# Tagged Questions

1answer
52 views

### Existence of periodic orthogonal basis in $L^2([0,1])$ which is not trigonometric?

Let $$\psi(x) := \sin(\pi x).$$ It is well-known that system $\{ \psi(n x) \}_{n \in \mathbb{N}}$ forms an orthogonal basis in $L^2([0,1])$. My question is the following: Are there other examples ...
2answers
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### Asymptotic approximation of binomial theorem

Binomial theorem is a very popular theorem that: $$(x + y) ^ n = \sum_{i=0}^n {n \choose i}x^i y^{n-i}$$ I am looking for any papers (the newer the better) where I can find any informations about ...
2answers
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4answers
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### 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) - ...
1answer
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### 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 ...
0answers
161 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 ...
1answer
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### 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 ...
0answers
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### 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 ...
1answer
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### What is “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 ...
5answers
559 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 ...