75,425 research outputs found
Estimation of a -monotone density: limit distribution theory and the spline connection
We study the asymptotic behavior of the Maximum Likelihood and Least Squares
Estimators of a -monotone density at a fixed point when .
We find that the th derivative of the estimators at converges at the
rate for . The limiting distribution depends
on an almost surely uniquely defined stochastic process that stays above
(below) the -fold integral of Brownian motion plus a deterministic drift
when is even (odd). Both the MLE and LSE are known to be splines of degree
with simple knots. Establishing the order of the random gap
, where denote two successive knots, is a key
ingredient of the proof of the main results. We show that this ``gap problem''
can be solved if a conjecture about the upper bound on the error in a
particular Hermite interpolation via odd-degree splines holds.Comment: Published in at http://dx.doi.org/10.1214/009053607000000262 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Bivariate Hermite subdivision
A subdivision scheme for constructing smooth surfaces interpolating scattered data in is proposed. It is also possible to impose derivative constraints in these points. In the case of functional data, i.e., data are given in a properly triangulated set of points from which none of the pairs and with coincide, it is proved that the resulting surface (function) is . The method is based on the construction of a sequence of continuous splines of degree 3. Another subdivision method, based on constructing a sequence of splines of degree 5 which are once differentiable, yields a function which is if the data are not 'too irregular'. Finally the approximation properties of the methods are investigated
Generalized Divided Differences, with Applications to Generalized B-Splines
Within the theory of spline functions, there was always great interest in the study of B-splines, i.e. splines with a finite support. In addition to the classical polynomial case, there exist also various approaches to the definition of B-splines from other function classes, such as trigonometrie or hyperbolic ones (cf. section 0). In the present paper we define B-splines from a rat her general function space, which covers almost all existing approaches as special cases. The only condition that the spaces under consideration must satisfy is that of being translation invariant, a fundamental property which we are going to define in section 2. Our definition of generalized B-splines is based on generalized divided differences, which go back to Popoviciu [14] and were further investigated by Mühlbach [10, 11]. In the first section of this paper we therefore study these operators and prove new results, such as a contour integral representation and a multistep-formula; the latter one expresses - in closed form - a generalized divided difference of order m+j by those of order m, for arbitratry j ∈ IN. This makes it possible to compute generalized divided differences recursively, even if the underlying function space is spanned by a non-complete Chebyshev system
Numerical solutions of matrix differential models using higher-order matrix splines
The final publication is available at http://link.springer.com/article/10.1007%2Fs00009-011-0159-zThis paper deals with the construction of approximate solution of first-order matrix linear differential equations using higher-order matrix splines. An estimation of the approximation error, an algorithm for its implementation and some illustrative examples are included. © 2011 Springer Basel AG.Defez Candel, E.; Hervás Jorge, A.; Ibáñez González, JJ.; Tung, MM. (2012). Numerical solutions of matrix differential models using higher-order matrix splines. Mediterranean Journal of Mathematics. 9(4):865-882. doi:10.1007/s00009-011-0159-zS86588294Al-Said E.A., Noor M.A.: Cubic splines method for a system of third-order boundary value problems. Appl. Math. Comput. 142, 195–204 (2003)Ascher U., Mattheij R., Russell R.: Numerical solutions of boundary value problems for ordinary differential equations. Prentice Hall, New Jersey, USA (1988)Barnett S.: Matrices in Control Theory. Van Nostrand, Reinhold (1971)Blanes S., Casas F., Oteo J.A., Ros J.: Magnus and Fer expansion for matrix differential equations: the convergence problem. J. Phys. Appl. 31, 259–268 (1998)Boggs P.T.: The solution of nonlinear systems of equations by a-stable integration techniques. SIAM J. Numer. Anal. 8(4), 767–785 (1971)Defez E., Hervás A., Law A., Villanueva-Oller J., Villanueva R.: Matrixcubic splines for progressive transmission of images. J. Math. Imaging Vision 17(1), 41–53 (2002)Defez E., Soler L., Hervás A., SantamarÃa C.: Numerical solutions of matrix differential models