Let $P_n$ be the set of degree $n$ polynomials that pass through $(0,1)$ and $(1,1)$ and are non-negative on the interval $[0,1]$ (but may be negative elsewhere).

Let $a_n = \min_{p\in P_n} \int_0^1 p(x)\,\mathrm{d}x$ and let $p_n$ be the polynomial that attains this minimum.

Are $a_n$ or $p_n$ known sequences? Is there some clever way to rephrase this question in terms of linear algebra and a known basis of polynomials?


2 Answers 2


Following Robert Israel's answer, we also scale everything to $[-1,1]$ (thus multiplying the result by 2). As he mentions, the optimal polynomial is always a square of some other polynomial, $p_{2n}=p_{2n+1}=q_n^2$, and $q_n$ is either even or odd (see Lemma below). So we are left to find the minimal $L_2[-1,1]$-norm of an odd/even polynomial $q_n$ such that $\deg q_n\leq n$ and $q_n(\pm1)=(\pm1)^n$. In other words (recall the division by 2 in the first line!), $a_{2n}=a_{2n+1}=d_n^2/2$, where $d_n$ is the distance from $0$ to the hyperplane defined by $q(1)=1$ in the space of all odd/even polynomials of degree at most $n$ with $L_2[-1,1]$-norm.

Now, this hyperplane is the affine hull of the Legendre polynomials $ P_i(x)=\frac1{2^ii!}\frac{d^i}{dx^i}(x^2-1)^i$, where $i$ has the same parity as $n$ (since we know that $ P_i(1)=1$ and $P_i(-1)=(-1)^i$). Next, by $\| P_i\|^2=\frac{2}{2i+1}$, our distance is $$ \left(\sum_{j\leq n/2}\| P_{n-2j}\|^{-2}\right)^{-1/2} =\left(\sum_{j\leq n/2}\frac{2(n-2j)+1}{2}\right)^{-1/2}=\sqrt{\frac4{(n+1)(n+2)}}, $$ attained at $$ q_n(x)=\left(\sum_{i\leq n/2}\frac{2}{2(n-2i)+1}\right)^{-1} \sum_{i\leq n/2}\frac{2P_{n-2i}(x)}{2(n-2i)+1}. $$ Thus the answer for the initial question is $a_{2n}=a_{2n+1}=\frac2{(n+1)(n+2)}$ and $p_{2n}=p_{2n+1}=q_n^2$.

Lemma. For every $n$, one of optimal polynomials on $[-1,1]$ is a square of an odd or even polynomial.

Proof. Let $r(x)$ be any polynomial.which is nonnegative on $[-1,1]$ with $r(\pm 1)=1$. We will replace it by some other polynomial which has the required form, has the same (or less) degree and the same values at $\pm 1$, is also nonnegative on $[-1,1]$, and is not worse in the integral sense. Due to the compactness argument, this yields the required result.

Firstly, replacing $r(x)$ by $\frac12(r(x)+r(-x))$, we may assume that $r$ is even (and thus has an even degree $2n$). Let $\pm c_1,\dots,\pm c_{n}$ be all complex roots of $r$ (regarding multiplicities); then $$ r(x)=\prod_{j=1}^n\frac{x^2-c_j^2}{1-c_j^2}, $$ due to $r(\pm 1)=1$.

Now, for all $c_j\notin[-1,1]$ we simultaneously perform the following procedure.

(a) If $c_j$ is real, then we replace $\pm c_j$ by $\pm x_j=0$. Notice that $$ \frac{|x^2-c_j^2|}{|1-c_j^2|}\geq 1\geq \frac{|x^2-0^2|}{|1-0^2|}. $$ for all $x\in[-1,1]$.

