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1.
David Elliott 《BIT Numerical Mathematics》1979,19(4):457-462
Hunter's (n+1)-point quadrature rule for the approximate evaluation of the Cauchy principal value integralf
1
–1 (w(x)f(x)/(x – ))dx, –1<<1, is based on approximatingf by the polynomial which interpolatesf at the point and then zeros of the orthogonal polynomialp
n
generated by the weight functionw. Sufficient conditions are given to ensure the convergence of a suitably chosen subsequence of the quadrature rules to the integral, whenf is Hölder continuous on [–1,1]. 相似文献
2.
H. Fiedler 《Aequationes Mathematicae》1986,31(1):294-299
LetX={x
1,x
2,...,
n
}I=[–1, 1] and
. ForfC
1(I) definef* byf–p
f
=f*, wherep
f
denotes the interpolation-polynomial off with respect toX. We state some properties of the operatorf f*. In particular, we treat the case whereX consists of the zeros of the Chebyshev polynomialT
n
(x) and obtain x
m
–p
x
m8eE
n–1(x
m
), whereE
n–1(f) denotes the sup-norm distance fromf to the polynomials of degree less thann. Finally we state a lower estimate forE
n
(f) that omits theassumptionf
(n+1)>0 in a similar estimate of Meinardus. 相似文献
3.
Ian H. Sloan 《BIT Numerical Mathematics》1978,18(1):91-102
Product-integration rules of the form
–1
1
k(x)f(x)dx
i
=1n
w
ni
f(x
ni
) are studied, with the points {w
ni
} chosen to be the zeros of certain orthogonal polynomials, and the weights {w
ni
} chosen to make the rule exact iff is any polynomial of degree less thann. If, in particular, the points are the Chebyshev points, and ifk L
p
[–1, 1] for somep>1, then it is shown that the rule converges to the exact result for all continuous functionsf. With this choice of points, the practical application of the rule is shown to be straightforward in many cases, and to yield satisfactory rates of convergence. The casek(x)=|–x|, >–1, is studied in detail. Results of a similar, but weaker, kind are also obtained for other choices of the points {x
ni
}. 相似文献
4.
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
5.
Among all integration rules with n points, it is well-known that n-point Gauss–Legendre quadrature rule∫−11f(x) dx∑i=1nwif(xi)has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n−1, where nodes xi are zeros of Legendre polynomial Pn(x), and wi's are corresponding weights.In this paper we are going to estimate numerical values of nodes xi and weights wi so that the absolute error of introduced quadrature rule is less than a preassigned tolerance ε0, say ε0=10−8, for monomial functionsf(x)=xj, j=0,1,…,2n+1.(Two monomials more than precision degree of Gauss–Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more accurate than the corresponding Gauss–Legendre rules. Some examples are given to show the numerical superiority of presented rules. 相似文献
6.
Stefano Serra 《BIT Numerical Mathematics》1996,36(1):135-142
In this paper we are concerned with the asymptotic behavior of the smallest eigenvalue
1
(n)
of symmetric (Hermitian)n ×n Toeplitz matricesT
n
(f) generated by an integrable functionf defined in [–, ]. In [7, 8, 11] it is shown that
1
(n)
tends to essinff =m
f
in the following way:
1
(n)
–m
f
1/n
2k
. These authors use three assumptions:A1)f –m
f
has a zero inx =x
0 of order 2k.A2)f is continuous and at leastC
2k
in a neighborhood ofx
0.A3)x =x
0 is the unique global minimum off in [–, ]. In [10] we have proved that the hypothesis of smoothnessA2 is not necessary and that the same result holds under the weaker assumption thatf L
1[–, ]. In this paper we further extend this theory to the case of a functionf L
1[–, ] having several global minima by suppressing the hypothesisA3 and by showing that the maximal order 2k of the zeros off –m
f
is the only parameter which characterizes the rate of convergence of
1
(n)
tom
f
. 相似文献
7.
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
8.
We obtain the exact values of extremal characteristics of a special form that connect the best polynomial approximations of functions f(x) ∈ L
2
r
(r ∈ ℤ+) and expressions containing moduli of continuity of the kth order ωk(f(r), t). Using these exact values, we generalize the Taikov result for inequalities that connect the best polynomial approximations and moduli of continuity of functions from L
2. For the classes
(k, r, Ψ*) defined by ω k(f
(r), t) and the majorant
, we determine the exact values of different widths in the space L2.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1458–1466, November, 2004. 相似文献
9.
