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1.
The aim of this note is to extend some classical results on the shape preserving approximation of real functions (of real variables) to functions with values in ordered vector spaces. 相似文献
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In this paper, we prove some inequalities involving the generalized trigonometric and hyperbolic functions. 相似文献
4.
张春苟 《数学的实践与认识》2005,(3)
讨论了 Meyer-Knig and Zeller算子的保形逼近问题 ,我们用基于算子特殊结构的分析方法得到了该算子的保单调性 .保凸性以及保形逆定理等保形性质 . 相似文献
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6.
José A. Raposo Javier Soria 《Proceedings of the American Mathematical Society》1997,125(4):1105-1109
We characterize the approach regions so that the non-tangential maximal function is of weak-type on potential spaces, for which we use a simple argument involving Carleson measure estimates.
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Carla Manni 《BIT Numerical Mathematics》2001,41(1):127-148
We propose a general parametric local approach for functional C
2 Hermite shape preserving interpolation. The constructed interpolant is a parametric curve which interpolate values, first and second derivatives of a given function and reproduces the behavior of the data. The method is detailed for parametric curves with piecewise cubic components. For the selected space necessary and sufficient conditions are derived to ensure the convexity of the constructed interpolant. Monotonicity is also studied. The approximation order is investigated for both cases. The use of a parametric curves to interpolate data from a function can be considered a disadvantage of the scheme. However, the simple structure of the used curve greatly reduces such a disadvantage. 相似文献
9.
In this article we furnish a representation of the solutions of some classes of first-order and second-order evolution problems as limit of iterates of classical sequences of approximating operators. The method is based on Trotter's theorem on the approximation of semigroups which is applied here also for the approximation of groups and cosine functions. We apply this method in spaces of continuous periodic functions and using some classical sequences of trigonometric polynomials. 相似文献
10.
N. Kleemann 《Mathematical Methods in the Applied Sciences》2012,35(12):1365-1391
This paper studies conical diffraction problems with non‐smooth grating structures. We prove the existence, uniqueness and regularity results for solutions in weighted Sobolev spaces of Kondratiev type. An a priori estimate that follows from these results is then used to prove shape differentiability of solutions. Finally, a characterization of the shape derivative as a solution of a modified transmission problem is given. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
11.
We prove a multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces, i.e., if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length. 相似文献
12.
Paolo Costantini Boris I. Kvasov Carla Manni 《Advances in Computational Mathematics》1999,11(4):331-354
A hyperbolic tension spline is defined as the solution of a differential multipoint boundary value problem. A discrete hyperbolic tension spline is obtained using the difference analogues of differential operators; its computation does not require exponential functions, even if its continuous extension is still a spline of hyperbolic type. We consider the basic computational aspects and show the main features of this approach. 相似文献
13.
This paper gives an affirmative answer to a conjecture given in [10]: the Bernstein basis has optimal shape preserving properties among all normalized totally positive bases for the space of polynomials of degree less than or equal ton over a compact interval. There is also a simple test to recognize normalized totally positive bases (which have good shape preserving properties), and the corresponding corner cutting algorithm to generate the Bézier polygon is also included. Among other properties, it is also proved that the Wronskian matrix of a totally positive basis on an interval [a, ) is also totally positive.Both authors were partially supported by DGICYT PS90-0121. 相似文献
14.
We show that for every potential space , there exists an approach region for which the associated maximal function is of weak-type, but the boundedness for the completed region is false, which is in contrast with the nontangential case.
15.
Approximations of Shape Metrics and Application to Shape
Warping and Empirical Shape Statistics 总被引:9,自引:0,他引:9
Guillaume Charpiat Olivier Faugeras Renaud Keriven 《Foundations of Computational Mathematics》2005,5(1):1-58
This paper proposes a framework for dealing with several problems related
to the analysis of shapes. Two related such problems are the definition of the
relevant set of shapes and that of defining a metric on it. Following a recent research
monograph by Delfour and Zolésio [11], we consider the characteristic functions
of the subsets of R2 and their distance functions. The L2 norm of the difference of
characteristic functions, the L and the W1,2 norms of the difference of distance
functions define interesting topologies, in particular the well-known Hausdorff distance.
Because of practical considerations arising from the fact that we deal with image shapes defined on finite grids of pixels, we restrict our attention to subsets of
2 of positive reach in the sense of Federer [16], with smooth boundaries of bounded
curvature. For this particular set of shapes we show that the three previous topologies
are equivalent. The next problem we consider is that of warping a shape onto another
by infinitesimal gradient descent, minimizing the corresponding distance. Because
the distance function involves an inf, it is not differentiable with respect to the shape.
We propose a family of smooth approximations of the distance function which are
continuous with respect to the Hausdorff topology, and hence with respect to the
other two topologies. We compute the corresponding Gâteaux derivatives. They define deformation flows that can be used to warp a shape onto another by solving an
initial value problem.We show several examples of this warping and prove properties
of our approximations that relate to the existence of local minima. We then use this
tool to produce computational definitions of the empirical mean and covariance of
a set of shape examples. They yield an analog of the notion of principal modes of
variation. We illustrate them on a variety of examples. 相似文献
16.
Carlos Cabrelli Sigrid B. Heineken Ursula M. Molter 《Proceedings of the American Mathematical Society》2006,134(6):1707-1718
We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular, these functions include all the homogeneous polynomials that are reproducible by the generator, which links this representation to the accuracy of the space. We completely characterize the class of homogeneous functions in the space and show that they can reproduce the generator. As a result we conclude that the homogeneous functions can be constructed from the vectors associated to the spectrum of the scale matrix (a finite square matrix with entries from the mask of the generator). Furthermore, we prove that the kernel of the transition operator has the same dimension as the kernel of this finite matrix. This relation provides an easy test for the linear independence of the integer translates of the generator. This could be potentially useful in applications to approximation theory, wavelet theory and sampling.
17.
G. A. Akishev 《Mathematical Notes》2007,81(3-4):287-290
We establish conditions for the convergence of double Fourier series in the trigonometric system of functions belonging to a symmetric space. 相似文献
18.
Shape preserving polynomial curves 总被引:3,自引:0,他引:3
We introduce particular systems of functions and study the properties of the associated Bézier-type curve for families of data points in the real affine space. The systems of functions are defined with the help of some linear and positive operators, which have specific properties: total positivity, nullity diminishing property and which are similar to the Bernstein polynomial operator. When the operators are polynomial, the curves are polynomial and their degrees are independent of the number of data points. Examples built with classical polynomial operators give algebraic curves written with the Jacobi polynomials, and trigonometric curves if the first and the last data points are identical. 相似文献
19.
We analyze the connection between two ideas of apparently different nature.
On one hand, the existence of an extended Chebyshev basis, which means that
the Hermite interpolation problem has always a unique solution. On the other
hand, the existence of a normalized totally positive basis, which means that
the space is suitable for design purposes. We prove that the intervals where
the existence of a normalized totally positive basis is guaranteed are those intervals where
the existence of an extended Chebyshev basis of the space of derivatives can be ensured.
We apply our results to the spaces C
n
generated by 1,t, , t
n-2, cos t, sin t.
In particular, C
5 is a space suitable for design which permits the exact reproduction of
remarkable parametric curves, including lines and circles with a single control polygon.
We prove that this space has the minimal dimension for this purpose. 相似文献
20.
We study the approximation of the classes
and
of periodic functions of several variables by multiple Fourier sums of fixed order constructed with regard to individual properties of functions from these classes. In a number of cases, such approximations allow us to achieve a better degree of approximation of the classes indicated above as compared to their approximation by staircase hyperbolic Fourier sums. 相似文献