → ∞ ∝
02π¦
f(θ) −
tn(
f θ)¦
p dθ = 0 for every
p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points
kπ/τ (
k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.
相似文献
8.
Let dλ(
t) be a given nonnegative measure on the real line
, with compact or infinite support, for which all moments
exist and are finite, and μ
0>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes
where σ=σ
n=(
s1,
s2,…,
sn) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness
dmax=2∑
ν=1nsν+2
n−1 if and only if
The proof of the uniqueness of the extremal nodes τ
1,τ
2,…,τ
n was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1–15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term
R(
f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τ
ν, ν=1,2,…,
n, which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.
相似文献
9.
In this paper we consider the problem of best approximation in ℓ
pn, 1<
p∞. If
hp, 1<
p<∞, denotes the best ℓ
p-approximation of the element
h
n from a proper affine subspace
K of
n,
hK, then lim
p→∞hp=
h∞*, where
h∞* is a best uniform approximation of
h from
K, the so-called strict uniform approximation. Our aim is to prove that for all
r
there are α
j
n, 1
jr, such that
, with γ
p(r)
n and γ
p(r)=
(
p−r−1).
相似文献
10.
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
11.
For a discrete-time Markov chain with finite state space {1, …,
r} we consider the joint distribution of the numbers of visits in states 1, …,
r−1 during the first
Nsteps or before the
Nth visit to
r. From the explicit expressions for the corresponding generating functions we obtain the limiting multivariate distributions as
N→∞ when state
rbecomes asymptotically absorbing and for
j=1, …,
r−1 the probability of a transition from
rto
jis of order 1/
N.
相似文献
12.
In the paper sufficient conditions are given under which the differential equation
y(n)=
f(
t,
y,…,
y(n−2))
g(
y(n−1)) has a singular solution
y :[
T,τ)→
R, τ<∞ fulfilling
相似文献
13.
Let {
u0,
u1,…
un − 1} and {
u0,
u1,…,
un} be Tchebycheff-systems of continuous functions on [
a,
b] and let
f ε
C[
a,
b] be generalized convex with respect to {
u0,
u1,…,
un − 1}. In a series of papers ([1], [2], [3])
D. Amir and Z. Ziegler discuss some properties of elements of best approximation to f from the linear spans of {
u0,
u1,…,
un − 1} and {
u0,
u1,…,
un} in the
Lp-norms, 1
p ∞, and show (under different conditions for different values of
p) that these properties, when valid for all subintervals of [
a,
b], can characterize generalized convex functions. Their methods of proof rely on characterizations of elements of best approximation in the
Lp-norms, specific for each value of
p. This work extends the above results to approximation in a wider class of norms, called “sign-monotone,” [6], which can be defined by the property: ¦
f(
x)¦ ¦
g(
x)¦,
f(
x)
g(
x) 0,
a x b, imply
f g . For sign-monotone norms in general, there is neither uniqueness of an element of best approximation, nor theorems characterizing it. Nevertheless, it is possible to derive many common properties of best approximants to generalized convex functions in these norms, by means of the necessary condition proved in [6]. For {
u0,
u1,…,
un} an Extended-Complete Tchebycheff-system and
f ε
C(n)[
a,
b] it is shown that the validity of any of these properties on all subintervals of [
a,
b], implies that f is generalized convex. In the special case of f monotone with respect to a positive function
u0(
x), a converse theorem is proved under less restrictive assumptions.
相似文献
14.
Upper and lower bounds for generalized Christoffel functions, called Freud-Christoffel functions, are obtained. These have the form λ
n,p(
W,
j,
x) =
infPWLp(R)/|
P(j)(
X)| where the infimum is taken over all polynomials
P(
x) of degree at most
n − 1. The upper and lower bounds for λ
n,p(
W,
j,
x) are obtained for all 0 <
p ∞ and
J = 0, 1, 2, 3,… for weights
W(
x) = exp(−
Q(
x)), where, among other things,
Q(
x) is bounded in [−
A,
A], and
Q″ is continuous in
β(−A, A) for some
A > 0. For
p = ∞, the lower bounds give a simple proof of local and global Markov-Bernstein inequalities. For
p = 2, the results remove some restrictions on
Q in Freud's work. The weights considered include
W(
x) = exp(− ¦
x¦
α/2), α > 0, and
W(
x) = exp(−
exp(¦
x¦
)), > 0.
相似文献
15.
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
16.
For a function
fLp[−1, 1], 0<
p<∞, with finitely many sign changes, we construct a sequence of polynomials
PnΠnwhich are copositive with
fand such that
f−
PnpCω(
f, (
n+1)
−1)
p, where
ω(
f,
t)
pdenotes the Ditzian–Totik modulus of continuity in
Lpmetric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, then
ωcannot be replaced by
ω2if 1<
p<∞. In fact, we show that even for positive approximation and all 0<
p<∞ the same conclusion is true. Also, some results for (co)positive spline approximation, exact in the same sense, are obtained.
相似文献
17.
Let
p > 1, and
dμ a positive finite Borel measure on the unit circle Γ: = {
z ε C: ¦
z¦ = 1}. Define the monic polynomial φ
n, p(
z)=
zn+…ε
Pn >(the set of polynomials of degree at most
n) satisfying
. Under certain conditions on
dμ, the asymptotics of φ
n, p(
z) for
z outside, on, or inside Γ are obtained (cf. Theorems 2.2 and 2.4). Zero distributions of φ
n, p are also discussed (cf. Theorems 3.1 and 3.2).
相似文献
18.
This note explains how to translate the author's old result on cyclic vectors of the multiple shift operator into the language of completeness theorems for integer translates. This translation, together with those results, turns out to be a source for many completeness theorems. In particular, there follows the existence of functions
f whose positive integer translates
f(
x−
k), where
k
+ are complete in the spaces
Cl0(
),
Lp(
),
Wlp(
), 2<
p<∞,
l=0, 1, …, as well as in their weighted and/or vector-valued analogues.
相似文献
19.
In this paper, we shall consider a class of neutral differential equations of the form
where τ (0, ∞), σ [0, ∞),
Q(
t) C([
t0, ∞),
R + ),
r(
t)
C([
t0, ∞), (0, ∞)) with
r(
t) nondecreasing on [
t0 − τ, ∞). We shall show that all positive solutions of ( * ) can be classified into four types, A, B, C, and D, and we shall obtain sufficient and necessary conditions for the existence of A-type, B-type, and D-type positive solutions of ( * ), respectively. A sufficient condition for the existence of C-type positive solutions of ( * ) is also given. Finally, we shall offer a sharp oscillation result for all solutions of ( * ). Our results generalize and improve those established in B. Yang and B. G. Zhang (
Funkcial. Ekvac.39 (1996), 347–362).
相似文献
20.
In this paper, the authors study the asymptotic behavior of solutions of second-order neutral type difference equations of the form
Δ2(yn+pyn−k)+f(n,yn−ℓ)=0,n
,
n ε, and
Δ2(yn+pyn−k)+f(n,yn−ℓ,Δyn−ℓ)=0,n
,
n ε using some difference inequalities. We establish conditions under which all nonoscillatory solutions are asymptotic to
an + b as
n → ∞ with a and
b ε .
相似文献