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Results tagged with hilbert-spaces
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user 75422
A Hilbert space $H$ is a real or complex vector space endowed with an inner product such that $H$ is a complete metric space when endowed with the norm induced by this inner product.
2
votes
Distribution of the RKHS norm of the posterior of a Gaussian process
Take the easiest example first: $f$ is brownian, $\eta=0$ (no noise) and $x_i=i/n$. Then $\mu_n$ is piecewise linear and $\int_0^1 \mu'_n(x)^2\ dx$ is a sum of $n$ squared unit variance independent Ga …
- 3,085
0
votes
Strictly convex norm on an infinite-dimensional Hilbert space
For an example, take $DN=$ the sum of $\mathbb R I$ and of the set of Hilbert-Schmidt operators $(Au)_k:=\sum_j a_{j,k}u_j$ with $\sum|a_{j,k}|^2<\infty$, and $N(cI+A)^2=c^2+\sum|a_{j,k}|^2$. It makes …
- 3,085
1
vote
Sum of two parts of a continuous stochastic process
Consider the Fourier basis of $L^2(0,2)$: $a_n(t)=cos(n\pi t/2)$ ($n\in \mathbb N$) and $b_n(t)=sin(n\pi t/2)$ ($n\in \mathbb N, n\geq 1$).
If you compute $a_n(t)+a_n(t+1)$ and $b_n(t)+b_n(t+1)$, dep …
- 3,085
12
votes
Accepted
Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\D...
"The" completion is not always a space of functions, for $N=1$ or $2$ for example it is a quotient $D^{-2}L^2/P_1$ (equivalence classes of functions $u\in H^2_{loc}$ with $\partial_i \partial_j u\in L …
- 3,085