2
votes
1answer
80 views

Approximation of functions in fractional Sobolev spaces

First, some background and motivation: Convergence estimates in finite elements are often of the form $\|u-u_h \|_{L^2} \leq Ch^m \|u\|_{H^m}$, where $h$ is the mesh norm, $u_h$ is some discrete ...
1
vote
1answer
48 views

Application of Stone-Weierstrass to approximate $f\in C(X\times Y,\mathbb{R})$ where $X$ and $Y$ are compact Hausdorff spaces?

On the wikipedia page for Stone-Weierstrass, the application section (http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem#Applications_2) says If $X$ and $Y$ are two compact Hausdorff ...
1
vote
1answer
52 views

approximate $[0, 1]$ continuous function with 2d basis.

everyone. I've been thinking of this problem when reading papers about Fourier series. I think I can state my question as follows: in the interval $[0, 1]$, I want to approximate an unknown ...
2
votes
1answer
57 views

Proof that Muckenhoupt's $A_q$ Condition Implies $A_p$ for $p<q$?

It is said $f\in A_p$ if it satisfies the following (Muckenhoupt's $A_p$) condition: ...
1
vote
1answer
46 views

Upper Bound of Sobolev norm by $L_2$ norm

A Paper by Madych and Potter states that if a function $f\in W_2^k(\mathbb{R})$ has evenly spaced zeroes (i.e. if $Z(f):=\{x:f(x)=0\}$, is such that $\underset{y\in\mathbb{R}}\sup ...
11
votes
4answers
595 views

Approximating continuous functions with polynomials

Given $\epsilon \gt 0$ and $f \in C^{0}[0,1]$, Weierstrass says that I can find at least one (how many? probably a lot?) polynomial $P$ which approximates f uniformly: $$\sup_{x \in [0,1]} |f(x) - ...
1
vote
1answer
37 views

Proof that interpolation converges; Reference request

I am interested in the mathematical justification for methods of approximating functions. In $x \in (C[a, b], ||\cdot||_{\infty})$ we know that we can get an arbitrarily good approximation by using ...
1
vote
0answers
134 views

Chebyshev Equioscillation Theorem in $L_{\infty}[a,b]$?

Let $a,b\in\mathbb{R}$, $a<b$. Consider \begin{align} C[a,b] & :=\{f\in\mathbb{R}^{[a,b]}:f\text{ is continuous}\}\text{,} \\ L_{\infty}[a,b] & :=\{f\in\mathbb{R}^{[a,b]}:f\text{ is ...
0
votes
1answer
67 views

Expansions of Hermite functions

I am wondering if someone knows good references. I am looking for expansions of Hermite functions, which gives connections between rates of decay and smoothness of coefficients. Thank you for your ...
1
vote
0answers
61 views

Books on Function approximation and Regression

Can you suggest books/articles on Function approximation Let me quote from the above wiki: Second, the target function, call it g, may be unknown; instead of an explicit formula, only a set of ...
4
votes
1answer
203 views

Approximation Theory

What exactly is "Approximation Theory"? If I read the wikipedia-article I doesn't get much clearer. Why are "pure" mathematicians interested in it? I see a lot of people that do harmonic analysis also ...
10
votes
5answers
539 views

Approximation theorems

The Weierstrass' approximation theorem for continuous functions on a compact space by using polynomials is well-known. As far as I know, there are some variants of this theorem, e.g. Stone-Weierstrass ...