# 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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### Does this “inverse Taylor series” exist in literature?

So next in my list of (my) useless ideas here's the idea of an "inverse Taylor series"...enjoy! Let $f$ be an analytic function, so that \begin{equation} f(x)=\sum_{n=0}^\infty a_n x^n \end{equation} ...
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### Prove that $\left\{x^{\lambda_i}\right\}_{i=0}^n$ is a Chebyshev system on $(0,\infty)$

The definition of Chebyshev system is as follows. A linearly independent system of $n$ basis functions $\varphi_1,\ldots,\varphi_n$ on $[a,b]$ is a Chebyshev system on $[a,b]$ if every non-trivial ...
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### How to approximate $f(x)=a\cdot e^{x}+b$?

There is common way in which one can approximate $f(x) = a\cdot e^{bx}$ . Just use $ln$ for both formula's sides and make it linear. What about $f(x) = a\cdot e^{bx}+c$ ? How to determine $a$, $b$ and ...
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### Error bounds for the simple function approximation theorem in the number of level curves

The simple function approximation theorem says that any bounded, measurable function on a compact subset of $\mathbb{R}$ is the point wise limit (almost everywhere) of a sequence of simple functions. ...
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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 ...