using cubic matrix splines. Comput. Math. Appl. 50, 693–699 (2005)Defez E., Soler L., Hervás A., Tung M.M.: Numerical solutions of matrix differential models using cubic matrix splines II. Mathematical and Computer Modelling 46, 657–669 (2007)Mazzia F., Trigiante A.S., Trigiante A.S.: B-spline linear multistep methods and their conitinuous extensions. SIAM J. Numer. Anal. 44(5), 1954–1973 (2006)Faddeyev L.D.: The inverse problem in the quantum theory of scattering. J. Math. Physics 4(1), 72–104 (1963)Flett, T.M.: Differential Analysis. Cambridge University Press (1980)Golub G.H., Loan C.F.V.: Matrix Computations, second edn. The Johns Hopkins University Press, Baltimore, MD, USA (1989)Graham A.: Kronecker products and matrix calculus with applications. John Wiley & Sons, New York, USA (1981)Jódar L., Cortés J.C.: Rational matrix approximation with a priori error bounds for non-symmetric matrix riccati equations with analytic coefficients. IMA J. Numer. Anal. 18(4), 545–561 (1998)Jódar L., Cortés J.C., Morera J.L.: Construction and computation of variable coefficient sylvester differential problems. Computers Maths. Appl. 32(8), 41–50 (1996)Jódar, L., Ponsoda, E.: Continuous numerical solutions and error bounds for matrix differential equations. In: Int. Proc. First Int. Colloq. Num. Anal., pp. 73–88. VSP, Utrecht, The Netherlands (1993)Jódar L., Ponsoda E.: Non-autonomous riccati-type matrix differential equations: Existence interval, construction of continuous numerical solutions and error bounds. IMA J. Numer. Anal. 15(1), 61–74 (1995)Loscalzo F.R., Talbot T.D.: Spline function approximations for solutions of ordinary differential equations. SIAM J. Numer. Anal. 4(3), 433–445 (1967)Marzulli P.: Global error estimates for the standard parallel shooting method. J. Comput. Appl. Math. 34, 233–241 (1991)Micula G., Revnic A.: An implicit numerical spline method for systems for ode’s. Appl. Math. Comput. 111, 121–132 (2000)Reid, W.T.: Riccati Differential Equations. Academic Press (1972)Rektorys, K.: The method of discretization in time and partial differential equations. D. Reidel Pub. Co., Dordrecht (1982)Scott, M.: Invariant imbedding and its Applications to Ordinary Differential Equations. Addison-Wesley (1973
Associated Primes of Spline Complexes
The spline complex whose top homology is
the algebra of mixed splines over the fan
was introduced by Schenck-Stillman in
[Schenck-Stillman 97] as a variant of a complex
of Billera [Billera 88]. In this paper we
analyze the associated primes of homology modules of this complex. In
particular, we show that all such primes are linear. We give two applications
to computations of dimensions. The first is a computation of the third
coefficient of the Hilbert polynomial of , including cases
where vanishing is imposed along arbitrary codimension one faces of the
boundary of , generalizing the computations in [Geramita-Schenck
98,McDonald-Schenck 09]. The second is a description of the fourth coefficient
of the Hilbert polynomial of for simplicial fans
. We use this to derive the result of Alfeld, Schumaker, and Whiteley
on the generic dimension of tetrahedral splines for
[Alfeld-Schumaker-Whiteley 93] and indicate via an example how this description
may be used to give the fourth coefficient in particular nongeneric
configurations.Comment: 40 pages, 10 figure
Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant
The purpose of this paper is to establish that for any compact, connected
C^{\infty} Riemannian manifold there exists a robust family of kernels of
increasing smoothness that are well suited for interpolation. They generate
Lagrange functions that are uniformly bounded and decay away from their center
at an exponential rate. An immediate corollary is that the corresponding
Lebesgue constant will be uniformly bounded with a constant whose only
dependence on the set of data sites is reflected in the mesh ratio, which
measures the uniformity of the data.
The analysis needed for these results was inspired by some fundamental work
of Matveev where the Sobolev decay of Lagrange functions associated with
certain kernels on \Omega \subset R^d was obtained. With a bit more work, one
establishes the following: Lebesgue constants associated with surface splines
and Sobolev splines are uniformly bounded on R^d provided the data sites \Xi
are quasi-uniformly distributed. The non-Euclidean case is more involved as the
geometry of the underlying surface comes into play. In addition to establishing
bounded Lebesgue constants in this setting, a "zeros lemma" for compact
Riemannian manifolds is established.Comment: 33 pages, 2 figures, new title, accepted for publication in SIAM J.
on Math. Ana
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