共查询到20条相似文献,搜索用时 15 毫秒
1.
Stanis
aw Lewanowicz 《Journal of Computational and Applied Mathematics》2003,150(2):193-327
Let {pk(x; q)} be any system of the q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the q-difference equation Dq(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {pk(1)(x;q)} be associated polynomials of the polynomials {pk(x; q)}. Explicit forms of the coefficients bn,k and cn,k in the expansions are given in terms of basic hypergeometric functions. Here k(x) equals xk if σ+(0)=0, or (x;q)k if σ+(1)=0, where σ+(x)σ(x)+(q−1)xτ(x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively.Writing the second-order nonhomogeneous q-difference equation satisfied by pn−1(1)(x;q) in a special form, recurrence relations (in k) for bn,k and cn,k are obtained in terms of σ and τ. 相似文献
2.
Sufficient conditions are found for the weak convergence of a weighted empirical process {(νn(C)/q(P(C))) 1 [P(C) λn]: C
}, indexed by a class
of sets and weighted by a function q of the size of each set. We find those functions q which allow weak convergence to a sample-continuous Gaussian process, and, given q, determine the fastest rate at which one may allow λn → 0. 相似文献
3.
The zeros of the Meixner polynomialmn(x; β, c) are real, distinct, and lie in (0, ∞). Letαn, sdenote thesth zero ofmn(nα; β, c), counted from the right; and letαn, sdenote thesth zero ofmn(nα; β, c), counted from the left. For each fixeds, asymptotic formulas are obtained for bothαn, sandαn, s, asn→∞. 相似文献
4.
M. G. Pleshakov 《Journal of Approximation Theory》1999,99(2):519
Let 2s points yi=−πy2s<…<y1<π be given. Using these points, we define the points yi for all integer indices i by the equality yi=yi+2s+2π. We shall write fΔ(1)(Y) if f is a 2π-periodic continuous function and f does not decrease on [yi, yi−1], if i is odd; and f does not increase on [yi, yi−1], if i is even. In this article the following Theorem 1—the comonotone analogue of Jackson's inequality—is proved.
1. If fΔ(1)(Y), then for each nonnegative integer n there is a trigonometric polynomial τn(x) of order n such that τnΔ(1)(Y), and |f(x)−πn(x)|c(s) ω(f; 1/(n+1)), x
, where ω(f; t) is the modulus of continuity of f, c(s)=const. Depending only on s. 相似文献
5.
Let I be a finite or infinite interval and dμ a measure on I. Assume that the weight function w(x)>0, w′(x) exists, and the function w′(x)/w(x) is non-increasing on I. Denote by ℓk's the fundamental polynomials of Lagrange interpolation on a set of nodes x1<x2<<xn in I. The weighted Lebesgue function type sum for 1≤i<j≤n and s≥1 is defined byIn this paper the exact lower bounds of Sn(x) on a “big set” of I and are obtained. Some applications are also given. 相似文献
6.
Kamenev-type oscillation criteria for delay difference equations 总被引:2,自引:0,他引:2
程金发 《数学物理学报(B辑英文版)》2007,27(3):574-580
Some oscillation criteria are established by Raccati transformation techniques for the following second-order nonlinear neutral difference equation Δ(pn(Δ(xn cnxn-τ))γ) qnxβn-σ=0,n=0,1,2…which extend and include several oscillation criteria in [11], and also correct a theorem and its proof in [10]. 相似文献
7.
Let Bn( f,q;x), n=1,2,… be q-Bernstein polynomials of a function f : [0,1]→C. The polynomials Bn( f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: |z|<q+} the rate of convergence of {Bn( f,q;x)} to f(x) in the norm of C[0,1] has the order q−n (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {Bnjn( f,q;x)}, where both n→∞ and jn→∞, are studied. It is shown that for q(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of jn→∞. 相似文献
8.
