fa.functional analysis – Lower-bounding the eigenvalues of a certain positive-semidefinite kernel matrix, as a function of the norm of the input matrix

Let $phi:(-1,1) to mathbb R$ be a function such that

  • $phi$ is $mathcal C^infty$ on $(-1,1)$.
  • $phi$ is continuous at $pm 1$.

For concreteness, and if it helps, In my specific problem I have $phi(t) := t cdot (pi – arccos(t)) + sqrt{1-t^2}$.

Now, given a $k times d$ matrix $U$ with linearly independent rows, consider the $k times k$ positive-semidefinite matrix $C_U=(c_{i,j})$ defined by $c_{i,j} := K_{phi}(u_i,u_j)$, where

$$
K_phi(x,y) := |x||y|phi(frac{x^top y}{|x||x|})
$$

Question. How express the eigenvalues of $C$ in terms of $U$ and $phi$ ?

I’m ultimated interested in lower-bounding $lambda_{min}(C_U)$ in terms of some norm of $U$ (e.g spectral norm or Frobenius norm).


Let $X$ be the $(d-1)$-dimensional unit-sphere in $mathbb R^d$, equipped with its uniform measure $sigma_{d-1}$, and consider the integral operator $T_phi: L^{2}(X) to L^2(X)$ defined by
$$
T_{phi}(f):x mapsto int K_{phi}(x,y)f(y)dsigma_{d-1}(y).
$$

It is easy to see that $T_phi$ is a compact positive-definite operator.

Question. Are the eigenvalues of $C_U$ be expressed as a function of (eigenvalues of) $K_{phi}$ ?

functional analysis – Norm of a linear form in a function space

I would appreciate some help with the following problem. Let $u, u_n:C((0,1))tomathbb{R}$ the linear forms defined by $$u(f)=int_0^1f(x)dxquad&quad u_n(f)=frac{1}{n}sum_{k=1}^nfleft(frac{k}{n}right).$$ I’m trying to prove that $$||u_n-u||=sup_{||f||_infty=1}|u_n(f)-u(f)|=2.$$

I already have that $||u_n-u||leq2$ and I’ve tried to show that the $sup$ is attained (I’m not sure that it is true) by using trigonometric functions but I haven’t been able to get anything. Could anyone give me a hint?

Thanks in advance.

linear algebra – Finding the norm of g

Consider polynomials $f,g in P^5(-1,1)$ satisfying $||f|| = sqrt{6}$, $||f+g||= 4$ and $||f-g||=2$. Where the norm is the $L^2$ norm.

What is $||g||$?

Originally I though to use the triangle inequality on $||f-g||=2$ to get that $||f||= sqrt{6} leq ||g||+2$ but I’m unsure that this is the right way to go. I don’t think we can use the triangle equality on $||f+g||$, we can but I’m unsure that it will help here. What can I do to find $||g||$

fa.functional analysis – Extension of Lipschitz functions that preserve the Frobenius norm of the Jacobian

Let $n,mge 1$ be integers and let $f:Eto R^m$ be $L$-Lipschitz for some subset $Esubset R^n$.

Kirszbraun’s theorem, https://en.wikipedia.org/wiki/Kirszbraun_theorem, states that there exists another $L$-Lipschitz function $bar f:R^nto R^m$ such that $f=bar f$ on $E$ and $bar f$ is called an extension of $f$.

I am wondering about similar extensions of $f$ that maintain the Frobenius norm of the gradient (instead of the operator norm in Kirszbraun’s theorem).

  1. Assume that $E$ is open. By Rademacher’s theorem, the Frechet derivative $D(x)$ (viewed as the matrix of size $mtimes n$ such that $f(x+h)=f(x) + D(x)h + o(|h|)$) exists at almost every $x$.
    Prove or disprove that there always exists an extension $bar f$ such that the essential supremum of the Frobenius norm $|bar D(x)|_F$ of the the Frechet derivative $bar D(x)$ of $bar f$ is no more than the essential supremum of $|D(x)|_F$.

  2. If the answer is positive, for which sets $E$ is it possible to ask for a similar extension? A nice feature of Kirszbraun’s theorem is that it allows for arbitrary sets $E$. Here, however, the Frechet derivatives would not even be defined if $E$ is for instance discrete.

pr.probability – Bounding $l^0$ norm of random quantity

There are many techniques in high dimensional probability for bounding quantities of the form

$$ mathbf{E}( sup_{s in S} X_s ) $$

where ${ X_s }$ are a family of random variables. In my research, I have run into a problem in the complete opposite direction i.e. bounding quantities of the form

$$ mathbf{E}( |{ s in S : X_s neq 0 }| ). $$

If we view the first equation as a bound on an $l^infty$ norm of the family ${ X_s }$, then the second equation can be viewed as a bound on the ”$l^0$ norm” of the family ${ X_s }$. What kind of techniques are there to bound quantities of this form? Is there perhaps a kind of `duality’ result that enables us to study one result in terms of the other? For simplicity, I only need to study such problems where the random variables ${ X_s }$ are integer valued and where $S$ is a finite set (though with the ${ X_s }$ depending on one another in some way that determines a kind of geometry on $S$ as in certain chaining type approaches).

Norm of a functional with restricted domain

Let $X$ be a normed linear space and $Y$ be a subspace of $X$. Let $f$ be a continuous functional and $g$ be a continuous functional such that $g(y)=0$ for all $yin Y$. For all $xin X$ there exist $yin Y$ such that $|f(x)-g(x)|leq |f(y)|$.

$f$ is a extenstion of $f-g$ restricted to $Y$. Maybe I can do use of the theorem of Hahn-Banach.

ordinary differential equations – Solution of dynamical systems bounded in norm

Is there a name for those non-linear dynamical systems whose solutions are not just bounded in norm but where the norms have a behaviour similar to this one? Indeed, the norms should increase for a finite time and then converge asymptotically to the zero solution. This is a sort of relaxation of asymptotic global stability of the origin, where instead of having an immediate decay in the norm we allow it to increase for a finite time interval. enter image description here

how to minimize l1-norm constrained by Frobenius norm

Let’s $A in mathbb{R}^{m times n}$ and $b in mathbb{R}^m $. ┬┐How to reduce that to a linear program?

begin{equation}
underset{x}{text{minimize}} | Ax-b |_1 \
s.t. | x |_{infty} leq 1
end{equation}

summation – How do I code an L2 Norm for a Finite Difference Scheme?

I have to compute the L2 norm of a Finite Difference Scheme applied to a second order differential equation. I set up the code to perform the Scheme with n = 20 and n = 200, and my input is

For(i = 1, i < n, i++, 
 u(i + 1) = (2*h^2 x(i)^2 + 2*h^2 u(i) + x(i)*h*u(i - 1) + 4*u(i) - 
      2*u(i - 1))/(2 + x(i)*h);

I know the formula for the L2 norm is ||U||_Uh = h*Sum((u(i)^2, {0, n}), but I don’t know how to get Mathematica to calculate the sum for me with each of the 20 or 200 u values. Help would be appreciated.

Discovery of norm in PDE

We have seen so many norms we need for PDE. For example, for elliptic PDE, we require a continuous version of $C^k$, i.e. $C^{k,alpha}$. Roughly speaking, under appropriate norm, we could capture the topological information we want. But a question (maybe too vague), how can we know what kind of norm we want in PDE?How can we invent the norm we want? I am just asking for a general idea.