共查询到20条相似文献,搜索用时 78 毫秒
1.
Francisco Luquin 《Mathematische Nachrichten》1996,181(1):261-275
Some results of A.O. Gelfond concerning monic polynomials of least deviation (uniform) from zero together with their derivatives are extended to certain Lp - norms and to several variables. As an application we extend a recent result of Xiaoming Huang to functions f∈C(n)n)([?1, 1]) which satisfy f(n)(x)≥1. 相似文献
2.
Let Hn be the nth Hermite polynomial, i.e., the nth orthogonal on
polynomial with respect to the weight w(x)=exp(−x2). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f||Hn| at the zeros of Hn+1, then for k=1,…,n we have f(k)Hn(k), where · is the
norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the
norm, and estimates for the expansion coefficients in the basis of Hermite polynomials. 相似文献
3.
We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x, y ∈ ℝn whose elementary symmetric polynomials satisfy ek(x) ≤ ek(y) (for 1 ≤ k < n) and en(x) = en(y) , the inequality ∑i(log xi)2 ≤ ∑i(log yi)2 holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function f : M ⊆ ℂn → ℝ with f(z) = ∑i(log zi)2 has nonnegative partial derivatives with respect to the elementary symmetric polynomials of z. We conclude by providing applications and wider connections of the SSLI. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
4.
I.A. Shevchuk 《Journal of Approximation Theory》1996,86(3):270-277
For each non-negative integerna functionf=fnis constructed such thatfhas a continuous and non-negative derivativef′ onI :=[−1, 1] and[formula]whereEn(f′) (E(1)n+1(f)) is the value of the best uniform approximation onIof the functionf′ (f) by arbitrary (monotone onI) algebraic polynomials of degree ?n(n+1). 相似文献
5.
The convergence properties of q-Bernstein polynomials are investigated. When q1 is fixed the generalized Bernstein polynomials
nf of f, a one parameter family of Bernstein polynomials, converge to f as n→∞ if f is a polynomial. It is proved that, if the parameter 0<q<1 is fixed, then
nf→f if and only if f is linear. The iterates of
nf are also considered. It is shown that
nMf converges to the linear interpolating polynomial for f at the endpoints of [0,1], for any fixed q>0, as the number of iterates M→∞. Moreover, the iterates of the Boolean sum of
nf converge to the interpolating polynomial for f at n+1 geometrically spaced nodes on [0,1]. 相似文献
6.
Paul-Jean Cahen 《代数通讯》2013,41(6):2231-2239
A one-dimensional, Noetherian, local domain D with maximal ideal 𝔪 and finite residue field was known to be an almost strong Skolem ring if analytically irreducible. It was unknown whether this condition is necessary. We show that it is at least necessary for D to be unibranched. After introducing a general notion of equalizing ideal, we show that, for k large enough, the ideals of the form 𝔐 k, a = {f ∈ Int(D) | f(a) ∈ 𝔪 k }, for a ∈ D, are distinct. This allows to show that the maximal ideals 𝔐 a = {f ∈ Int(D) | f(a) ∈ 𝔪}, although not necessarily distinct, are never finitely generated. 相似文献
7.
In this paper we define the relation of analytic equivalence of functions at infinity. We prove that if the ?ojasiewicz exponent at infinity of the gradient of a polynomial f∈R[x1,…,xn] is greater or equal to k−1, then there exists ε>0 such that for every polynomial P∈R[x1,…,xn] of degree less or equal to k, whose coefficients of monomials of degree k are less or equal ε, the polynomials f and f+P are analytically equivalent at infinity. 相似文献
8.
Eitan Lapidot 《Journal of Approximation Theory》1984,40(4):298-300
It is known that the Bernstein polynomials of a function f defined on [0, 1 ] preserve its convexity properties, i.e., if f(n) 0 then for m n, (Bmf)(n) 0. Moreover, if f is n-convex then (Bmf)(n) 0. While the converse is not true, we show that if f is bounded on (a, b) and if for every subinterval [α, β] (a, b) the nth derivative of the mth Bernstein polynomial of f on [α, β] is nonnegative then f is n-convex. 相似文献
9.
