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1.
The problem of finding a best Lp-approximation (1 ≤ p < ∞) to a function in Lp from a special subcone of generalized n-convex functions induced by an ECT-system is considered. Tchebycheff splines with a countably infinite number of knots are introduced and best approximations are characterized in terms of local best approximations by these splines. Various properties of best approximations and their uniqueness in L1 are investigated. Some special results for generalized monotone and convex cases are obtained.  相似文献   

2.
A set of results concerning goodness of approximation and convergence in norm is given for L and L1 approximation of multivariate functions on hypercubes. Firstly the trigonometric polynomial formed by taking a partial sum of a multivariate Fourier series and the algebraic polynomials formed either by taking a partial sum of a multivariate Chebyshev series of the first kind or by interpolating at a tensor product of Chebyshev polynomial zeros are all shown to be near-best L approximations. Secondly the trigonometric and algebraic polynomials formed by taking, respectively, a partial sum of a multivariate Fourier series and a partial sum of a multivariate Chebyshev series of the second kind are both shown to be hear-best L1 approximations. In all the cases considered, the relative distance of a near-best approximation from a corresponding best approximation is shown to be at most of the order of Π log nj, where nj (j = 1, 2,…, N) are the respective degrees of approximation in the N individual variables. Moreover, convergence in the relevant norm is established for all the sequences of near-best approximations under consideration, subject to appropriate restrictions on the function space.  相似文献   

3.
We consider the problem of approximating a given f from Lp [0, ∞) by means of the family Vn(S) of exponential sums; Vn(S) denotes the set of all possible solutions of all possible nth order linear homogeneous differential equations with constant coefficients for which the roots of the corresponding characteristic polynomials all lie in the set S. We establish the existence of best approximations, show that the distance from a given f to Vn(S) decreases to zero as n becomes infinite, and characterize such best approximations with a first-order necessary condition. In so doing we extend previously known results that apply in Lp[0, 1].  相似文献   

4.
Three possible definitions are proposed for best simultaneous L1 approximation to n continuous real-valued functions, and the relation between best simultaneous approximations and best L1 approximations to the arithmetic mean of the n functions is discussed.  相似文献   

5.
The n-widths of the unit ball Ap of the Hardy space Hp in Lq( −1, 1) are determined asymptotically. It is shown that for 1 ≤ q < p ≤∞ there exist constants k1 and k2 such that [formula]≤ dn(Ap, Lq(−1, 1)),dn(Ap, Lq(−1, 1)), δn(Ap, Lq(−1, 1))[formula]where dn, dn, and δn denote the Kolmogorov, Gel′fand and linear n-widths, respectively. This result is an improvement of estimates previously obtained by Burchard and Höllig and by the author.  相似文献   

6.
Let ƒbe a continuous function and sn be the polynomial of degree at mostn of best L2(μ)-approximation to ƒon [-1,1]. Let Zn(ƒ):=\s{xε[-1,1]:ƒ(x)−sn(x) = 0\s}. Under mild conditions on the measure μ, we prove that Zn(ƒ) is dense in [-1,1]. This answers a question posed independently by A. Kroó and V. Tikhomiroff. It also provides an analogue of the results of Kadec and Tashev (for L∞) and Kroó and Peherstorfer (for L1) for least squares approximation.  相似文献   

7.
The almost everywhere convergence rates of Fourier–Laplace series are given for functions in certain subclasses of L2n−1) defined in terms of moduli of continuity.  相似文献   

8.
On a simplex SRd, the best polynomial approximation is En()Lp(S)=Inf{PnLp(S): Pn of total degree n}. The Durrmeyer modification, Mn, of the Bernstein operator is a bounded operator on Lp(S) and has many “nice” properties, most notably commutativity and self-adjointness. In this paper, relations between Mn−z.dfnc;Lp(S) and E[√n]()Lp(S) will be given by weak inequalities will imply, for 0<α<1 and 1≤p≤∞, En()Lp(S)=O(n-2α)Mn−z.dfnc;Lp(S)=O(n). We also see how the fact that P(DLp(S) for the appropriate P(D) affects directional smoothness.  相似文献   

9.
In appropriate function space settings, it is proved that the Fourier, Taylor, and Laurent series projections are minimal in all Lp norms (1 p ∞). This result unifies and extends known results for the Fourier, Taylor, and Laurent projections in L and for the Fourier projection in L1. The proof is based on a generalisation of a kernel summation formula due to Berman.  相似文献   

