Optimal Polynomial Prediction Measures and Extremal Polynomial Growth
Keyword(s):
Abstract We show that the problem of finding the measure supported on a compact set $$K\subset \mathbb {C}$$ K ⊂ C such that the variance of the least squares predictor by polynomials of degree at most n at a point $$z_0\in \mathbb {C}^d\backslash K$$ z 0 ∈ C d \ K is a minimum is equivalent to the problem of finding the polynomial of degree at most n, bounded by 1 on K, with extremal growth at $$z_0.$$ z 0 . We use this to find the polynomials of extremal growth for $$[-1,1]\subset \mathbb {C}$$ [ - 1 , 1 ] ⊂ C at a purely imaginary point. The related problem on the extremal growth of real polynomials was studied by Erdős (Bull Am Math Soc 53:1169–1176, 1947).
2010 ◽
Vol 08
(05)
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pp. 721-754
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1990 ◽
Vol 147
(2)
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pp. 351-362
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1973 ◽
Vol 1
(2)
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pp. 167-181
1974 ◽
Vol 3
(9)
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pp. 909-912
1975 ◽
Vol 4
(2)
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pp. 193-202