Let us consider polynomials as functions on $[0,1]$, and so define \begin{align*} \|f\|_2 &= \sqrt{\int_0^1f(x)^2\,dx} \\ \|f\|_\infty &= \max\{|f(x)|: 0 \leq x\leq 1\}. \end{align*} I am interested in the ratio of these norms. It is easy to see that $\|f\|_2\leq\|f\|_\infty$, with equality only for constant polynomials. In the opposite direction, put $$ f_d(x) = \sum_{i=0}^d \frac{(d+1+i)!}{(d-i)!i!(i+1)!}(-x)^i. $$ Experiments make it clear that $\|f_d\|_2=1$ and $\|f_d\|_\infty=(d+1)$ and that $f_d$ maximises the ratio $\|f\|_\infty/\|f\|_2$ among polynomials of degree $d$. These facts must surely be known. Can anyone point me to a reference? Do the polynomials $f_d(x)$ have a standard name?
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asked Dec 12, 2017 at 16:39
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3$\begingroup$ Have a look here $\endgroup$– user111Commented Dec 12, 2017 at 18:31
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1$\begingroup$ @user111 If you want to promote your comment to an answer, then I will accept it. $\endgroup$– Neil StricklandCommented Dec 12, 2017 at 18:40
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1 Answer
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So, I write this as an answer rather than a comment to close the question.
This is problem VI.103 in volume 2 of Polya and Szego. For the interval $[-1,1]$, the extremal polynomial is $$\frac{P_{n}(x)-P_{n+1}(x)}{1-x},$$ where $P_{n}$ is the Legendre polynomials of degree $n$ (with normalization $\int P_{n}^{2}(x)dx=2/(2n+1)$). Making use of the explicit formula $$P_{n}(x)= \sum _ { k = 0} ^ { n } \binom{ n } { k }\binom{ n + k }{ k } \left( \frac { x - 1} { 2} \right) ^ { k },$$ and the change of variables $x=1-2y$, it is straightforward to check that it gives indeed your formula on the segment $[0,1]$.
