Paper of the day: 12/03/14

“ON THE MAXIMUM OF THE FUNDAMENTAL FUNCTIONS OF THE ULTRASPHERICAL POLYNOMIALS”, P. ERDOS, 1944

http://www.renyi.hu/~p_erdos/1944-05.pdf

Main theorem of the paper:
Let { -1 \leq x_1 < x_2 < \dots < x_n \leq 1 } be the roots of the ultraspherical polynomial {P_n^{(\alpha)}(x) } with { 0 \leq \alpha \leq 3/2 }. Let { l_k^{(n)}(x) = \dfrac{P_n^{(\alpha)}(x)}{P_n^{'(\alpha)}(x_k) (x-x_k)} } be the fundamental polynomial of the Lagrange interpolation. Then { max_{k = 1,2, \dots, n, -1 \leq x \leq 1} | l_k^{(n)}(x)| =l_1^{(k)}(-1) =l_n^{(n)}(-1)}.

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