Lengths Of Subintervals Research Articles

A collocation method for eigenvalue problems

Approximations to the eigenvalues ofmth-order linear eigenvalue problems are determined by using a collocation method with piecewise-polynomial functions of degreem+d possessingm continuous derivatives as basis functions. It is shown that for a general class of problems the error in these approximations is at leastO([h(II)]d+1) whereh(II) is the maximal subinterval length. The question of stability of the discretized problems is also considered. It is not assumed that the eigenvalue problems are in any sense self-adjoint.

Collocation at Gaussian Points

Approximations to an isolated solution of an mth order nonlinear ordinary differential equation with m linear side conditions are determined. These approximations are piecewise polynomial functions of order $m + k$ (degree less than $m + k$) possessing $m - 1$ continuous derivatives. They are determined by collocation, i.e., by the requirement that they satisfy the differential equation at k points in each subinterval, together with the m side conditions. If the solution of the sufficiently smooth differential equation problem has $m + 2k$ continuous derivatives and if the collocation points are the zeroes of the kth Legendre polynomial relative to each subinterval, then the global error in these approximations is $O(| \Delta |^{m + k} )$ with $| \Delta |$ the maximum subinterval length. Moreover, at the ends of each subinterval, the approximation and its first $m - 1$ derivatives are $O(| \Delta |^{2k} )$ accurate. The solution of the nonlinear collocation equations may itself be approximated by solving ...

Optimal spline digital simulators of analog filters

A now approach is presented for obtaining a linear timevarying recursive digital filter which will optimally simulate the behavior of a linear time-varying analog filter. The property of the best approximation of linear functionals by means of generalized spline functions is invoked in the derivation of the results. In this approach, the optimal digital simulator is, viewed as a min-max estimator of the samples of the analog filter output based on the samples of its input. These samples are not required to appear at uniformly spaced instants of time. The assumption is made that the analog filter input belongs to a class <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Omega_{a}</tex> of signals, each member of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Omega_{a}</tex> being generated by a known differential dynamical system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Lambda</tex> forced by some input whose energy is less than or equal to a constant <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha^{2}</tex> . By interpolating the samples of the analog filter input by a generalized spline associated with the differential operator pertaining to the system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Lambda</tex> , the structure of the optimal digital simulator is derived. Error bounds are derived as functions of the maximum sampling subinterval length and the theory is illustrated by means of examples.

Asymptotical independence of the lengths of subintervals of a randomly partitioned interval; a sample from S. Ikeda's work

Asymptotical independence of the lengths of subintervals of a randomly partitioned interval; a sample from S. Ikeda's work