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### Duality argument to get $L^\infty-L^2$ inequality

In page 79 of Davies's book on Heat Kernels and spectral theory, the author proves that
$$\lVert e^{-Ht}f \rVert_2 \leq c_1t^{-\mu/ 4}\lVert f \rVert_1$$
where the norms are $L^p$ norms. He states
...

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26 views

### How do we prove that a specific kernel is positive definite (case of logarithm)? [on hold]

I have a problem proving that some specific kernels are positive definite. In general, I can find the answer quickly enough but here I have a specific case involving a logartihm :
$K : ...

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353 views

### Which sections of $T^*M\odot T^*M$ have reproducing kernel “primitives”?

Given a smooth reproducing kernel $\kappa:M\times M\rightarrow \mathbb{R}$ on a manifold $M$, we can construct a section, $\alpha_{\kappa}$, of the symmetric tensor product $T^*M\odot T^*M$ by taking ...

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### On the positive definiteness of RBF kernel with DTW distance

Consider an RBF kernel that is defined as $K(\mathbf{x},\mathbf{y}) = \exp\left(-\dfrac{d^2(\mathbf{x},\mathbf{y})}{2\sigma^2}\right)$, where $d(\mathbf{x},\mathbf{y})$ is usually chosen as the ...

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111 views

### On proof of the conditionally negative definiteness of a kernel

Let the kernel be $f(\mathbf{x},\mathbf{y}) = \arccos(\mathbf{x}^T \mathbf{y})$, where $\mathbf{x}$ and $\mathbf{y}$ are $\ell_2$ normalized vectors of the same dimensionality, and $\arccos(\cdot): ...

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42 views

### Solving Fredholm Integral Equations of the first kind

I am looking to solve a Fredholm Integral Equation of 1st kind of the form:
$\int_{a}^b h(u-u')g(u')du' = f(u)$
where $h$ is a general function.
I know from the Hilbert-Schmidt theorem that all I ...

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175 views

### system with solutions $\{x-a:0\leqslant a\leqslant z-1\}$ [closed]

What must be $F$ there where $0=F(1,x,0)=F(x-0,x,z)=F(x-1,x,z)=F(x-2,x,z)=F(x-3,x,z)=$ $\dots$ $=f(x-z-1,x,z)=0$?
Define $F$ in the domain where a continuous function exists that behaves so for ...

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648 views

### Doubts on Reproducing Kernel Hilbert Spaces and orthogonal decomposition

I'm a CS student and I'm trying to learn RKHS theory to understand the passages made in this paper .
Among the bibliography I'm using there are "On the mathematical fundamentals of learning" and ...

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81 views

### Injective inclusion map from RKHS function space to $L_p(\mu)$

Let $X$ be a measurable space, $\mu$ be a $\sigma$-finite measure on $X$, and $H$ be a separable reproducing kernel Hilbert space over $X$ with a measurable kernel $k$.
At a certain part in a proof I ...

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### Measurability for disintegration of a kernel

Let $(x, A) \mapsto P(x, A)$ be a probability kernel whose "target" (wikipedia terminology) is a product space $Y \times Z$, and say both $Y$ and $Z$ are compact metric spaces. For every $x$ there is ...

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923 views

### Proving that a specific kernel is positive definite

Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite.
We are interested in ...

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55 views

### Efficient evaluation of multidimensional kernel density estimate

Edit I have copied this discussion to the stats community site here, since I feel it is more relevant. Please feel free to close this in due course.
I've seen a reasonable amount of literature about ...

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250 views

### Karhunen-Loeve expansion for discrete-time process

Is there a Karhunen-Loeve theorem for discrete-time process?
For example, let $\left\{X_i\right\}$ be a sequence of independent random variable which are uniformly distributed on the set $\{-1,1\}$. ...

