Packet 6: Tossup 19

Gutknecht and Trefethen developed a numerical method for this task that involves studying the eigenvalues of a Hankel matrix, called the Carathéodory–Fejér or “CF” method. This task’s difficulty is characterized by “lethargy theorems” developed by Sergei Bernstein, who provided a constructive proof of a theorem for this task. This task can be done if and only if exponents lie in a combinatorially so-called “large set,” according to the Müntz–Szász theorem. “Best” methods for this task, such as the Remez exchange algorithm, minimize the uniform norm. (10[1])Stone generalized a theorem about this task first described using uniformly (-5[1])convergent (10[1])polynomials (10[2])by Weierstrass. (10[3])This task (10[1])may use a linear combination (-5[1])of (-5[1])Chebyshev (-5[1])polynomials or a truncated (-5[1])Taylor (10[1]-5[1])series. (-5[2])For 10 points, (10[1])what task involves finding a close match (10[3])for (10[1])a given (10[1])function? (10[2])■END■ (10[4]0[2])

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