A note on some peculiar nonlinear extremal phenomena of the Chebyshev polynomials

Details A note on some peculiar nonlinear extremal phenomena of the Chebyshev polynomials A note on some peculiar nonlinear extremal phenomena of the Chebyshev polynomials

1994-06-07

arxiv.org/abs/math/9406222v1

Mathematics

Classical Analysis and ODEs

Scientific paper

We consider the problem of maximizing the sum of squares of the leading coefficients of polynomials $P_{i_1}(x),\ldots ,P_{i_m}(x)$ (where $P_j(x)$ is a polynomial of degree $j$) under the restriction that the sup-norm of $\sum_{j=1}^m P_{i_j}^2(x)$ is bounded on the interval $[-b,b]$ ($b>0$). A complete solution of the problem is presented using duality theory of convex analysis and the theory of canonical moments. It turns out, that contrary to many other extremal problems the structure of the solution will depend heavily on the size of the interval $[-b,b]$.

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