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# Questions tagged [hermite-polynomials]

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### Approximation to the $n$-th derivative using reproducing kernels.

For integrable functions defined on the real line, the normalized gaussian function approximates the convolution identity, Dirac Delta, in the sense that if $$g(t):=N_0e^{-x²}$$ (denoting the ...
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### Can $\frac{d^k}{dx^k} e^{\frac{x^2}{2}}$ be written in terms of Hermite polynomials?

We know that $\frac{d^k}{dx^k} e^{-\frac{x^2}{2}}$ can be written in terms of Hermite polynomials as \begin{align} \frac{d^k}{x^k} e^{-\frac{x^2}{2}}= (-1)^k e^{-\frac{x^2}{2}} H_{e_k}(x) \end{align}...
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### Inner product of Hermite polynomial

How to prove the Fourier Hermite series $$\int H_n(x) f(x) \phi(x) dx=\int f^{(n)}(x) \phi(x) dx$$ where $\phi(x)=\frac{1}{\sqrt{2\pi}} e^{-x^2/2}$, $f^{(n)}(x)$ is the $n$th derivative of the ...