# 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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### Can a neural network with ReLU activation represents exactly all $B$-bounded and $L$-Lipschitz $K$-max-affine functions?

A max-affine function is defined as the maximum over a set of affine functions, which is always convex. More specifically, we define a $K$-max-affine function $f:\mathbb{R}^d\to\mathbb{R}$ that can be ...
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### Relation between the local maxima and the local minima for approximating the generalized Laguerre polynomial

I have already asked my question in the link below: Minima approximation for Laguerre polynomials I have suggested to anyone to give me the approximations of the minima for the Laguerre polynomial, ...
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### Find weak approximation by smooth unit vector fields for Sobolev fields on manifold

I am considering the Sobolev space of unit tangent vector fields on a compact manifold: $Γ_{W^{1,2}}(M, UTM)$. I would like to approximate those weakly with smooth vector fields ($Γ_{C^∞}(M, UTM)$). ...
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### Series expansion for gaussian-like function

I need a series expansion to describe a general gaussian-like (bell shaped) function. I couldn't find a rigorous definition of "bell shaped" online but in essence the function should have ...
1 vote
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### Distribution of quadratic polynomials mod $n$ and $n^2$

Suppose $n$ is odd, then both equations $x^2 = D \; mod \;n$ and $x^2 = D \; mod \;n^2$ have the same number of solutions for fixed $D$ coprime to $n$. What can be said about the relationship between ...
1 vote
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### Fast decaying Fourier coefficients for indicator function

Let $0 \leq a < b \leq 1$. I wanted to compute the Fourier series expansion of the indicator function $f = \chi_{[a, b]}$ of the interval $[a, b]$, as $f(x) = \sum_{k\geq 0}a_k e(kx)$. My ...
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### Approximating exp(x) with a piecewise-linear function accurately

I am looking for the best way to approximate $\exp(x)$ on a finite domain $[0,M]$ with a piecewise-linear function. My initial approach is to take $K$ evenly-spaced segments from $0$ to $M$. For each ...
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### How to numerically compute $x \ln x$ and related functions near $0$?

I was recently trying to find a numerical solution to a thermodynamics problem and the expression $x\ln x$ appeared in one of the computations. I did not have to find its value very near $0$, so the ...
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