# Questions tagged [approximation-theory]

Approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby.

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### Multi-dimensional uniform approximation results

Recently I've been investigating results from approximation theory, especially the uniform approximation by polynomials. I find most of the interesting results are for one-dimensional, uni-variate ...
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### Does every vector space have a Hamel basis ? And is every linear comination representation finite?

If $V$ is a linear space, then a set $B$ of linearly independent vectors in $V$ that span $V$ is called a Hamel basis for $V$. Does every infinite dimensional vector space have a Hamel basis ? My ...
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### function parameterization with known sums!

I want to find a function, let's say $y= a x + b$ but I don't have sample $(x,y)$ pairs but what I have is samples of following form $((x_1, x_2, ..., x_n), \sum_{i=1}^n y_i)$ where n is also a known ...
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### Robustness of a model to learnt parameters

There is a recent push to study how sensitive a model is to small changes in its input. This has also been studied from an adversarial point of view: e.g what is the smallest input perturbation that ...
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### Wavelet expansions

I'm looking into wavelets to approximate a known square-integrable function $$f(x) = \sum_{j,k} a_{j,k} \times 2^{j/2}\psi(2^j x - k), \qquad a_{j,k}=2^{j/2}\int f(x) \psi(2^jx-k)dx$$ and I'm happy ...
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### Polynomials can approximate the identity (proof)

From Terence Tao's Analysis II: I'm having problems with 14.8.2.(c), since it seems like the choice of $N$ depends on $c$ (and vice versa). The only proof I have relies on the convergence of the ...
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### Approximation theorems and sketches of their proofs

I would like to collect (with support of users here) approximation theorems and sketches of their proofs. Each answer would give one approximation theorem and sketch of a proof. Any other proof of ...
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### Is there a class of functions that have a cyclical or constant derivative?

I'm working on approximating functions in A.I., and I noticed that everyday functions seem to, at some point, have either a cyclical, or constant, derivative. For example, a straight line has a ...
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### Can we approximate elements in $L^2$ via smooth maps while preserving a pointwise constraint on the derivatives?

Let $\mathbb{D}^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball. Suppose we are given an open connected* subset $U \subseteq \mathbb{D}^n$ of full measure in $\mathbb{D}^n$, and a ...
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### Do all convergent sub-sequences of a sequence in a compact set $\mathcal K$ converge to the same element in $\mathcal K$?

I came upon this when trying to understand the proof to "Theorem on Existence of Best Approximations in a Metric Space" as given by Cheney (1981). Let $\mathcal K$ denote a compact set in a metric ...
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### Can we approximate a smooth function by a continuous and nowhere smooth function uniformly?

From Stone-Weierstrass approximation theorem we know that we can approximate a continuous(no matter differentiable or not) by a polynomial function uniformly within a compact interval domain. ...
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### Best approximation of a real number with two functions

There's two functions, called $F(x)$ and $G(x)$, where $F(x)>x,1<G(x) < x$ ,$F'(x)>0, G'(x)>0$ on $(2,\infty)$, and $F(x),G(x)\in(2,\infty)$. Given $x, y\in (2,\infty)$, and now I want ...
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### Approximating a Discrete Function with a Polynomial

Consider a discrete function $f : \{x_1,\dots,x_n\} \to [a,b]$ that, for some $\omega>0$, satisfies $$\max_i |f_i-f_{i-1}| \le \omega. \tag1$$ Let $q$ be the polynomial of order $m<n$ that, ...
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### Monotonicity of function involving ratios of modified Bessel function of first kind

I am a scientist who is trying to come up with some analytic solutions for a system that I only have approximate answers to and I have run into the problem of proving that the following function is ...
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### $||\phi-\phi_\epsilon||_{L^1(\mu)}<\epsilon$ and $||\phi-\tilde{\phi}_\epsilon ||_{L^1(v\mu)}<\epsilon$, then $\tilde{\phi}_\epsilon= \phi_\epsilon$?

Let $\Omega\subset \mathbb{R}^n$ be an open and bounded set. Let $\mu:\mathcal{B}(\Omega)\to [0,+\infty)$ a bounded Radon measure and let $\varphi, \, v \in L^1(\Omega,\,\mu)$, $v\geq 0$. Then $v\mu$ ...
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### “Taylor series” is to “Volterra series” as “Padé approximant” is to _________?

Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it. Volterra ...
### How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?
How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...