L. Brutman, Lebesgue functions for polynomial interpolation|a survey, Ann. Numer. Math., 4 (1997), pp. 111-128. Home/Search Document Details and Download
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An Extension of Matlab to Continuous Functions and Operators - Battles, Trefethen (2004) (1 citation) (Correct)
.... N is the Lebesgue constant for the given set of interpolation points, i.e. the 1 norm of the mapping from data in these points to their degree N polynomial interpolant on [ 1; 1] 16] The proof of (i) is completed by noting that for the set of points (1. 2) N is bounded by 1 (2= log N [2]. Result (ii) can be proved by transplanting the interpolation problem to one of Fourier ( trigonometric) interpolation on an equispaced grid and using the Poisson ( aliasing) formula together with the fact that a function of bounded variation has a Fourier transform that decreases at least ....
....jxj the system quits with N = 2 after about 1 sec. For any function, one has the option of forcing the system to use a xed value of N by a command such as f = chebfun( abs(x) 1000) If f were a column vector in MATLAB, we could evaluate it at various indices by commands like f(1) or f([1 2 3]) For a chebfun, the appropriate analogue is that f should be evaluated at the corresponding arguments, not indices. For example, if f is the x chebfun de ned above, we get ans = 125.0000 f( 0.5) If g is the sin(5 x) chebfun, we get (after executing format long) g(0: 05: 2) ans = ....
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L. Brutman, Lebesgue functions for polynomial interpolation|a survey, Ann. Numer. Math., 4 (1997), pp. 111-128.
Barycentric Lagrange Interpolation - Berrut, Trefethen (1 citation) (Correct)
....C( 1; 1] denotes the space of all continuous functions on [ 1; 1] This number can be shown to be a function of the Lagrange polynomials j of (2. 1) 46, 56] n = max j j (x)j : For some sets of interpolation points, the values of n have been determined or estimated theoretically; see [15], 25, p. 121] and [35] For any nodes, we can use the computed weights (3.2) to yield a lower bound: n 2n max 0 j n jw j j min 0 j n jw j j : 9.2) This inequality can be derived by applying Markov s inequality on the size of derivatives of polynomials [11] Notice that it quanti es ....
L. Brutman, Lebesgue functions for polynomial interpolation|a survey, in The heritage of P. L. Chebyshev: A Festschrift in honor of the 70th birthday of T. J. Rivlin, Ann. Numer. Math., 4 (1997), pp. 111-127.
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