# 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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### Smooth approximation of Hölder functions "from below"

We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1].$ I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\mathbb R)$ such ...
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### Approximation of pseudogeometric progression

Let $f_n(x)=1+x+x^{\sqrt{2}}+x^{\sqrt{3}}+x^{\sqrt{4}}+\cdots+x^{\sqrt{n}}$ be a sequence of functions on the interval $[0, 1]$. Is there a good closed form approximation for such a function ( ...
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### Rate of uniform approximation by piecewise constant functions

Definitions and Notation: Fix a positive constant $M>0$ with positive integers $m,n$ and the standard orthonormal basis $e_1,\dots,e_n$ of $\mathbb{R}^n$. For every positive integer $N$, define the ...
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### Uniform norm bounds for linear approximation of 1-Lipschitz functions

This problem seems like it should be quite easy/standard, but I've not found a solution written down anywhere. Consider the set of 1-Lipschitz functions on the $[0,d]$ interval. Define the linear ...
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### Show that $\frac{1}{n} \sum_{i=1}^n a_i \operatorname{erf} \left( \frac{b_i-x}{\sqrt{2}} \right) \to x$ for some sequence $\{a_n\}$ and $\{b_n\}$

Consider the following function \begin{align} f_n(x)=\frac{1}{n} \sum_{i=1}^n a_i \operatorname{erf} \left( \frac{b_i-x}{\sqrt{2}} \right) \end{align} where $\operatorname{erf}$ is the error ...
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### A function is of bounded variation if and only if the errors of its best approximation by trigonometric polynomials satisfy $\sum\frac{e_n}n<\infty$?

Let $\mathcal P_n$ be the set of trigonometric polynomials of degree less than or equal to $n$ and let $\lVert\cdot\rVert_\infty$ be the supremum norm. The error of the best approximation of $f$ of ...
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### Under what conditions is the least-squares approximation bounded with the same Lipschitz gradient constants?

Let $f(x):\mathbb{R}^K\Longrightarrow \mathbb{R}^L$ denote a multivariate continuously differentiable function. All the partial derivatives of $f$ (all its Jacobian elements) are bounded from above ...
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Is there any result like the Bramble-Hilbert lemma for Bochner spaces? More specifically: let $H$ be a (e.g.) Hilbert space, $I\subset \mathbb R$ a bounded interval, and $L \in \mathcal L(H^k(I;H), Y)... 3 votes 1 answer 287 views ### Smooth approximation of the$\max\{0,x\}$function with controlled derivatives Motivation/Hand-Wavy Question: In this post, it was asked what the best local approximation of$f(x):=\max\{0,x\}$is by a polynomial of a given degree; with the answer provided by Chebyshev's ... 3 votes 0 answers 144 views ### Chebyshev-like polynomials [closed] In some approximation problems I'm working on, the errors turned out to be polynomials of various degrees whose graphs on the interval$[-1,1]$look like this: As you can see, these things look a bit ... 1 vote 0 answers 72 views ### Does a matrix product have an upper bound on the largest coefficient? Let$A$and$B$be two$n\times n$random matrices. Matrix$A$has coefficients taken from a normal distribution$ \mathcal{N}(\mu_A,\sigma_A)$, and matrix$B$has coefficients taken from$ \mathcal{N}...
Main Question Let $f:[0,1]\to [0,1]$ be continuous, let $B_n(f)$ be the $n$-th degree Bernstein polynomial of $f$, and let $r\ge 3$. Given certain assumptions on $f$, what is an explicit and tight ...