In this paper, we study orthogonal polynomials with respect to the bilinear form (f, g)
S
= V(f) A
V(g)
T
+ <u, f
(N)
g
(N)V(f) =(f(c
0), f "(c
0), ..., f
(n – 1)
0(c
0), ..., f(c
p
), f "(c
p
), ..., f
(n – 1)
p(c
p
))
u is a regular linear functional on the linear space P of real polynomials, c
0, c
1, ..., c
p
are distinct real numbers, n
0, n
1, ..., n
p
are positive integer numbers, N=n
0+n
1+...+n
p
, and A is a N × N real matrix with all its principal submatrices nonsingular. We establish relations with the theory of interpolation and approximation. 相似文献
10.
The first part of this paper is devoted to the study of FN{\Phi_N} the orthogonal polynomials on the circle, with respect to a weight of type f = (1 − cos θ)
α
c where c is a sufficiently smooth function and ${\alpha > -\frac{1}{2}}${\alpha > -\frac{1}{2}}. We obtain an asymptotic expansion of the coefficients F*(p)N(1){\Phi^{*(p)}_{N}(1)} for all integer p where F*N{\Phi^*_N} is defined by
F*N (z) = zN [`(F)]N(\frac1z) (z 1 0){\Phi^*_N (z) = z^N \bar \Phi_N(\frac{1}{z})\ (z \not=0)}. These results allow us to obtain an asymptotic expansion of the associated Christofel–Darboux kernel, and to compute the
distribution of the eigenvalues of a family of random unitary matrices. The proof of the results related to the orthogonal
polynomials are essentially based on the inversion of the Toeplitz matrix associated to the symbol f. 相似文献
11.
Cao Jiading 《分析论及其应用》1989,5(2):99-109
Let an≥0 and F(u)∈C [0,1], Sikkema constructed polynomials:
, ifα
n
≡0, then Bn (0, F, x) are Bernstein polynomials.
Let
, we constructe new polynomials in this paper:
Q
n
(k)
(α
n
,f(t))=d
k
/dx
k
B
n+k
(α
n
,F
k
(u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα
n
≡0, k=1, then Qn
(1) (0, f(t), x) are Kantorovič polynomials Pn(f). Ifα
n
=0, k=2, then Qn
(2), (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is:
Theorem 2. Let 1≤p≤∞, in order that for every f∈LP [0, 1],
, it is sufficient and necessary that
,
§ 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define[1–2],[8–10]:
.
As usual, for the space Lp [a,b](1≤p<∞), we have
and L[a, b]=l1[a, b].
Letα
n
⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials
[3] [4].
The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports. 相似文献
12.
13.
Soulaymane Korry 《Israel Journal of Mathematics》2003,133(1):357-367
Letp∈(1, +∞) ands ∈ (0, +∞) be two real numbers, and letH
p
s
(ℝ
n
) denote the Sobolev space defined with Bessel potentials. We give a classA of operators, such thatB
s,p
-almost all points ℝ
n
are Lebesgue points ofT(f), for allf ∈H
p
s
(ℝ
n
) and allT ∈A (B
s,p
denotes the Bessel capacity); this extends the result of Bagby and Ziemer (cf. [2], [15]) and Bojarski-Hajlasz [4], valid
wheneverT is the identity operator. Furthermore, we describe an interesting special subclassC ofA (C contains the Hardy-Littlewood maximal operator, Littlewood-Paley square functions and the absolute value operatorT: f→|f|) such that, for everyf ∈H
p
s
(ℝ
n
) and everyT ∈C, T(f) is quasiuniformly continuous in ℝ
n
; this yields an improvement of the Meyers result [10] which asserts that everyf ∈H
p
s
(ℝ
n
) is quasicontinuous. However,T (f) does not belong, in general, toH
p
s
(ℝ
n
) wheneverT ∈C ands≥1+1/p (cf. Bourdaud-Kateb [5] or Korry [7]). 相似文献
14.
