共查询到20条相似文献,搜索用时 62 毫秒
1.
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
2.
Hans-Peter Blatt Ren Grothmann Ralitza Kovacheva 《Journal of Approximation Theory》2004,126(2):157-170
Let E be a compact set in
with connected complement and positive logarithmic capacity. For any f continuous on E and analytic in the interior of E, we consider the distribution of extreme points of the error of best uniform polynomial approximation on E. Let Λ=(nj) be a subsequence of
such that nj+1/nj→1. If, for nΛ, An( f)∂E denotes the set of extreme points of the error function, we prove that there is a subsequence Λ′ of Λ such that the distribution of any (n+2)th Fekete point set
of An( f) tends weakly to the equilibrium distribution on E as n→∞ in Λ′. Furthermore, we prove a discrepancy result for the distribution of the point sets
if the boundary of E is smooth enough. 相似文献
3.
We prove that for f ε E = C(G) or Lp(G), 1 p < ∞, where G is any compact connected Lie group, and for n 1, there is a trigonometric polynomial tn on G of degree n so that f − tnE Crωr(n−1,f). Here ωr(t, f) denotes the rth modulus of continuity of f. Using this and sharp estimates of the Lebesgue constants recently obtained by Giulini and Travaglini, we obtain “best possible” criteria for the norm convergence of the Fourier series of f. 相似文献
4.
M. P. H. Wolff 《Journal of Approximation Theory》2001,113(2):229
Let E be a Banach space over
and let the densely defined closed linear operator A:
(A)E→E be discretely approximated by the sequence ((An,
(An)))n
of operators An where each An is densely defined in the Banach space Fn. Let σa(A) be the approximate point spectrum of A and let σ(An) denote the -pseudospectrum of An. Generalizing our own result, we show that σa(A)lim inf σ(An)=n
∩kn σ(Ak) holds for every >0. We deduce that then for every compact set K
limn dist(σa(A)∩K, σa(An))=0 provided there exists M>0 such that (λ−An)−1M dist(λ, σ(An))−1 holds for every n and every λ in the resolvent set ρ(An) of An. We finally treat the problem under which conditions σa(A) can be approximated from below. More precisely we investigate the problem: Under which assumptions does ∩>0 ∩n
kn σ, a(Ak)σa(A) hold where σ, a(A) denotes the -approximate pseudospectrum? 相似文献
5.
A recent method of Soundararajan enables one to obtain improved Ω-result for finite series of the form ∑nf(n) cos (2πλnx+β) where 0≤λ1≤λ2≤. . . and β are real numbers and the coefficients f(n) are all non-negative. In this paper, Soundararajan’s method is adapted to obtain improved Ω-result for E(t), the remainder term in the mean-square formula for the Riemann zeta-function on the critical line. The Atkinson series for E(t) is of the above type, but with an oscillating factor (−1)n attached to each of its terms. 相似文献
6.
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
7.
This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRn, n2, and such that exp(βK(x))Lloc1(Ω), β>0. We show that there exist two universal constants c1(n),c2(n) with the following property: Let f be a mapping in Wloc1,1(Ω,Rn) with |Df(x)|nK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L1 log−c1(n)βL. Then automatically J(x,f) is locally in L1 logc2(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings. 相似文献
8.
F. Vogl 《Journal of Approximation Theory》2000,107(2):281
For a class of analytic functions f(z) defined by Laplace–Stieltjes integrals the uniform convergence on compact subsets of the complex plane of the Bruwier series (B-series) ∑∞n=0 λn(f)
, λn(f)=f(n)(nc)+cf(n+1)(nc), generated by f(z) and the uniform approximation of the generating function f(z) by its B-series in cones |arg z|<
is shown. 相似文献
9.
M. A. Botto 《Journal of Approximation Theory》1976,16(4):347-365
We investigate two sequences of polynomial operators, H2n − 2(A1,f; x) and H2n − 3(A2,f; x), of degrees 2n − 2 and 2n − 3, respectively, defined by interpolatory conditions similar to those of the classical Hermite-Féjer interpolators H2n − 1(f, x). If H2n − 2(A1,f; x) and H2n − 3(A2,f; x) are based on the zeros of the jacobi polynomials Pn(α,β)(x), their convergence behaviour is similar to that of H2n − 1(f;, x). If they are based on the zeros of (1 − x2)Tn − 2(x), their convergence behaviour is better, in some sense, than that of H2n − 1(f, x). 相似文献
10.
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 → ∞ ∝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
11.
Vasiliy A. Prokhorov 《Journal of Approximation Theory》2000,107(2):337
Let E be a compact set in the extended complex plane C and let f be holomorphic on E. Denote by ρn the distance from f to the class of all rational functions of order at most n, measured with respect to the uniform norm on E. We obtain results characterizing the relationship between estimates of lim infn→∞ ρ1/nn and lim supn→∞ ρ1/nn. 相似文献
12.
A. A. Dovgoshei 《Mathematical Notes》1995,58(3):921-927
For an arbitrary compact setK⊂ℂ, we relate the order and the type of an entire functionf to the sequenceE
n
(f,K) of best polynomial approximations to this function onK.
Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 355–364, September, 1995. 相似文献
13.
