共查询到20条相似文献,搜索用时 62 毫秒
1.
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
2.
Let μ be a probability measure on [− a, a], a > 0, and let x0ε[− a, a], f ε Cn([−2a, 2a]), n 0 even. Using moment methods we derive best upper bounds to ¦∫−aa ([f(x0 + y) + f(x0 − y)]/2) μ(dy) − f(x0)¦, leading to sharp inequalities that are attainable and involve the second modulus of continuity of f(n) or an upper bound of it. 相似文献
3.
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
4.
L∞ estimates are derived for the oscillatory integral ∫+0∞e−i(xλ + (1/m) tλm)a(λ) dλ, where 2 ≤ m
and (x, t)
×
+. The amplitude a(λ) can be oscillatory, e.g., a(λ) = eit
(λ) with
(λ) a polynomial of degree ≤ m − 1, or it can be of polynomial type, e.g., a(λ) = (1 + λ)k with 0 ≤ k ≤
(m − 2). The estimates are applied to the study of solutions of certain linear pseudodifferential equations, of the generalized Schrödinger or Airy type, and of associated semilinear equations. 相似文献
5.
H. S. Jung 《Journal of Approximation Theory》2004,127(2):155-177
Let wλ(x)(1−x2)λ−1/2 and Pn(λ) be the ultraspherical polynomials with respect to wλ(x). Then we denote En+1(λ) the Stieltjes polynomials with respect to wλ(x) satisfyingIn this paper, we give estimates for the first and second derivatives of the Stieltjes polynomials En+1(λ) and the product En+1(λ)Pn(λ) by obtaining the asymptotic differential relations. Moreover, using these differential relations we estimate the second derivatives of En+1(λ)(x) and En+1(λ)(x)Pn(λ)(x) at the zeros of En+1(λ)(x) and the product En+1(λ)(x)Pn(λ)(x), respectively. 相似文献
6.
We consider the Tikhonov regularizer fλ of a smooth function f ε H2m[0, 1], defined as the solution (see [1]) to We prove that if f(j)(0) = f(j)(1) = 0, J = m, …, k < 2m − 1, then ¦f − fλ¦j2 Rλ(2k − 2j + 3)/2m, J = 0, …, m. A detailed analysis is given of the effect of the boundary on convergence rates. 相似文献
7.
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
8.
W. A. J. Luxemburg 《Journal of Approximation Theory》1975,13(4):363-374
The purpose of this paper is to show that for a certain class of functions f which are analytic in the complex plane possibly minus (−∞, −1], the Abel series f(0) + Σn = 1∞ f(n)(nβ) z(z − nβ)n − 1/n! is convergent for all β>0. Its sum is an entire function of exponential type and can be evaluated in terms of f. Furthermore, it is shown that the Abel series of f for small β>0 approximates f uniformly in half-planes of the form Re(z) − 1 + δ, δ>0. At the end of the paper some special cases are discussed. 相似文献
9.
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(− exp(¦x¦)), > 0. 相似文献
10.
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
11.
Let A = (aij) be an n × n Toeplitz matrix with bandwidth k + 1, K = r + s, that is, aij = aj−i, i, J = 1,… ,n, ai = 0 if i > s and if i < -r. We compute p(λ)= det(A - λI), as well as p(λ)/p′(λ), where p′(λ) is the first derivative of p(λ), by using O(k log k log n) arithmetic operations. Moreover, if ai are m × m matrices, so that A is a banded Toeplitz block matrix, then we compute p(λ), as well as p(λ)/p′(λ), by using O(m3k(log2 k + log n) + m2k log k log n) arithmetic operations. The algorithms can be extended to the computation of det(A − λB) and of its first derivative, where both A and B are banded Toeplitz matrices. The algorithms may be used as a basis for iterative solution of the eigenvalue problem for the matrix A and of the generalized eigenvalue problem for A and B. 相似文献
12.
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
13.
For a fixed integer m ≥ 0, and for n = 1, 2, 3, ..., let λ2m, n(x) denote the Lebesgue function associated with (0, 1,..., 2m) Hermite-Fejér polynomial interpolation at the Chebyshev nodes {cos[(2k−1) π/(2n)]: k=1, 2, ..., n}. We examine the Lebesgue constant Λ2m, n max{λ2m, n(x): −1 ≤ x ≤ 1}, and show that Λ2m, n = λm, n(1), thereby generalising a result of H. Ehlich and K. Zeller for Lagrange interpolation on the Chebyshev nodes. As well, the infinite term in the asymptotic expansion of Λ2m, n) as n → ∞ is obtained, and this result is extended to give a complete asymptotic expansion for Λ2, n. 相似文献
14.
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
15.
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
16.
M. Deza 《Journal of Combinatorial Theory, Series A》1976,20(3):306-318
Le nombre maximal de lignes de matrices seront désignées par:
- 1. (a) R(k, λ) si chaque ligne est une permutation de nombres 1, 2,…, k et si chaque deux lignes différentes coïncide selon λ positions;
- 2. (b) S0(k, λ) si le nombre de colonnes est k et si chaque deux lignes différentes coïncide selon λ positions et si, en plus, il existe une colonne avec les éléments y1, y2, y3, ou y1 = y2 ≠ y3;
- 3. (c) T0(k, λ) si c'est une (0, 1)-matrice et si chaque ligne contient k unités et si chaque deux lignes différentes contient les unités selon λ positions et si, en plus, il existe une colonne avec les éléments 1, 1, 0.
17.
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.