Express a sum of dot products via dot product I am going to find somehow explicit feature mapping for Gaussian kernel in SVM for one dimensional feature space, i.e. find $\boldsymbol \phi \colon \mathbb R \to \mathbb R^\infty$ such that:
$$
\forall x,y\in \mathbb R: \exp (-\gamma |x - y|) = \boldsymbol \phi (x)\cdot \boldsymbol \phi (y)
$$
My suggestions:
$$
\exp (-\gamma |x - y|) = \sum _{n=0}^\infty \frac{(-\gamma )^n}{n!} \left|\sum _{k=0}^n \binom nk x^ky^{n-k}\right|=\ldots
$$
Can one please help me with that?
 A: I don't see a way to get rid of the absolute value in your suggestion. 
I would take another route: There is a general statement (see for instance Th. 6.11 in [1], it is a refinement of Bochner's theorem) that gives the structure of the Hilbert space associated to translation invariant kernels. Namely, if $\kappa$ is a continuous $L^1$ positive definite function, let us define
$$
\mathcal{H} = \bigl\{f\in L^2,\; \mathrm{T}f/\sqrt{\mathrm{T}\kappa}\in L^2\bigr\},
$$
where $\mathrm{T}f$ is the Fourier transform of $f$. Define the bilinear form
$$
\langle f,g\rangle_{\mathcal{H}} = \frac{1}{2\pi}\int \frac{\mathrm{T}f(\omega)\overline{\mathrm{T}g(\omega)}}{\mathrm{T}\kappa(\omega)}\,\mathrm{d}\omega.
$$
Then $\mathcal{H}$ is a Hilbert space with inner product $\langle\cdot,\cdot\rangle_{\mathcal{H}}$ and reproducing kernel $\kappa(\cdot - \cdot)$.
In our setting, $\kappa(x)=\exp(-\beta \lvert x\rvert)$ (and the associated kernel is often called the Laplace kernel, or Laplacian kernel). It is a positive definite function, continuous and integrable, hence we can apply our result:
$$
\mathcal{H} = \bigl\{f\in L^2,\; \int \lvert \mathrm{T} f(\omega)\rvert^2 (\omega^2+\beta^2)\,\mathrm{d}\omega < \infty\bigr\},
$$
endowed with
$$
\langle f,g\rangle_{\mathcal{H}} = \int \mathrm{T}f(\omega) \overline{\mathrm{T}g(\omega)}(\omega^2+\beta^2)\,\mathrm{d}\omega.
$$
[1]: Holger Wendland, Scattered data approximation, Cambridge University press (2005)