10.
Forγ(0, 1/2] we constructn-dimensional polynomial subspacesYnofC[−1, 1] andL1(−1, 1) such that the relative projection constantsλ(Yn, C[−1, 1]) andλ(Yn, L1(−1, 1)) grow asnγ. These subspaces are spanned by Chebyshev polynomials of the first and second kind, respectively. The spacesL1w(α, βwherewα, βis the weight function of the Jacobi polynomials and (α, β){(−1/2, −1/2), (−1/2, 0), (0, −1/2)} are also studied.  相似文献   

11.
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   

12.
Let 0<p<∞ and 0α<β2π. We prove that for n1 and trigonometric polynomials sn of degree n, we have

cnpβα |sn(θ)|p dθ, where c is independent of α, β, n, sn. The essential feature is the uniformity in [α,β] of the estimate and the fact that as [α,β] approaches [0,2π], we recover the Lp Markov inequality. The result may be viewed as the complete Lp form of Videnskii's inequalities, improving earlier work of the second author.  相似文献   

13.
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   

14.
Let X1, X2, …, Xn be random vectors that take values in a compact set in Rd, d ≥ 1. Let Y1, Y2, …, Yn be random variables (“the responses”) which conditionally on X1 = x1, …, Xn = xn are independent with densities f(y | xi, θ(xi)), i = 1, …, n. Assuming that θ lives in a sup-norm compact space Θq,d of real valued functions, an optimal L1-consistent estimator of θ is constructed via empirical measures. The rate of convergence of the estimator to the true parameter θ depends on Kolmogorov's entropy of Θq,d.  相似文献   

15.
Let (Ω, ,P) be a measurable space and a sub-σ-lattice of the σ-algebra . For XL1(Ω, ,P) we denote by P X the set of conditional 1-mean (or best approximants) of X given L1( ) (the set of all -measurable and integrable functions). In this paper, we obtain characterizations of the elements in P X, similar to those obtained by Landers and Rogge for conditional s-means with 1<s<∞. Moreover, using these characterizations we can extend the operator P to a bigger space L0(Ω, ,P). When, in certain sense, n goes to , we will be able to prove theorems about convergence and we will obtain bounds for the maximal function supnP nX. A sharper characterization of conditional 1-means for certain particular σ-lattice was proved in previous papers. In the last section of this paper we generalize those results to all totally ordered σ-lattices.  相似文献   

16.
This paper gives a class of functions having the excellent property that Lp, n f=Snf for all n and all p,1≤p≤∞, Lp, nf and Snf being the best approximations to f in the Lp norm with weight wp and the partial sum of the orthogonal expansion of f, respectively. Supported by National Naturol Science Fundation of China  相似文献   

17.
We study a large class of infinite variance time series that display long memory. They can be represented as linear processes (infinite order moving averages) with coefficients that decay slowly to zero and with innovations that are in the domain of attraction of a stable distribution with index 1 < α < 2 (stable fractional ARIMA is a particular example). Assume that the coefficients of the linear process depend on an unknown parameter vector β which is to be estimated from a series of length n. We show that a Whittle-type estimator βn for β is consistent (βn converges to the true value β0 in probability as n → ∞), and, under some additional conditions, we characterize the limiting distribution of the rescaled differences (n/logn)1/gan − β0).  相似文献   

18.
Laurent–Padé (Chebyshev) rational approximants P m (w,w –1)/Q n (w,w –1) of Clenshaw–Lord type [2,1] are defined, such that the Laurent series of P m /Q n matches that of a given function f(w,w –1) up to terms of order w ±(m+n), based only on knowledge of the Laurent series coefficients of f up to terms in w ±(m+n). This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent series of P m matches that of Q n f up to terms of order w ±(m+n), but based on knowledge of the series coefficients of f up to terms in w ±(m+2n). The Clenshaw–Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of the Padé–Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations are obtainable for all m0, n0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé–Chebyshev approximants. Remarkably similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw–Lord type methods, thus validating the use of either.  相似文献   

19.
Lα (0 α 1) is a class of infinitely divisible distributions defined by restricting the measure in the Levy-Khinchin formula to a special form. When α = 1, Lα is just the classical class L. Several properties for Lα classes, which are similar to the most important properties for the class L, are established. Also, a conjecture of Wolfe about unimodality of some Lα distributions is disproved by giving a counterexample.  相似文献   

20.
It is shown that an algebraic polynomial of degree k−1 which interpolates ak-monotone functionfatkpoints, sufficiently approximates it, even if the points of interpolation are close to each other. It is well known that this result is not true in general for non-k-monotone functions. As an application, we prove a (positive) result on simultaneous approximation of ak-monotone function and its derivatives inLp, 0<p<1, metric, and also show that the rate of the best algebraic approximation ofk-monotone functions (with bounded (k−2)nd derivatives inLp, 1<p<∞, iso(nk/p).  相似文献   

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