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31.
In 1985, Khan, R. A. established the asymptotic formulas of operators of probabilistic type inL1, space by introducing a newLp-norm. The purpose of this paper is to study the asymptotic rate of these operators, inLp (p>1) spaces. Project supported by the National Natural Science Foundation of China  相似文献   
32.
We show that the remainder in Ruijsenaars’ asymptotic expansion of the logarithm of Barnes double gamma function gives rise to a completely monotone function. Fourier expansions of the multiple Bernoulli polynomials are also obtained. Research supported by the Carlsberg Foundation.  相似文献   
33.
Summary In the present work we extent the results in [RS] on CHIP, i.e. Cardinal Hermite Interpolation by the span of translates of directional derivatives of a box spline. These directional derivatives are that ones which define the type of the Hermite Interpolation. We admit here several (linearly independent) directions with multiplicities instead of one direction as in [RS]. Under the same assumptions on the smoothness of the box spline and its defining matrixT we can prove as in [RS]: CHIP has a system of fundamental solutions which are inL L 2 together with its directional derivatives mentioned above. Moreover, for data sequences inl p ( d ), 1p2, there is a spline function inL p, 1/p+1/p=1, which solves CHIP.Research supported in part by NSERC Canada under Grant # A7687. This research was completed while this author was supported by a grant from the Deutscher Akademischer Austauschdienst  相似文献   
34.
Summary A certain class of entire functionsF(s) of order zero which are asymptotically equal to the sum of just two neighbouring terms of their power series when |s| with |args| < – for any fixed > 0, is investigated. Which two terms one has to take, depends upons. It is shown that these functions have infinitely many negative zeros, and the asymptotic behaviour of the zeros is also determined.  相似文献   
35.
LetC m be a compound quadrature formula, i.e.C m is obtained by dividing the interval of integration [a, b] intom subintervals of equal length, and applying the same quadrature formulaQ n to every subinterval. LetR m be the corresponding error functional. Iff (r) > 0 impliesR m [f] > 0 (orR m [f] < 0),=" then=" we=" say=">C m is positive definite (or negative definite, respectively) of orderr. This is the case for most of the well-known quadrature formulas. The assumption thatf (r) > 0 may be weakened to the requirement that all divided differences of orderr off are non-negative. Thenf is calledr-convex. Now letC m be positive definite or negative definite of orderr, and letf be continuous andr-convex. We prove the following direct and inverse theorems for the errorR m [f], where , denotes the modulus of continuity of orderr:
  相似文献   
36.
Summary We prove that the error inn-point Gaussian quadrature, with respect to the standard weight functionw1, is of best possible orderO(n –2) for every bounded convex function. This result solves an open problem proposed by H. Braß and published in the problem section of the proceedings of the 2. Conference on Numerical Integration held in 1981 at the Mathematisches Forschungsinstitut Oberwolfach (Hämmerlin 1982; Problem 2). Furthermore, we investigate this problem for positive quadrature rules and for general product quadrature. In particular, for the special class of Jacobian weight functionsw , (x)=(1–x)(1+x), we show that the above result for Gaussian quadrature is not valid precisely ifw , is unbounded.Dedicated to Prof. H. Braß on the occasion of his 55th birthday  相似文献   
37.
Summary We propose and analyse a method of estimating the poles near the unit circleT of a functionG whose values are given at a grid of points onT: we give an algorithm for performing this estimation and prove a convergence theorem. The method is to identify the phase for an estimate by considering the peaks of the absolute value ofG onT, and then to estimate the modulus by seeking a bestL 2 fit toG over a small arc by a first order rational function. These pole estimates lead to the construction of a basis ofL 2 which is well suited to the numerical representation of the Hankel operator with symbolG and thereby to the numerical solution of the Nehari problem (computing the bestH , analytic, approximation toG relative to theL norm), as analysed in [HY]. We present the results of numerical tests of these algorithms.Partially supported by grants from the AFOSR and NSF  相似文献   
38.
An exact formula for the various measure dimensions of attractors associated with contracting similitudes is given. An example is constructed showing that for more general affine maps the various measure dimensions are not always equal.Communicated by Michael F. Barnsley.  相似文献   
39.
We consider some self-affine fractal functions previously studied by Barnsleyet al. The graphs of these functions are invariant under certain affine scalings, and we extend their definition to allow the use of nonlinear scalings. The Hölder exponent,h, for these fractal functions is calculated and we show that there is a larger Hölder exponent,h , defined at almost every point (with respect to Lebesgue measure). For a class of such functions defined using linear affinities these exponents are related to the box dimensionD B of the graph byh2–D Bh .Communicated by Michael F. Barnsley.  相似文献   
40.
Much recent work has been done to investigate convergence of modified continued fractions (MCF's), following the proof by Thron and Waadeland [35] in 1980 that a limit-periodic MCFK(a n , 1;x 1), with andnth approximant
  相似文献   
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