It is well known by a classical result of Bourgain–Fremlin–Talagrand that if K is a pointwise compact set of Borel functions on a Polish space then given any cluster point f of a sequence (fn)nω in K one can extract a subsequence (fnk)kω converging to f. In the present work we prove that this extraction can be achieved in a “Borel way.” This will prove in particular that the notion of analytic subspace of a separable Rosenthal compacta is absolute and does not depend on the particular choice of a dense sequence. 相似文献
14.
Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ2 the set of convex functions f ∈ ℂ[−,1]. Also, let E
n
(f) and E
n
(2) (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ2 by algebraic polynomials of degree < n, respectively. Clearly, En (f) ≦ E
n
(2) (f), and Lorentz and Zeller proved that the inverse inequality E
n
(2) (f) ≦ cE
n
(f) is invalid even with the constant c = c(f) which depends on the function f ∈ Δ2.
In this paper we prove, for every α > 0 and function f ∈ Δ2, that
where c(α) is a constant depending only on α. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (−1,1) is also investigated. It turns out that there are substantial differences between the cases
s≦ 1 and s ≧ 2.
Dedicated to Jóska Szabados on his 70th birthday 相似文献
15.
Christoph Hamburger 《Advances in Mathematics》2005,190(2):360-424
We study the nonlinear Hodge system dω=0 and δ(ρ(|ω|2)ω)=0 for an exterior form ω on a compact oriented Riemannian manifold M, where ρ(Q) is a given positive function. The solutions are called ρ-harmonic forms. They are the stationary points on cohomology classes of the functional
with e′(Q)=ρ(Q)/2. The ρ-codifferential of a form ω is defined as δρω=ρ−1δ(ρω) with ρ=ρ(|ω|2).We evolve a given closed form ω0 by the nonlinear heat flow system
for a time-dependent exterior form ω(x,t) on M. This system is the differential of the normalized gradient flow
for E(ω) with ω=ω0+du. Under a technical assumption on the function 2ρ′(Q)Q/ρ(Q), we show that the nonlinear heat flow system
, with initial condition ω(·,0)=ω0, has a unique solution for all times, which converges to a ρ-harmonic form in the cohomology class of ω0. This yields a nonlinear Hodge theorem that every cohomology class of M has a unique ρ-harmonic representative. 相似文献
16.
For a functionfLp[−1, 1], 0<p<∞, with finitely many sign changes, we construct a sequence of polynomialsPnΠnwhich are copositive withfand such that f−PnpCω(f, (n+1)−1)p, whereω(f, t)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.
V. A. Prokhorov 《Journal of Approximation Theory》2002,116(2):380-396
The paper deals with problems relating to the theory of Hankel operators. Let G be a bounded simple connected domain with the boundary Γ consisting of a closed analytic Jordan curve. Denote by
n,p(G), 1p<∞, the class of all meromorphic functions on G that can be represented in the form h=β/α, where β belongs to the Smirnov class Ep(G), α is a polynomial degree at most n, α0. We obtain estimates of s-numbers of the Hankel operator Af constructed from fLp(Γ), 1p<∞, in terms of the best approximation Δn,p of f in the space Lp(Γ) by functions belonging to the class
n,p(G). 相似文献
18.
C. E. Chidume E. U. Ofoedu H. Zegeye 《Journal of Mathematical Analysis and Applications》2003,280(2):364-374
Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T :K→E be an asymptotically nonexpansive nonself-map with sequence {kn}n1[1,∞), limkn=1, F(T):={xK: Tx=x}≠. Suppose {xn}n1 is generated iteratively by where {αn}n1(0,1) is such that ε<1−αn<1−ε for some ε>0. It is proved that (I−T) is demiclosed at 0. Moreover, if ∑n1(kn2−1)<∞ and T is completely continuous, strong convergence of {xn} to some x*F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {xn} to some x*F(T) is obtained. 相似文献
19.
Toshiyasu Arai 《Mathematical Logic Quarterly》2002,48(1):125-130
For α < ε0, Nα denotes the number of occurrences of ω in the Cantor normal form of α with the base ω. For a binary number-theoretic function f let B(K; f) denote the length n of the longest descending chain (α0, …, αn–1) of ordinals <ε0 such that for all i < n, Nαi ≤ f (K, i). Simpson [2] called ε0 as slowly well ordered when B (K; f) is totally defined for f (K; i) = K · (i+ 1). Let |n| denote the binary length of the natural number n, and |n|k the k-times iterate of the logarithmic function |n|. For a unary function h let L(K; h) denote the function B (K; h0(K; i)) with h0(K, i) = K + |i| · |i|h(i). In this note we show, inspired from Weiermann [4], that, under a reasonable condition on h, the functionL (K; h) is primitive recursive in the inverse h–1 and vice versa. 相似文献
20.
We consider a strictly convex domain D
n and m holomorphic functions, φ1,…, φm, in a domain
. We set V = {z ε Ω: φ1(z) = ··· = φm(z) = 0}, M = V ∩ D and ∂M = V ∩ ∂D. Under the assumptions that the variety V has no singular point on ∂M and that V meets ∂D transversally we construct an explicit kernel K(ζ, z) defined for ζ ε ∂M and z ε D so that the integral operator Ef(z) = ∝ ζ ε ∂M f(ζ) K(ζ, z) (z ε D), defined for f ε H∞(M) (using the boundary values f(ζ) for a.e. ζ ε ∂M), is an extension operator, i.e., Ef(z) = f(z) for z ε M and furthermore E is a bounded operator from H∞ to H∞(D). 相似文献