共查询到20条相似文献,搜索用时 31 毫秒
1.
Summary We seek a approximation to a zero of an infinitely differentiable functionf: [0, 1] such thatf(0)0 andf(1)0. It is known that the error of the bisection method usingn function evaluations is 2–(n+1). If the information used are function values, then it is known that bisection information and the bisection algorithm are optimal. Traub and Woniakowski conjectured in [5] that the bisection information and algorithm are optimal even if far more general information is permitted. They permit adaptive (sequential) evaluations of arbitrary linear functionals and arbitrary transformations of this information as algorithms. This conjecture was established in [2]. That is forn fixed, the bisection information and algorithm are optimal in the worst case setting. Thus nothing is lost by restricting oneself to function values.One may then ask whether bisection is nearly optimal in theasymptotic worst case sense, that is,possesses asymptotically nearly the best rate of convergence. Methods converging fast asymptotically, like Newton or secant type, are of course, widely used in scientific computation. We prove that the answer to this question is positive for the classF of functions having zeros ofinfinite multiplicity and information consisting of evaluations of continuous linear functionals. Assuming that everyf inF has zeroes withbounded multiplicity, there are known hybrid methods which have at least quadratic rate of convergence asn tends to infinity, see e.g., Brent [1], Traub [4] and Sect. 1. 相似文献
2.
Summary We seek the zero of a continuous increasing functionf: [0, 1] [–1, 1] such thatf(0)=–1 andf(1)=1. It is known that the bisection method makes optimal use ofn function evaluations within a worst case analysis. In this paper we study the average error with respect to the natural measure of Graf et al. (1986). We prove that the bisection method is not optimal on the average. Actually, the average error of the bisection method is about (1/2)
n
, while the average error of the optimal method is less than
n
with some <1/2. 相似文献
3.
Wolfgang Reichel 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2003,54(5):822-838
We consider the equation
(pu)-qu+wu
= f
in (0,1) subject to homogenous boundary conditions at
x = 0
and
x = 1, e.g.,
u(0)
= u(1) = 0. Let
1
be the first eigenvalue of the corresponding Sturm-Liouville problem. If
f 0
but
0 then it is known that there exists
> 0 (independent on
f) such that for
(1,
1 + ]
any solution
u
must be negative. This so-called
uniform anti-maximum principle
(UAMP)
goes back to Clément, Peletier [4]. In this paper we
establish the sharp values of
for which (UAMP) holds. The same phenomenon, including sharp values of
, can be shown for the radially symmetric
p-Laplacian on balls and annuli in
n
provided
1 n <
p. The results are illustrated by
explicitly computed examples. 相似文献
4.
Ayelet Lindenstrauss 《K-Theory》1996,10(3):239-265
Topological Hochschild homology is calculated for the rings /p[x]/(f(x)) (where p is prime and f(x) /p[x] any polynomial), [x]/(x
n) and [x]/(x
n–1). A spectral sequence argument is used for calculating the homology of the topological Hochschild homology spectrum, from which its stable homotopy structure can be read off since the spectrum is known for a priori reasons to be a restricted product of Eilenberg-MacLane spectra. 相似文献
5.
Summary Let be a compactly supported function on
s
andS () the linear space withgenerator ; that is,S () is the linear span of the multiinteger translates of . It is well known that corresponding to a generator there are infinitely many quasi-interpolation formulas. A characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications, and in addition, provides a clear formulation of the difficult problem of minimally supported quasi-interpolants. We introduce a generalization of interpolation called -interpolation and a notion of higher order quasi-interpolation called -approximation. A characterization of -approximants similar to that of quasi-interpolants is obtained with similar applications. Among these applications are estimating least-squares approximants without matrix inversion, surface fitting to incomplete or semi-scattered discrete data, and constructing generators with one-point quasi-interpolation formulas. It will be seen that the exact values of the generator at the multi-integers
s
facilitates the above study. An algorithm to yield this information for box splines is discussed.Supported by the National Science Foundation and the U.S. Army Research Office 相似文献
6.
