## 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?

## 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.

## 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.