13,427 research outputs found
Approximation error of the Lagrange reconstructing polynomial
The reconstruction approach [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009)
82--126] for the numerical approximation of is based on the
construction of a dual function whose sliding averages over the interval
are equal to (assuming
an homogeneous grid of cell-size ). We study the deconvolution
problem [Harten A., Engquist B., Osher S., Chakravarthy S.R.: {\em J. Comp.
Phys.} {\bf 71} (1987) 231--303] which relates the Taylor polynomials of
and , and obtain its explicit solution, by introducing rational numbers
defined by a recurrence relation, or determined by their generating
function, , related with the reconstruction pair of . We
then apply these results to the specific case of Lagrange-interpolation-based
polynomial reconstruction, and determine explicitly the approximation error of
the Lagrange reconstructing polynomial (whose sliding averages are equal to the
Lagrange interpolating polynomial) on an arbitrary stencil defined on a
homogeneous grid.Comment: 31 pages, 1 table; revised version to appear in J. Approx. Theor
The Effect of Quadrature Errors in the Computation of L^2 Piecewise Polynomial Approximations
In this paper we investigate the L^2 piecewise polynomial approximation problem. L^2 bounds for the derivatives of the error in approximating sufficiently smooth functions by polynomial splines follow immediately from the analogous results for polynomial spline interpolation. We derive L^2 bounds for the errors introduced by the use of two types of quadrature rules for the numerical computation of L^2 piecewise polynomial approximations. These bounds enable us to present some asymptotic results and to examine the consistent convergence of appropriately chosen sequences of such approximations. Some numerical results are also included
Laguerre polynomials and the inverse Laplace transform using discrete data
We consider the problem of finding a function defined on from a
countable set of values of its Laplace transform. The problem is severely
ill-posed. We shall use the expansion of the function in a series of Laguerre
polynomials to convert the problem in an analytic interpolation problem. Then,
using the coefficients of Lagrange polynomials we shall construct a stable
approximation solution.Comment: 14 page
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