Zhi-Wei Sun 《Journal of Algebra》2001,240(2):223
A residue class a + n
with weight λ is denoted by λ, a, n. For a finite system
= {λs, as, ns}ks = 1 of such triples, the periodic map w
(x) = ∑ns|x − as λs is called the covering map of
. Some interesting identities for those
with a fixed covering map have been known; in this paper we mainly determine all those functions f : Ω →
such that ∑ks = 1 λsf(as + ns
) depends only on w
where Ω denotes the family of all residue classes. We also study algebraic structures related to such maps f, and periods of arithmetical functions ψ(x) = ∑ks = 1 λse2πiasx/ns and ω(x) = |{1 ≤ s ≤ k : (x + as, ns) = 1}|. 相似文献
9.
David Paget 《Journal of Approximation Theory》1988,54(3)
Let f ε Cn+1[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials qk associated with a distribution dα on [−1, 1]. It is shown that if qn+1/qn max(qn+1(1)/qn(1), −qn+1(−1)/qn(−1)), then f − H[f] fn + 1 · qn+1/qn + 1(n + 1), where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)α (1 + t)β dt, α, β > −1, the condition on qn+1/qn is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2. 相似文献
10.
In this paper, we discuss properties of the ω,q-Bernstein polynomials introduced by S. Lewanowicz and P. Woźny in [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63–78], where fC[0,1], ω,q>0, ω≠1,q−1,…,q−n+1. When ω=0, we recover the q-Bernstein polynomials introduced by [G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518]; when q=1, we recover the classical Bernstein polynomials. We compute the second moment of , and demonstrate that if f is convex and ω,q(0,1) or (1,∞), then are monotonically decreasing in n for all x[0,1]. We prove that for ω(0,1), qn(0,1], the sequence converges to f uniformly on [0,1] for each fC[0,1] if and only if limn→∞qn=1. For fixed ω,q(0,1), we prove that the sequence converges for each fC[0,1] and obtain the estimates for the rate of convergence of by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions. 相似文献
11.
Let I be a finite interval, r
and ρ(t)=dist{t, ∂I}, tI. Denote by Δs+Wrp, α, 0α<∞, the class of functions x on I with the seminorm x(r)ραLp1 for which Δsτx, τ>0, is nonnegative on I. We obtain two-sided estimates of the Kolmogorov widths dn(Δs+Wrp, α)Lq and of the linear widths dn(Δs+Wrp, α)linLq, s=0, 1, …, r+1. 相似文献
12.
The main purpose of this article is to establish nearly optimal results concerning the rate of almost everywhere convergence of the Gauss–Weierstrass, Abel–Poisson, and Bochner–Riesz means of the one-dimensional Fourier integral. A typical result for these means is the following: If the function f belongs to the Besov spaceBsp, p, 1<p<∞, 0<s<1, thenTmtf (x)−f(x)=ox(ts) a.e. ast→0+. 相似文献
13.
In the setting of doubling metric measure spaces with a 1-Poincaré inequality, we show that sets of Orlicz Φ-capacity zero have generalized Hausdorff h-measure zero provided thatwhere Θ−1 is the inverse of the function Θ(t)=Φ(t)/t, and s is the “upper dimension” of the metric measure space. This condition is a generalization of a well known condition in Rn. For spaces satisfying the weaker q-Poincaré inequality, we obtain a similar but slightly more restrictive condition. Several examples are also provided. 相似文献
14.
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
15.
We give a direct formulation of the invariant polynomials μGq(n)(, Δi,;, xi,i + 1,) characterizing U(n) tensor operators p, q, …, q, 0, …, 0 in terms of the symmetric functions Sλ known as Schur functions. To this end, we show after the change of variables Δi = γi − δi and xi, i + 1 = δi − δi + 1 thatμGq(n)(,Δi;, xi, i + 1,) becomes an integral linear combination of products of Schur functions Sα(, γi,) · Sβ(, δi,) in the variables {γ1,…, γn} and {δ1,…, δn}, respectively. That is, we give a direct proof that μGq(n)(,Δi,;, xi, i + 1,) is a bisymmetric polynomial with integer coefficients in the variables {γ1,…, γn} and {δ1,…, δn}. By making further use of basic properties of Schur functions such as the Littlewood-Richardson rule, we prove several remarkable new symmetries for the yet more general bisymmetric polynomials μmGq(n)(γ1,…, γn; δ1,…, δm). These new symmetries enable us to give an explicit formula for both μmG1(n)(γ; δ) and 1G2(n)(γ; δ). In addition, we describe both algebraic and numerical integration methods for deriving general polynomial formulas for μmGq(n)(γ; δ). 相似文献
16.
