Let t=min{a1,a2,…,am−1} and b=a1+a2++am−1t. 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.
Weierstrass'' Theorem in Weighted Sobolev Spaces     
Jos M. Rodríguez 《Journal of Approximation Theory》2001,108(2):119
We characterize the set of functions which can be approximated by polynomials with the following norm

for a big class of weights w0w1, …, wk  相似文献   

7.
On the L convergence of Lagrange interpolating entire functions of exponential type     
Q. I. Rahman  P. Vrtesi 《Journal of Approximation Theory》1992,69(3)
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 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that limn → ∞0¦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.
Quadrature formulae connected to σ-orthogonal polynomials     
Gradimir V. Milovanovi  Miodrag M. Spalevi 《Journal of Computational and Applied Mathematics》2002,140(1-2)
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 μ0>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes
where σ=σn=(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+2n−1 if and only if
The proof of the uniqueness of the extremal nodes τ12,…,τn 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 τν, 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.
Asymptotic Behaviour of Best lp-Approximations from Affi ne Subspaces     
J. M. Quesada  J. Martinez-Moreno  J. Navas 《Journal of Approximation Theory》2002,118(2):275-289
In this paper we consider the problem of best approximation in ℓpn, 1<p∞. If hp, 1<p<∞, denotes the best ℓp-approximation of the element h n from a proper affine subspace K of n, hK, then limp→∞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 αj n, 1jr, such that

, with γp(r) n and γp(r)= (pr−1).  相似文献   

10.
Curves of positive solutions of boundary value problems on time-scales     
Fordyce A. Davidson  Bryan P. Rynne   《Journal of Mathematical Analysis and Applications》2004,300(2):491-504
Let TR be a time-scale, with a=infT, b=supT. We consider the nonlinear boundary value problem
(2)
(4)
u(a)=u(b)=0,
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 0λ<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.
Joint Distributions of the Numbers of Visits for Finite-State Markov Chains     
Wolfgang Stadje 《Journal of multivariate analysis》1999,70(2):157
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 firstNsteps or before theNth visit tor. From the explicit expressions for the corresponding generating functions we obtain the limiting multivariate distributions asN→∞ when staterbecomes asymptotically absorbing and forj=1, …, r−1 the probability of a transition fromrtojis of order 1/N.  相似文献   

12.
On existence of singular solutions     
Miroslav Bartu ek 《Journal of Mathematical Analysis and Applications》2003,280(2):232-240
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.
Characterization of generalized convex functions by their best approximation in sign-monotone norms     
E. Kimchi 《Journal of Approximation Theory》1978,24(4):350-360
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.
Estimates of Freud-Christoffel functions for some weights with the whole real line as support     
D. S. Lubinsky 《Journal of Approximation Theory》1985,44(4):343-379
Upper and lower bounds for generalized Christoffel functions, called Freud-Christoffel functions, are obtained. These have the form λn,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 λn,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(− expx¦)), > 0.  相似文献   

15.
Markov-Type Inequalities for Products of Müntz Polynomials     
Tams Erdlyi 《Journal of Approximation Theory》2001,112(2):171
Let Λ(λj)j=0 be a sequence of distinct real numbers. The span of {xλ0xλ1, …, xλn} over is denoted by Mn(Λ)span{xλ0xλ1, …, 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. Let

Then

18(n+m+1)(λnm).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 [ab](0, ∞).  相似文献   

16.
On Positive and Copositive Polynomial and Spline Approximation inLp[−1, 1], 0<p<∞     
Y. K. Hu  K. A. Kopotun  X. M. Yu 《Journal of Approximation Theory》1996,86(3):320-334
For a functionfLp[−1, 1], 0<p<∞, with finitely many sign changes, we construct a sequence of polynomialsPnΠnwhich are copositive withfand such that fPnp(f, (n+1)−1)p, whereω(ft)pdenotes the Ditzian–Totik modulus of continuity inLpmetric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, thenωcannot be replaced byω2if 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.
Asymptotics for Lp extremal polynomials on the unit circle     
X. Li  K. Pan 《Journal of Approximation Theory》1991,67(3)
Let p > 1, and dμ a positive finite Borel measure on the unit circle Γ: = {z ε C: ¦z¦ = 1}. Define the monic polynomial φn, p(z)=zn+…εPn >(the set of polynomials of degree at most n) satisfying
. Under certain conditions on dμ, the asymptotics of φn, p(z) for z outside, on, or inside Γ are obtained (cf. Theorems 2.2 and 2.4). Zero distributions of φn, p are also discussed (cf. Theorems 3.1 and 3.2).  相似文献   

18.
Remarks Concerning Completeness of Translates in Function Spaces     
Nikolai Nikolski 《Journal of Approximation Theory》1999,98(2):61
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(xk), 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.
The Existence of Positive Solutions to Neutral Differential Equations     
Ravi P. Agarwal  X.H. Tang  Z.C. Wang 《Journal of Mathematical Analysis and Applications》1999,240(2):1
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.
The asymptotic behavior of nonoscillatory solutions of nonlinear neutral type difference equations     
E. Thandapani  R. Arul  P.S. Raja 《Mathematical and Computer Modelling》2004,39(13):1457-1465
In this paper, the authors study the asymptotic behavior of solutions of second-order neutral type difference equations of the form
Δ2(yn+pynk)+f(n,yn)=0,n
, n ε, and
Δ2(yn+pynk)+f(n,yn,Δyn)=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   

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1.
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−axg(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 ex 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
a1x1+a2x2++am−1xm−1=xm.
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