The kernels tag has no usage guidance.

**6**

votes

**1**answer

298 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 ...

**8**

votes

**0**answers

178 views

### 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=
...

**6**

votes

**0**answers

437 views

### 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 ...

**4**

votes

**0**answers

72 views

### 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 ...

**3**

votes

**0**answers

74 views

### 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 ...

**2**

votes

**0**answers

93 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 ...

**2**

votes

**0**answers

128 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 ...

**2**

votes

**0**answers

152 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 ...

**1**

vote

**0**answers

24 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 ...

**1**

vote

**0**answers

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 ...

**0**

votes

**0**answers

30 views

### Implicit feature space of Power Kernel

For the polynomial kernel, $K(x,y) = (x^Ty+c)^d$, the implicit feature space $\phi$ for which $K(x,y) = \phi(x)^T \phi(y)$ is of finite dimension and well known [1][2].
It is also well known that the ...

**0**

votes

**0**answers

35 views

### Multiple integral of the resolvent kernel

I am trying to implent Volterra equations using resolvent kernel.To do this, the iterative kernel
$$K_i(x,y) = \int\limits_x^y K_1(y,t)K_{i-1}(t, x)dt. $$
should be calculated. However, it is not ...

**0**

votes

**0**answers

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 ...

**0**

votes

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157 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 ...