Primitive polynomial with three coefficients prescribed 总被引:1,自引:1,他引:0
The authors proved in Fan and Han (Finite Field Appl., in press) that, for any given (a1,a2,a3)Fq3, there exists a primitive polynomial f(x)=xn−σ1xn−1++(−1)nσn over Fq of degree n with the first three coefficients σ1,σ2,σ3 prescribed as a1,a2,a3 when n8. But the methods in Fan and Han (in press) are not effective for the case of n=7. Mills (Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl., in press) resolves the n=7 case for finite fields of characteristic at least 5. In this paper, we deal with the remaining cases and prove that there exists a primitive polynomial of degree 7 over Fq with the first three coefficient prescribed where the characteristic of Fq is 2 or 3. 相似文献
17.
Michael Voit 《Journal of multivariate analysis》1996,59(2):230-248
Forn2, let (μxτ, n)τ0be the distributions of the Brownian motion on the unit sphereSn
n+1starting in some pointxSn. This paper supplements results of Saloff-Coste concerning the rate of convergence ofμxτ, nto the uniform distributionUnonSnforτ→∞ depending on the dimensionn. We show that,[formula]forτn:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measuresμxτ, nby measures which are known explicitly via Poisson kernels onSn, and which tend, after suitable projections and dilatations, to normal distributions on
forn→∞. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups. 相似文献
18.
Let
be the classical middle-third Cantor set and let μ
be the Cantor measure. Set s = log 2/log 3. We will determine by an explicit formula for every point x
the upper and lower s-densities Θ*s(μ
, x), Θ*s(μ
, x) of the Cantor measure at the point x, in terms of the 3-adic expansion of x. We show that there exists a countable set F
such that 9(Θ*s(μ
, x))− 1/s + (Θ*s(μ
, x))− 1/s = 16 holds for x
\F. Furthermore, for μC almost all x, Θ*s(μ
, X) − 2 · 4− s and Θ*s(μ
, x) = 4− s. As an application, we will show that the s-dimensional packing measure of the middle-third Cantor set
is 4s. 相似文献
19.
Leszek Plaskota 《Journal of Approximation Theory》1998,93(3):501-515
We consider the average caseL∞-approximation of functions fromCr([0, 1]) with respect to ther-fold Wiener measure. An approximation is based onnfunction evaluations in the presence of Gaussian noise with varianceσ2>0. We show that the n th minimal average error is of ordern−(2r+1)/(4r+4) ln1/2 n, and that it can be attained either by the piecewise polynomial approximation using repetitive observations, or by the smoothing spline approximation using non-repetitive observations. This completes the already known results forLq-approximation withq<∞ andσ0, and forL∞-approximation withσ=0. 相似文献
20.
Daniel S. Moak 《Aequationes Mathematicae》1980,21(1):179-191
F. H. Jackson defined aq analogue of the gamma function which extends theq-factorial (n!)
q
=1(1+q)(1+q+q
2)...(1+q+q
2+...+q
n–1) to positivex. Askey studied this function and obtained analogues of most of the classical facts about the gamma function, for 0<q<1. He proved an analogue of the Bohr-Mollerup theorem, which states that a logarithmically convex function satisfyingf(1)=1 andf(x+1)=[(q
x
–1)/(q–1)]f(x) is in fact theq-gamma function He also studied the behavior of
q
asq changes and showed that asq1–, theq-gamma function becomes the ordinary gamma function forx>0.I proved many of these results forq>1. The current paper contains a study of the behavior of
q
(x) forx<0 and allq>0. In addition to some basic properties of
q
, we will study the behavior of the sequence {x
n
(q)} of critical points asn orq changes. 相似文献