Prof. Dr. W. Kratz 《Monatshefte für Mathematik》1981,91(4):275-289
Given semi-normsf andg on
n
and a real number >0. Then the successive minima off under the constraintg are defined by
j
: = inf {: there existj linear independent vectors inZ
n
withf andg}. The main theorem of this paper (Lagrange multiplier theorem) states that the successive minima of a certainnorm h on
n
(without constraints) coincide with the
j
's up to bounded factors. Moreover, this norm is constructed explicitly. Using Minkowski's wellknown theorem on successive minima and our result certain inequalities on simultaneous Diophantine approximations are derived. 相似文献
7.
We establish conditions for the existence of a solution of the interpolation problem f(
n
) = b
n in the class of functions f analytic in the unit disk and such that
Here, : [1; +) (0; +) is an increasing function convex with respect to lnt on the interval [1; +) and such that lnt = o((t)), t . 相似文献
0} \right)\;\left( {\forall z,\;|\;z\;| < 1} \right):\;\;\left| {f\left( z \right)} \right|\;\; \leqslant \;\;\;\exp \left( {c_1 \eta \left( {\frac{{c_1 }}{{1 - \left| z \right|}}} \right)} \right).$$ " align="middle" vspace="20%" border="0"> |
8.
Using the well known properties of thes-stage implicit Runge-Kutta methods for first order differential equations, single step methods of arbitrary order can be obtained for the direct integration of the general second order initial value problemsy=f(x, y, y),y(x
o)=y
o,y(x
o)=y
o. These methods when applied to the test equationy+2y+
2
y=0, ,0, +>0, are superstable with the exception of a finite number of isolated values ofh. These methods can be successfully used for solving singular perturbation problems for which f/y and/or f/y are negative and large. Numerical results demonstrate the efficiency of these methods. 相似文献
9.
Let be a probability measure on a locally compact groupG. A real Borel functionf onG is called -harmonic if it satisfies the convolution equation *f=f. Given that isnonsingular with its translates, we show that the bounded -harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If is nondegenerate and absolutely continuous, we solve the more general equation *= for positive measure on those groups which are metrizable and separable.Supported by Hong Kong RGC Earmarked Grant and CUHK Direct Grant 相似文献
10.
Characterization theorem of generalized polynomial of best approximation having bounded coefficients
Shusheng Xu 《应用数学学报(英文版)》1989,5(4):361-366
Let the set of generalized polynomials having bounded coefficients beK={p=
jgj.
j
j
j,j=1, 2, ...,n}, whereg
1,g
2, ...,g
n are linearly independent continuous functions defined on the interval [a, b],
j,
j are extended real numbers satisfying
j<+,
j>-, and
j
j. Assume thatf is a continuous function defined on a compact setX [a, b]. This paper gives the characterization theorem forp being the best uniform approximation tof fromK, and points out that the characterization theorem can be applied in calculating the approximate solution of best approximation tof fromK. 相似文献
11.
O. D. Tsereteli 《Mathematical Notes》1968,4(4):768-770
For any functionf of L(0, 2), we prove that there is a function L(0, 2) such that ¦(x)¦ = ¦f(x)¦ almost everywhere and L(0, 2), where is the conjugate of.Translated from Matematicheskie Zametki, Vol. 4, No. 4, pp. 461–465, October, 1968. 相似文献
12.
V. V. Zhuk 《Mathematical Notes》1968,4(1):500-508
Let n (f) and Pr (f) be, respectively, the Fejer and Poisson means of the Fourier series of the functionf. The present work considers problems associated with the rapidity of approximation of a continuous 2-periodic function by means of Fejer and Poisson processes, and gives, in particular, an upper bound to the deviation of the Fejer and Poisson processes from the function in terms of moduli of continuity, and a lower bound to n (f)–f in terms of functionals composed of best approximations to the functionf; in addition, some relationships among the quantities Pr (f)–f and n (f)–f are established.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 21–32, July, 1968. 相似文献
13.
