[{"id":37,"title":"Is numerical analysis statisitcal at heart?","sha":"b4b4bea3213964f132dd13f780ff870d","user_id":57,"state":"published","body":"**[Interpolation and approximation of continuous functions by Hermite polynomials](https://statisticalnumericalanalysis.wordpress.com/2011/12/08/interpolation-and-approximation-of-continuous-functions-by-hermite-polynomials/)**\r\n\r\nThe [Chebyshev points](http://en.wikipedia.org/wiki/Chebyshev_nodes) in [-1,1] are the zeros of the [Chebyshev polynomials](http://en.wikipedia.org/wiki/Chebyshev_polynomials). \r\n\r\nFejér uniformly approximated arbitrary continuous functions f:[-1,1]\u0026rarr; \u0026#8477; by use of [Hermite polynomials](http://en.wikipedia.org/wiki/Hermite_polynomials) at Chebyshev points ([Interpolation and Approximation, P. J. Davis, 1963, pp. 118-121](http://www.amazon.com/Interpolation-Approximation-Dover-Books-Mathematics/dp/0486624951/ref=sr_1_1?ie=UTF8\u0026qid=1423937109\u0026sr=8-1\u0026keywords=Interpolation+and+Approximation)). These are polynomials that agree with the function at the Chebyshev points and whose derivative there is 0.\r\n\r\nChebyshev points have a \u0026#946;(1/2, 1/2) distribution. The generic orbits of the chaotic function 4x(1-x) also have a \u0026#946;(1/2, 1/2) distribution ([1](http://en.wikipedia.org/wiki/Logistic_map#Chaos_and_the_logistic_map)), and a large enough finite portion of such an orbit lies very close to the the Chebyshev points: could we use these points in place of Chebyshev points? More generally:\r\n\r\n**Question**: If f:[-1,1] \u0026rarr; \u0026#8477; is a continuous function and for each integer n, \u0026#915;(n) is a random sample of n points from a \u0026#946; (1/2,1/2) distribution on [-1,1] do the Hermite-Fejér polynomials corresponding to the grid points \u0026#915;(n) converge uniformly to f with probability 1?\r\n\r\nThe answer is likely \"yes\": [\"Polynomial Approximation\" by Gordon K. Smyth, pp. 5-6](http://www.statsci.org/smyth/pubs/EoB/bap064-.pdf).\r\n\r\nCan we formulate other ideas of numerical analysis as statistical theorems (with convergence in probability)? ([2](https://statisticalnumericalanalysis.wordpress.com/)).\r\n","created_at":"2015-02-14T18:23:32.159Z","updated_at":"2019-04-20T18:29:35.902Z","zenodo_id":22276,"doi":"http://dx.doi.org/10.5281/zenodo.15135","subject":null,"tags":["interpolation","approximation","fejer","hermite","chebyshev"],"vote_count":6,"view_count":1,"score":6.1,"deleted":false,"muted":false,"tweeted":false,"attachment_file_name":null,"attachment_content_type":null,"attachment_file_size":null,"attachment_updated_at":null}]