# Questions tagged [kernels]

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### Sum of squared Dirichlet kernels

Consider the sum $$S_n=\sum_{h=1}^{n-1}D_{q_n}^2\Bigl(\frac{2\pi h}n\Bigr) =\sum_{h=1}^{n-1}\frac{\sin^2((2q_n+1)\pi h/n)}{\sin^2(\pi h/n)},$$ where $D_{q_n}$ is the Dirichlet kernel and $q_n$ is a ...
• 401
1 vote
103 views

### Estimator for the conditional expectation operator with convergence rate in operator norm

Let $X$ and $Z$ be two random variables defined on the same probability space, taking values in euclidian spaces $E_X$ and $E_Z$, with distributions $\pi$ and $\nu$, respectively. Let $L^2(\pi)$ ...
• 111
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### Gaussian lower heat kernel bounds on non-convex bounded domain

I am looking for a proof the following theorem. Let $U \subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary and $p(x,y,t)$ be the Neumann heat kernel. Then there exist a constant $C>0$ ...
• 61
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### Orthonormal bases in RKHSs via interpolating sequences

Definitions and setting Let $\mathcal{H}$ be a separable, infinite-dimensional, reproducing kernel Hilbert space on a nonempty set $X$. As usual, denote the reproducing kernel on $\mathcal{H}$ by $K$ ...
• 4,961
1 vote
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• 1,866
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### In the proof of Neural Tangent Kernel stays constant in infinite width limit, why the norm of the dual mapping operator equals operator norm of kernel

For a fixed distribution $p^{in}$ on the input space $\mathbb{R}^{n_0}$, consider a function space $\mathcal{F}$ defined as $\{{f: \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_L}}\}$. On this space, ...
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### Pair of positive harmonic functions with negative inner product in Drury-Arveson space

Define a reproducing kernel on the Euclidean ball in $\mathbb{C}^d$ by $$k(z,w)=\frac{1}{1-\langle z,w\rangle}+\frac{1}{1-\langle w,z\rangle}-1.$$ Call the corresponding real reproducing kernel ...
• 1,429
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### Sufficient conditions to ensure that a function $P(x,y) := \langle \pi(x),\pi(y)\rangle$ can be represented as $P(x,y) = \phi(\langle x,y\rangle)$

Let $n$ be a positive integer. Give a nonempty collection $\mathcal S$ of subsets of $[n] := \{1,2,\ldots,n\}$, define an nonlinear dot-product by \begin{eqnarray} \langle x,y\rangle_{\mathcal S} := \...
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