For a certain class of discrete approximation operators Bnf defined on an interval I and including, e.g., the Bernstein polynomials, we prove that for all f ε C(I), the ordinary moduli of continuity of Bnf and f satisfy ω(Bnf; h) cω(f; h), N = 1,2,…, 0 < h < ∞, with a universal constant c > 0. A similar result is shown to hold for a different modulus of continuity which is suitable for functions of polynomial growth on unbounded intervals. Some special operators are discussed in this connection. 相似文献
15.
Yong-Zhuo Chen 《Positivity》2012,16(1):97-106
We apply the Thompson’s metric to study the global stability of the equilibium of the following difference equation
yn = \fracf2m+12m+1 (yn-k1r, yn-k2r, ..., yn-k2m+1r)f2m2m+1 (yn-k1r, yn-k2r, ..., yn-k2m+1r), n = 0,1,2, ?, y_{n} = \frac{f_{2m+1}^{2m+1} (y_{n-k_{1}}^r, y_{n-k_{2}}^r, \dots, y_{n-k_{2m+1}}^r)}{f_{2m}^{2m+1} (y_{n-k_{1}}^r, y_{n-k_{2}}^r, \dots, y_{n-k_{2m+1}}^r)}, \;\;\;\; n = 0,1,2, \ldots, 相似文献
16.
S. P. Zhou 《Israel Journal of Mathematics》1992,78(1):75-83
The present paper gives a converse result by showing that there exists a functionf ∈C
[−1,1], which satisfies that sgn(x)f(x) ≥ 0 forx ∈ [−1, 1], such that {fx75-1} whereE
n
(0)
(f, 1) is the best approximation of degreen tof by polynomials which are copositive with it, that is, polynomialsP withP(x(f(x) ≥ 0 for allx ∈ [−1, 1],E
n(f) is the ordinary best polynomial approximation off of degreen. 相似文献
17.
Li Banghe 《Commentarii Mathematici Helvetici》1982,57(1):135-144
Under the assumption of (f, M
n
,N
2n−1) being trivial, the classification of immersions homotopic tof: M
n
→N
2n−1 is obtained in many cases. The triviality of (f, M
n
,P
2n−1) is proved for anyM
n
andf.
LetM, N be differentiable manifolds of dimensionn and2n−1 respectively. For a mapf: M → N, denote byI[M, N]
f
the set of regular homotopy classes of immersions homotopic tof. It has been proved in [1] that, whenn>1,I[M, N]
f
is nonempty for anyf. In this paper we will determine the setI[M, N]
f
in some cases.
For example, ifN=P
2n−1 or more generally, the lens spacesS
m
2n−1
=Z
m
/S
2n−1,M is any orientablen-manifold or nonorientable butn≡0, 1, 3 mod 4, then, for anyf, theI[M, N]
f is determined completely.
WhenN=R
2n−1, the setI[M, N] of regular homotopy classes of all immersions has been enumerated by James and Thomas in [2] and McClendon in [3] forn>3. Applying our results toN=R
2n−1 we obtain their results again, except for the casen≡2 mod 4 andM nonorientable.
Whenn=3, McClendon's results cannot be used. Our results include the casesn=3,M orientable or not (for orientableM, I[M, R5] is known by Wu [4]). 相似文献
18.
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
19.
《Journal of Algebra》1999,211(2):562-577
LetRbe a Krull ring with quotient fieldKanda1,…,aninR. If and only if theaiare pairwise incongruent mod every height 1 prime ideal of infinite index inRdoes there exist for all valuesb1,…,bninRan interpolating integer-valued polynomial, i.e., anf ∈ K[x] withf(ai) = biandf(R) ⊆ R.IfSis an infinite subring of a discrete valuation ringRvwith quotient fieldKanda1,…,aninSare pairwise incongruent mod allMkv ∩ Sof infinite index inS, we also determine the minimald(depending on the distribution of theaiamong residue classes of the idealsMkv ∩ S) such that for allb1,…,bn ∈ Rvthere exists a polynomialf ∈ K[x] of degree at mostdwithf(ai) = biandf(S) ⊆ Rv. 相似文献
20.
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
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