(b) If $c_j$ is non-real, then we choose $x_j\in[-1,1]$ such that $\frac{|c_j-1|}{|c_j+1|}=\frac{|x_j-1|}{|x_j+1|}$. Notice that all complex $z$ with $\frac{|c_j-z|}{|x_j-z|}=\frac{|c_j-1|}{|x_j-1|}$ form a circle passing through $-1$ and $1$, and the segment $[-1,1]$ is inside this circle. Therefore, for every $x\in[-1,1]$ we have $$ \frac{|c_j-x|}{|x_j-x|}\geq\frac{|c_j-1|}{|x_j-1|}, $$ thus $$ \frac{|x^2-c_j^2|}{|1-c_j^2|} =\frac{|c_j-x|}{|c_j-1|}\cdot\frac{|c_j+x|}{|c_j+1|} \geq \frac{|x_j-x|}{|x_j-1|}\cdot\frac{|x_j+x|}{|x_j+1|} =\frac{|x^2-x_j^2|}{|1-x_j^2|}. $$ So, we replace $\pm c_j$ and $\pm\bar c_j$ by $\pm x_j$ and $\pm x_j$ (or simply $\pm c_j$ by $\pm x_j$ if $c_j$ is purely imaginary).

After this procedure has been applied, we obtain a new polynomial whose roots are in $[-1,1]$ and have even multiplicities, and its values at $\pm1$ are equal to $1$. So it is a square of some polynomial which is even/odd (since the roots are still split into pairs of opposite numbers). On the other hand, its values at every $x\in[-1,1]$ do not exceed the values of $r$ at the same points, as was showed above. So the obtained polynomial is not worse in the integral sense, as required. The lemma is proved.

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    $\begingroup$ Sorry to be dense, but please say why it has to be the square of another polynomial. I can see that any zeros in the critical interval have even multiplicity, but why can't the polynomial be negative elsewhere (as allowed by OP)? $\endgroup$ Oct 9, 2015 at 22:54
  • $\begingroup$ The last part of the answer (after `finally') is about it; I'll try to expand it. In short: If our polynomial has a non-real root, or if it has real roots outside $[-1,1]$, then we may replace these roots by some root on $[-1,1]$ so as to lower the values of the polynomial at all points in $(-1,1)$(and keep it nonnegative). $\endgroup$ Oct 10, 2015 at 7:05
  • $\begingroup$ Expanded (or, I would say, rewritten). $\endgroup$ Oct 10, 2015 at 7:47

For better symmetry, let me change the boundary points to $(-1,1)$ and $(1,1)$. Then it is clear that we may assume $p_n$ is an even function (and thus you really want $P_n$ to be polynomials of degree at most $n$, otherwise you'll be likely to have an infimum rather than an actual minimum): $p_{2k+1} = p_{2k}$. I get $$ \eqalign{p_0(x) = 1, & a_0 = 2\cr p_2(x) = x^2, & a_2 = \dfrac{2}{3}\cr p_4(x) = \dfrac{(5 x^2 - 1)^2}{16},& a_4 = \dfrac{1}{3}\cr p_6(x) = \dfrac{x^2}{16} (7 x^2 - 3)^2,& a_6 = \dfrac{1}{5}\cr p_8(x) = \dfrac{1}{64} (21 x^4 - 14 x^2 + 1)^2, & a_8 = \dfrac{2}{15}\cr p_{10}(x) = \dfrac{x^2}{64} (33 x^4 - 30 x^2 + 5)^2, & a_{10} = \dfrac{2}{21}\cr p_{12}(x) = \dfrac{1}{4096} (429 x^6 - 495 x^4 + 135 x^2 - 5)^2, & a_{12} = \dfrac{1}{14}}$$ Well, so far it looks like $a_{2n} = \dfrac{4}{(n+1)(n+2)}$

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    $\begingroup$ How were these computed? $\endgroup$ Oct 9, 2015 at 0:03
  • $\begingroup$ The leading coefficients look like Catalan numbers when the polynomials are normalized to $\sqrt{2^{2n} p_{2n}(x)}$. $\endgroup$ Oct 9, 2015 at 1:51
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    $\begingroup$ The next coefficients match $-{2n \choose n-2}$. The whole triangle looks like oeis.org/A234950 read skewed, with alternating signs. See oeis.org/A062991 for a signed version. $\endgroup$ Oct 9, 2015 at 3:43
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    $\begingroup$ For $n$ even, $$p_{2n} = \prod_{i=1}^{n/2} \dfrac{(x^2 -c_i^2)^2}{(1-c_i^2)^2}$$ For $n$ odd, there's also a factor of $x^2$. Differentiate $a_{2n} =\int_{-1}^1 p_{2n}(x)\; dx$ with respect to each $c_i$, solve, check which solution gives the least $a_{2n}$. $\endgroup$ Oct 9, 2015 at 4:49

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