We derive the approximation on [0, 1] of functionsf(x) by interpolating spline-functions sr(f; x) of degree 2r+1 and defect r+1 (r=1, 2,...). Exact estimates for ¦f(x)–sr(f; x) ¦ and f(x)–sr(f; x)|c on the class WmH for m=1, r=1, 2, ..., and m=2, 3, r=1 for the case of convex (t),are derived.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 483–494, May, 1971. 相似文献
10.
B. I. Golubov 《Functional Analysis and Its Applications》2005,39(2):135-139
For functions in the Lebesgue space L(ℝ+), a modified strong dyadic integral J
α and a modified strong dyadic derivative D
(α) of fractional order α > 0 are introduced. For a given function f ∈ L(ℝ+), criteria for the existence of these integrals and derivatives are obtained. A countable set of eigenfunctions for the operators J
α and D
(α) is indicated. The formulas D
(α)(J
α(f)) = f and J
α(D
(α)(f)) = f are proved for each α > 0 under the condition that
. We prove that the linear operator
is unbounded, where
is the natural domain of J
α. A similar statement for the operator
is proved. A modified dyadic derivative d
(α)(f)(x) and a modified dyadic integral j
α(f)(x) are also defined for a function f ∈ L(ℝ+) and a given point x ∈ ℝ+. The formulas d
(α)(J
α(f))(x) = f(x) and j
α(D
(α)(f)) = f(x) are shown to be valid at each dyadic Lebesgue point x ∈ ℝ+ of f.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 64–70, 2005Original Russian Text Copyright © by B. I. GolubovSupported by the Russian Foundation for Basic Research (grant no. 05-01-00206). 相似文献
11.
Dr. Herbert E. Salzer 《Numerische Mathematik》1971,18(2):144-153
We considern-point Lagrange-Hermite extrapolation forf(x), x>1, based uponf(x
i
),i=1(1)n, –1x
i
1, including non-distinct pointsx
i
in confluent formulas involving derivatives. The problem is to find the pointsx
i
that minimize the factor
in the remainderP
n
(x)f
(n)()/n, –1<<x subject to the condition|P
n
(x)|M, –1x1,2–n+1M2
n
. The solution is significant only when a single set of pointsx
i
suffices for everyx>1. The problem is here completely solved forn=1(1)4. Forn>4 it may be conjectured that there is a single minimal
, 0 rn, whererr(M) is a non-decreasing function ofM, P
n
(–1)=(–1)
n
M, and for 0rn–2, thej-th extremumP
n
(x
e, j
)=(–1)
n–j
M,j=1(1)n–r–1 (except forM=M
r
,r=1(1)n–1, whenj=1(1)n–r). 相似文献
12.
Prof. Dr. Gilbert Helmberg 《Monatshefte für Mathematik》1982,94(3):213-248
Let the continuous broken linear transformationf of the unit interval into itself satisfyingf(0)=f(1)=0 be determined by the coordinates of its peak pointP (x, y). The topological entropyh off, as a function of (x, y), is zero outside the triangle max (x, 1–x)<y1. Inside it is shown to be nonzero, continuous, monotonically increasing both iny/x and iny/(1–x) and to assume its maximum log2 ony=1. The level curvec(h
0) of constant corresponding entropyh
0>0 is a continuous curve joining the two diagonalsy=x andy=1–x in whichh has discontinuities (jumping to zero). For 1/2log2<hlog 2 the curvesc(h) pass through (0,1) withy=1 as a tangent and fill up the area bounded below by the parabolay
2=1–x on whichh(x,y)=1/2 log 2. For 1/2 log 2 <h log 2 the curvec(h) is the image of the arc ofc(2h) between the hyperbolax
2–xy2x+1=0 and the diagonaly=1–x under the transformation
. 相似文献
13.
Isha Bagai B. L. S. Prakasa Rao 《Annals of the Institute of Statistical Mathematics》1995,47(2):253-266
Let {X
n
,n1} be a strictly stationary sequence of associated random variables defined on a probability space (,B, P) with probability density functionf(x) and failure rate functionr(x) forX
1. Letf
n
(x) be a kerneltype estimator off(x) based onX
1,...,X
n
. Properties off
n
(x) are studied. Pointwise strong consistency and strong uniform consistency are established under a certain set of conditions. An estimatorr
n
(x) ofr(x) based onf
n
(x) andF
n
(x), the empirical survival function, is proposed. The estimatorr
n
(x) is shown to be pointwise strongly consistent as well as uniformly strongly consistent over some sets. 相似文献
14.
