Solution of The Problem
Under the above conditions, there exists a solution to the problem for any given set of data points {xk, p(xk)} as long as N, the number of data points, is not larger than the number of coefficients in the polynomial, i.e., N ≤ 2n+1 (a solution may or may not exist if N>2n+1 depending upon the particular set of data points). Moreover, the interpolating polynomial is unique if and only if the number of adjustable coefficients is equal to the number of data points, i.e., N = 2n + 1. In the remainder of this article, we will assume this condition to hold true.
The solution can be written in a form similar to the Lagrange formula for polynomial interpolation:
This can be shown to be a trigonometric polynomial by employing the multiple-angle formula and other identities for sin ½(x − xm).
Read more about this topic: Trigonometric Interpolation
Famous quotes containing the words solution of the, solution of, solution and/or problem:
“The Settlement ... is an experimental effort to aid in the solution of the social and industrial problems which are engendered by the modern conditions of life in a great city. It insists that these problems are not confined to any one portion of the city. It is an attempt to relieve, at the same time, the overaccumulation at one end of society and the destitution at the other ...”
—Jane Addams (1860–1935)
“The truth of the thoughts that are here set forth seems to me unassailable and definitive. I therefore believe myself to have found, on all essential points, the final solution of the problems. And if I am not mistaken in this belief, then the second thing in which the value of this work consists is that it shows how little is achieved when these problems are solved.”
—Ludwig Wittgenstein (1889–1951)
“What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
—Boris Pasternak (1890–1960)
“If we parents accept that problems are an essential part of life’s challenges, rather than reacting to every problem as if something has gone wrong with universe that’s supposed to be perfect, we can demonstrate serenity and confidence in problem solving for our kids....By telling them that we know they have a problem and we know they can solve it, we can pass on a realistic attitude as well as empower our children with self-confidence and a sense of their own worth.”
—Barbara Coloroso (20th century)