Let D
N
, G
M
be two open sets, E D and F G two compact sets which satisfy the condition (H) (that is a harmonic condition similar to Leja"s condition). We find an open set
N+M
such that each separately harmonic function f : X : = (D× F) (E × G) (i.e.: for all x in E, f(x,.) is harmonic on G; for all y in F, f(., y) is harmonic on D) extends to a harmonic function on . 相似文献
14.
Živorad Tomovski 《Analysis Mathematica》2003,29(2):165-170
In this paper equivalent classes of the classes M'
and S'
p
r, p >1, 0,r {0,1,2,...,[]} defined by Sheng [5] are obtained. Then it is shown that the classes of Fourier coefficients S
p, S'
p(case r==0) and S
p(), p>1, defined by . V. Stanojevi, V. B. Stanojevi Sheng and the author of the present note are identical. As a corollary of this result, the L
1-estimate for cosine series, obtained in [10], is refined. 相似文献
15.
Smoothness of aC
-functionf is measured by (Carleman) sequence {M
k}
0
; we sayfC
M
[0, 1] if|f
(k)
(t)|CR
k
M
k,k=0, 1, ... withC, R>0. A typical statement proven in this paper isTHEOREM: Let u, b be two C
-functions on [0, 1]such that (a) u=u
2+b, (b) |b
(k)
(t)|CR
k
(k!)
, >1,k–.Then |u(k)(t)|C1Rk((k–1)!),k–.The first author acknowledges the hospitality of Mathematical Research Institute of the Ohio State University during his one month visit there in the spring of 1999 相似文献
16.
Bolesław Gaweł 《Aequationes Mathematicae》1996,52(1):55-71
Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)).
No assumption on the iterative behaviour off is imposed. 相似文献
17.
Héctor H. Cuenya 《Analysis Mathematica》2000,26(3):175-186
Let (, A, ) be a measure space, a function seminorm on M, the space of measurable functions on , and M
the space {f M : (f) < }. Every Borel measurable function : [0, ) [0, ) induces a function : M M by (f)(x) = (|f(x)|). We introduce the concepts of -factor and -invariant space. If is a -subadditive seminorm function, we give, under suitable conditions over , necessary and sufficient conditions in order that M
be invariant and prove the existence of -factors for . We also give a characterization of the best -factor for a -subadditive function seminorm when is -finite. All these results generalize those about multiplicativity factors for function seminorms proved earlier. 相似文献
18.
Klaus Schmidt 《Probability Theory and Related Fields》1984,68(1):75-95
Summary LetT be a non-singular ergodic automorphism of a Lebesgue space (X,L,) and letf: X be a measurable function. We define the notion of recurrence of such a functionf and introduce the recurrence setR(f)={:f– is recurrent}. If
, then R()={0}, but in general recurrence sets can be very complicated. We prove various conditions for a number to lie in R(f) and, more generally, forR(f) to be non-empty. The results in this paper have applications to the theory of random walks with stationary increments. 相似文献
19.
Summary We investigate generalizations of the classical Jensen and Chebyshev inequalities. On one hand, we restrict the class of functions and on the other we enlarge the class of measures which are allowed. As an example, consider the inequality (J)(f(x) d) A (f(x) d, d d = 1. Iff is an arbitrary nonnegativeL
x
function, this holds if 0, is convex andA = 1. Iff is monotone the measure need not be positive for (J) to hold for all convex withA = 1. If has higher monotonicity, e.g., is also convex, then we get a version of (J) withA < 1 and measures that need not be positive. 相似文献
20.
L. V. Zhizhiashvili 《Mathematical Notes》1968,4(3):655-661
The following theorem was proved by M. Riesz: Iff(x) L(–,),f(x) 0 and the conjugate functionf (x) is also integrable on [-, ], thenf(x) L log+L. The analog of this theorem for functions of several variables is established.Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 269–280, November, 1968. 相似文献