Letf
0(x) be a function of one variable with a simple zero atr
0. An iteration scheme is said to be locally convergent if, for some initial approximationsx
1, ...,x
s nearr
0 and all functionsf which are sufficiently close (in a certain sense) tof
0, the scheme generates a sequence {x
k} which lies nearr
0 and converges to a zeror off. The order of convergence of the scheme is the infimum of the order of convergence of {x
k} for all such functionsf. We study iteration schemes which are locally convergent and use only evaluations off,f, ...,f
[d] atx
1, ...,x
k–1 to determinex
k, and we show that no such scheme has order greater thand+2. This bound is the best possible, for it is attained by certain schemes based on polynomial interpolation.This work was supported (in part) by the Office of Naval Research under contract numbers N0014-69-C-0023 and N0014-71-C-0112. 相似文献
15.
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
16.
J. L. García-Roig 《Aequationes Mathematicae》1991,41(1):2-12
Summary We first characterize all the ultrametric functionsf on (assuming both thatf(–x)=f(x) and thatf(x)=0 if and only ifx=0) and then, among these functions, we describe those satisfying the functional equationf(x)·f(x
–1)=1, for all nonzerox in .Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth 相似文献
17.
M. H. A. Davis 《Probability Theory and Related Fields》1980,54(2):125-139
Summary This paper concerns the nonlinear filtering problem of calculating estimates E[f(xt)¦y s, st] where {x
t} is a Markov process with infinitesimal generator A and {y
t} is an observation process given by dy
t=h(xt)dt +dwtwhere {w
t} is a Brownian motion. If h(xt) is a semimartingale then an unnormalized version of this estimate can be expressed in terms of a semigroup T
s,t
y
obtained by a certain y-dependent multiplicative functional transformation of the signal process {x
t}. The objective of this paper is to investigate this transformation and in particular to show that under very general conditions its extended generator is A
t
y
f=ey(t)h(A– 1/2h2)(e–y(t)h
f).Work partially supported by the U.S. Department of Energy (Contract ET-76-C-01-2295) at the Massachusetts Institute of Technology 相似文献
18.
Yingguang Shi 《应用数学学报(英文版)》1995,11(3):240-254
Necessary and sufficient conditions for the regularity andq-regularity of (0,1,...,m–2,m) interpolation on the zeros of (1–x
2)P
n
–2/(,)
(x) (,>–1) in a manageable form are established, whereP
n
–2/(,)
(x) stands for the (n–2)th Jacobi polynomial. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, we show that under a mild assumption if the problem of (0,1,...,m–2,m) interpolation has an infinity of solutions then the general form of the solutions isf
0(x)+C f(x) with an arbitrary constantC.This work is supported by the National Natural Science Foundation of China. 相似文献
19.
Suppose thatX
l
,..., X
n
are samples drawn from a population with density functionf andf
n
(x)=f
n
(x;X
l
,..., X
n
is an estimate off(x), Denote bym
nr
=|f
n
(x)–f(n)|
r
dx andM
nr
=E(m
nr) the Integratedr-th Order Error and Mean Integratedr-th Order Error off
n
for somer1 (whenr=2,they are the familiar and widely studied ISE and MISE), In this paper the same necessary and sufficient condition for
and
a.s. is obtained whenf
n
(x) is the ordinary histogram estimator.The Project supported by National Natural Science Foundation of China. 相似文献
20.
For integrals
–1
1
w(x)f(x)dx with
and with analytic integrands, we consider the determination of optimal abscissasx
i
o
and weightsA
i
o
, for a fixedn, which minimize the errorE
n
(f)=
–1
1
w(x)f(x)dx –
i
=1n
A
i
f(x
i
) over an appropriate Hilbert spaceH
2(E
; w(z)) of analytic functions. Simultaneously, we consider the simpler problem of determining intermediate-optimal weightsA
i
*, corresponding to (preassigned) Gaussian abscissasx
i
G
, which minimize the quadrature error. For eachw(x), the intermediate-optimal weightsA
i
* are obtained explicitly, and these come out proportional to the corresponding Gaussian weightsA
i
G
. In each case,A
i
G
=A
i
*+O(
–4n
), . For
, a complete explicit solution for optimal abscissas and weights is given; in fact, the set {x
i
G
,A
i
*;i=1,...,n} to provides the optimal abscissas and weights. For otherw(x), we study the closeness of the set {x
i
G
,A
i
*;i=1,...,n} to the optimal solution {x
i
o
,A
i
o
;i=1,...,n} in terms of
n
(), the maximum absolute remainder in the second set ofn normal equations. In each case,
n
() is, at least, of the order of
–4n
for large. 相似文献