Borel Measure such that integrating a polynomial yields the derivative at a point

Question

Does there exist a signed regular Borel measure such that

$$ \int_0^1 p(x) d\mu(x) = p'(0) $$

for all polynomials of at most degree $N$ for some fixed $N$. This seems similar to a Dirac measure at a point. If it were instead asking for the integral to yield $p(0)$, I would suggest letting $\mu = \delta_0$. That is, $\mu(E) = 1$ iff $0 \in E$. However, this is slightly different and I'm a bit unsure of it. It's been a while since I've done any real analysis, so I've forgotten quite a bit. I took a look back at my old textbook and didn't see anything too similar. If anyone could give me a pointer in the right direction, that would be great. I'm also kind of curious if changing the integration interval from [0,1] to all of $\mathbb{R}$ changes anything or if the validity of the statement is altered by allowing it to be for all polynomials, instead of just polynomials of at most some degree.

Thanks!

Answer 1

Yes there does. In fact you can take any $N+1$ distinct points in $[0,1]$ and have the measure supported there. Suppose your points are $x_1, \ldots, x_{N+1}$. Let $e_j(x)$ be the unique polynomial of degree $\le N$ with $e_j(x_i) = 1$ for $i=j$, $0$ otherwise: explicitly $$e_j(x) = \prod_{i\in \{1,\ldots,N+1\} \backslash \{j\}} \frac{x - x_i}{x_j - x_i}$$
Note that for any polynomial $p$ of degree $\le N$, $p = \sum\limits_{j=1}^{N+1} p(x_j) e_j$. So you can take $\mu = \sum\limits_{j=1}^{N+1} e'_j(0) \delta_{x_j}$ and get $$ \int_0^1 p \ d\mu = \sum_{j=1}^{N+1} e'_j(0) p(x_j) = p'(0)$$

Answer 2

As long as you're only interested in polynomials of degree less than or equal to some fixed $N$, then you can take $\mu$ itself to be a polynomial times $dx$ because you're looking at a finite dimensional inner product space. In this case, you can also regard it as a measure on the line which is supported on an interval.

My guess is that if you want to get polynomials of arbitrary degree, it won't be possible. A proof of that may proceed by something like RobJohn's suggestion -- if the measure were compactly supported, then, by Stone-Weierstrass, its values on continuous functions (and hence Borel sets) would be determined by its values on polynomials. Now the measure you're looking for is assumed to have finite moments of any order, but not necessarily compactly supported.

Answer 3

Yes, there are any number of nice choices. If you take $d\mu(x)$ to be a polynomial (times $dx$), say $\sum_{i=0}^{n}b_i x^{i} dx$, and integrate it against $p(x)=\sum_{i=0}^{n} a_i x^i$, your requirement is that $$ \int_{0}^{1} p(x)d\mu(x) = \int_{0}^{1} \sum_{i,j=0}^{n} a_i b_j x^{i+j} = \sum_{i,j=0}^{n}\frac{a_i b_j}{i+j+1} $$ must be equal to $a_1$. As a matrix equation, this is $$\langle a|\mathcal{\hat{H}}|b \rangle = \langle a | 0,1,0,0,\dots\rangle = \langle a | e_2 \rangle,$$ where $\mathcal{\hat{H}}$ is the Hilbert matrix of order $n+1$. The solution is clearly $b=\mathcal{\hat{H}}^{-1}|e_2\rangle$, or $b_i = (\mathcal{\hat{H}}^{-1})_{i+1,2}$. This has a known closed-form solution, $$ b_{i} = (-1)^{i+1}(i+2)(i+1)^2{{n+i+1}\choose{n-1}}{{n+2}\choose{n-i}} $$ (barring any transcription and renumbering errors), which is interesting in that the desired coefficients are all integers.