How small can one make the derivatives of an interpolating function?
نویسنده
چکیده
is finite, and that K(1) = 1, K(2) = 2. For k > 2, Favard gives no quantitative information about K(k). An estimate for the supremum under the additional restriction that only uniform t be considered can be found in Jerome and Schumaker [5]. Their argument was extended by Golomb [4] as far as it will go, viz., to include nonuniform t’s whose global mesh ratio Rt := maxi ∆ti/mini ∆ti is bounded. It is the purpose of the present paper to show how Favard’s argument can be used to obtain upper bounds for K(k). Further, an upper bound for K(k) is also obtained by a completely different method which, incidentally, also provides a simple proof of a theorem concerning the existence of H–extensions, thereby simplifying and extending three theorems of Golomb [4]. A lower bound for K(k) is also given. The author’s interest in the numbers K(k) was sparked by a question about them from H–O. Kreiss, who apparently was looking for a shortcut in computing error bounds for a given finite difference approximation to the solution of an ordinary differential equation. A bound on K(k) allows to bound the kth derivative (and therefore all lower derivatives) of some smooth interpolant f to given data f(t1), . . . , f(tn+k) in terms of the computable absolutely biggest kth divided difference without actually constructing and then bounding such an interpolant and its derivatives.
منابع مشابه
Evaluation of New MOND Interpolating Function with Rotation Curves of Galaxies
The rotation curves of a sample of 46 low- and high-surface brightness galaxies are considered in the context of Milgrom's modi_ed dynamics (MOND) to test a new interpolating function proposed by Zhao et al. (2010) [1] and compare with the results of simple interpolating function. The predicted rotation curves are calculated from the total baryonic matter based on the B-band surface photometry,...
متن کاملارزیابی رژیم‌های مختلف ارزی در یک اقتصاد باز کوچک
This research is an attempt to study about the efficiency of different exchange-rate regimes based on reduces in loss function of central bank in a form of Dynamic Stochastic General Equilibrium (DSGE), considering an oil exporting economy such as Iran's Economy. Thus, after adjusting a pattern, considering the structural characteristic of Iran’s Economy as oil exporting country and obtaining m...
متن کاملgH-differentiable of the 2th-order functions interpolating
Fuzzy Hermite interpolation of 5th degree generalizes Lagrange interpolation by fitting a polynomial to a function f that not only interpolates f at each knot but also interpolates two number of consecutive Generalized Hukuhara derivatives of f at each knot. The provided solution for the 5th degree fuzzy Hermite interpolation problem in this paper is based on cardinal basis functions linear com...
متن کاملNumerical solution of linear control systems using interpolation scaling functions
The current paper proposes a technique for the numerical solution of linear control systems.The method is based on Galerkin method, which uses the interpolating scaling functions. For a highly accurate connection between functions and their derivatives, an operational matrix for the derivatives is established to reduce the problem to a set of algebraic equations. Several test problems are given...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1975