Integral of Hermite Polynomials.

Question

Let $f \in L^2(\mathbb{R}, (1/\sqrt{2 \pi})\exp(-x^2/2))$ be such that $0 \leq f(x) \leq 1$. We know that the (normalized) Hermite polynomials are a complete orthonormal basis for this space. Therefore we let \begin{equation} f_n(x) = \sum_{i=0}^n \langle f, H_i\rangle\ H_i(x) = \sum_{i=0}^n c_i H_i(x), \end{equation} where \begin{equation} c_i = \langle f, H_i \rangle = \int H_i(x) f(x) \frac{1}{\sqrt{2 \pi}} \exp(-x^2/2)\ \mathrm{d} x = \int H_i(x) f(x)\ \phi(x) \mathrm{d} x, \end{equation} where $\phi(x) = (1/\sqrt{2 \pi}) \exp(-x^2/2)$. Consider now the integral \begin{equation} d_i = \int H_i(x) f(x)\ \phi(x-\mu) \mathrm{d} x \end{equation} $d_i$ is the Hermite coefficient with respect to a translated Gaussian.

Question: Is there an upper bound for $|c_i - d_i|$? Ideally it would be something in the form $|c_i - d_i| \leq A c_i$.

My guess would be that there should exist some fixed constant $A$ such that \begin{equation} |c_i - d_i| \leq A |c_i| \exp(\mu^2). \end{equation}

I know that this question could be formulated only in terms of integrals but I thought that providing more context could be helpful.

Any pointers to relevant literature are welcome.

Answer 1

A partial answer. We have

$$ c_i-d_i = \int H_i(x)f(x)\left(1-e^{\mu(x-\frac{\mu}{2})}\right)\phi(x)dx $$ hence by the Cauchy-Schwarz inequality $$ \left|c_i-d_i\right|\leq \sqrt{\left(e^{\mu^2}-1\right)\int H_i(x)^2f(x)^2\phi(x)\,dx}. $$ Since $f(x)\in[0,1]$ the inner integral is bounded by $$ \int H_i(x)^2 \phi(x)\,dx = 1$$ and $\left|c_i-d_i\right|\leq e^{\mu^2/2}$, which maybe is enough for your purposes.
I am not sure on the route to follow for deriving an upper bound involving $|c_i|$, but this past question may be useful in the localization of the zeroes of $H_i(x)$.