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97 views

### Is every covariance operator the covariance of a measure?

Let $X$ be a topological linear space over $\mathbb R$ which is complete and Hausdorff with a dual space that separates points. Let $k : X^* \to X$ be an arbitrary covariance operator. i.e., any ...

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261 views

### Do kernels provide a basis for a RKHS?

Let $H$ be a Reproducing Kernel Hilbert Space with elements $f:X\rightarrow \mathbb{C}$, with kernel $K(x, y)$. My question is whether, for some choice of $x_i\in X$, it is the case that $u_i:=K(x_i, ...

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### Level sets of linear combinations of Gaussians

I am trying to work out whether level sets of linear combinations of Gaussian functions are unique.
For a given integer $n\ge 1$, fix $n$ points $x_i\in\mathbb{R}^d$ and $\sigma>0$. Let ...

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351 views

### Kernel of AB if $[A,B]=0$ and $AB\neq0$? [closed]

I have found similar results here and mathematics stack exchange but they all imposed specific conditions that don't suit this problem in particular. The problem is as follows.
Let A,B be square ...

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135 views

### Gaussian upper bounds on Heat kernel

Let $M$ be a complete Riemannian manifold with Ricci curvature bounded below and let $k(t, x, y)$ be the heat kernel of the Laplace-Beltrami operator.
It seems to be standard that for any compact ...

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### Operators from $L^{\infty}$ to $L^{\infty}$

If $T$ defined as $T(f)(x)=\int K(x,y)f(y)dy$, where $K(x,y)$ is locally integrable, is bounded from $L^{\infty}$ to $L^{\infty}$, how can we show that $\|\int|K(\cdot,y)|dy\|_{L^{\infty}}\le ...

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113 views

### Rank of Conjugate Closure of a Subset

How does one find the rank of a conjugate closure of a subset?
In particular, I am studying this group: Let $K_n$ be the group with $n$ generators $x_1,\cdots,x_n$ satisfying the relation that each ...

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### Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using >this< formula.
The following type of matrix has a similar structure, and should also have a one-dimensional kernel.
$V=
...

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159 views

### Finite rank free modules over PIDs

I have two finite rank free modules $M,N$ over a principal ideal domain (the ring of formal complex power series to be exact) and a homomorphism between these two modules
$\phi:M\rightarrow N$. Under ...

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405 views

### Integral kernel of form $e^{-<x,y>^2}$

Let $K(x,y): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ given by $K(x,y) = e^{-< x,y>^2}$ where $<\cdot,\cdot>$ denote the canonical inner product. Define integral operator ...

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514 views

### Eigenfunctions and eigenvalues of the product of two exponential kernels

Consider the following exponential kernel:
$k(x_1, x_2) = \exp\left(\frac{|x_1 - x_2|}{L}\right)$,
which is symmetric and non-negative definite. By virtue of Mercer's theorem, we have
$k(x_1, x_2) ...

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157 views

### Cameron-Martin like RKHS

Hello,
I know that $k(x,y)=min(x,y)$ is the reproducing kernel of the Cameron Martin space of all i.i.d. RVs of Brownian motion at different times, with the $cov$ inner product.
What is the RKHS ...

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374 views

### Does anybody know an estimation of L4 norm of fejer kernel ?

Hi, I need an estimation or an exact closed form expression for the following integral
$\int_{0}^{2\pi} K_N^4(s) ds $
where $K_N(s)= \frac{1}{N2\pi} (\frac{sin(Ns/2)}{sin(s/2)})^2$, the Fejer ...

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### The log kernel and Bochner Theorem

I was wondering if it possible to find a measure $\eta$ on $\mathbb{R}$ such that
$$
L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t)
$$
for every $x\in [0,1/2]$.
On a structural ground, this ...

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890 views

### Kernel width in Kernel density estimation

Hi,
I am doing some Kernel density estimation, with a weighted points set (ie., each sample has a weight which is not necessary one), in N dimensions.
Also, these samples are just in a metric space ...