# Tagged Questions

**2**

votes

**0**answers

128 views

### When is it possible to split a non-linear operator into a composition of a linear and local one?

Let $A: L^2(R^n)\to L^2(R^n)$ be a non-linear operator. Is it known when it's possible to split $A$ into a composition of a linear operator $B: L^2(R^n)\to (L^2(R^n))^k$ and a local operator $C: ...

**1**

vote

**3**answers

241 views

### What are the basis functions for a product space?

Let $X=L^1\left([0,1]^3\right)$,
for numerical purpose, what are the possible basis function for $X$?
In finite element method, the basis functions are tooth functions, or polynomial functions.
Is ...

**0**

votes

**1**answer

120 views

### How to handle a scalar product in an integral?

I am having a problem with a certain inequality I try to understand. I think it's just a basic idia (/trick) I'm missing, but I can't seem to find it.
Here's a simplification of the problem:
$ ...

**3**

votes

**1**answer

452 views

### Stability in algebraic geometry

Suppose I have a collection of polynomials with multiple variables (more polynomials than variables, say), and I'm given noisy versions the values of these polynomials at a certain unknown point. I ...

**2**

votes

**0**answers

75 views

### Application and relevance of Sobolev gradients

The Sobolev gradient concept has been developed in the 1970s, with a first publication in 1985, and an introduction can be found at: Ranka
I would like to learn how strong the impact of Sobolev ...

**1**

vote

**1**answer

195 views

### How are real-analytic functions encoded in computer algebra?

I would like to know how are encoded the real-analytic functions on the interval by the computers. When I think in a real-analytic function I just think in a composition of the ''typical'' analytic ...

**3**

votes

**1**answer

261 views

### Lebesgue constant as condition number of polynomial interpolation

Let $T = \{ x_0,\ldots,x_n \}$ be a set of $n+1$ different points in the real interval $[a,b]$. Let $X_T$ be the associated interpolation operator on $C[a,b]$: it takes a function $f \in C[a,b]$ into ...

**4**

votes

**1**answer

700 views

### Smoothing L1 norm, Huber vs Conjugate

I'm trying to minimize a convex (not necessarily strictly convex) function involving an L1 norm (similar to lasso), which makes it non-differentiable at some points. So I'd like to smooth it and treat ...

**0**

votes

**1**answer

213 views

### Discrete Sobolev space of $R^n$ valued maps

Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega ...

**0**

votes

**0**answers

122 views

### Can we separate Toeplitz matrices for negative and positive eigenvalues?

Consider a Toeplitz matrix T which has both positive and negative eigenvalues. Can we prove that there exist two Toeplitz matrix T1 and T2 such that T1+T2=T and T1 has only one positive Eigenvalues ...

**3**

votes

**2**answers

209 views

### Convergence rate of an iterative process

I have the following iterative process
$$a_n=a_{n-1}(1-\phi(a_{n-1})),\quad 0< a_0<1,$$
where $\phi(x)$ is a continuous increasing function, $\phi(0)=0$, and if $x\in(0,1)$ then $0< ...

**5**

votes

**0**answers

124 views

### reference for perturbation of projection result

Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...

**5**

votes

**1**answer

391 views

### Approximating high-dimensional integrals by low-dimensional ones

This question is motivated by the following naive one: suppose we have a nice subset $X$ of some Euclidean space, say a polyhedron, and a nice $\mathbb{R}$-valued function $f$ on this subset, say a ...

**13**

votes

**1**answer

777 views

### Who introduced the notion of “stability” in numerical analysis?

I am preparing a lecture course on the applications of operator theory where I intended to make some numerical analysis application. I was wondering about this question while browsing the literature I ...

**3**

votes

**1**answer

413 views

### How to calculate a Fredholm index numerically

How can one calculate the index of a Fredholm operator numerically ?
In numerically calculations one uses always finte dimensional spaces.
But linear operators on finite dimensional spaces have ...

**2**

votes

**4**answers

2k views

### Why do we want to have orthogonal bases in decompositions?

In the decompositions I encountered so far, we all had orthogonal set of bases. For example in Singular Value Decomposition, we had orthogonal singular right and left vectors, in [discrete] cosine ...

**2**

votes

**2**answers

763 views

### Do the Euler method's approximations always approach the true solution?

Let $B$ be a Banach space and $f : [0,+\infty)\times B \to B$ be a continuous function which is Lipschitz continuous in the second argument with Lipschitz constant $L$ (which does not depend on the ...

**2**

votes

**4**answers

330 views

### When do functions near F have zeros near a zero of F?

Consider a sequence of functions $F_n : \mathbb{R}^d \to \mathbb{R}^d$, a function $F: \mathbb{R}^d \to \mathbb{R}^d$, and an $\mathbf{x} \in \mathbb{R}^d$ so that $F(\mathbf{x}) = \mathbf{0}$. In ...

**2**

votes

**4**answers

184 views

### How to compare finite point sets in normed spaces?

I want to define a "distance" between two subsets $A, B$ of a normed space $(V, \|\cdot\|)$ both with (at most) $n$ elements. A straightforward way for me to do this would be to define
$$ d(A, B) := ...