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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What is the collection of functions that a given finite neural network can approximate with ease?
To my understanding, one version of the universal approximation theorem runs as follows: Let $\Phi$ be the family of (trained) feedforward neural networks of bounded width, arbitrary depth, and mild ...
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Integral divergence implies summation divergence
Assume $f:(0,1)\times (0,1) \to \mathbb{R}$ is a nonnegative continiuos bivariate function such that
$$\int_0^1 \int_0^1 f^2(x,y) dx dy = \infty,\quad \int_0^1 \int_0^1 f(x,y) dx dy = 1.$$
Can we ...
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Proving the high precision of the series correct to at least half a billion digits
I recently learned about this high-precision series. It is claimed that it is correct to at least half a billion digits. I am curious to know how it works.
$$
\sum_{n=1}^\infty\frac{\left\lfloor n e^{\...
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Problem in A Course in Approximation Theory by Cheney and Light
In chapter 2 of "A Course in Approximation Theory" by Cheney and Light, problem 11 states:
How large can the coefficients be in a polynomial $p$ of degree at most $n$, if $p$ satisfies the ...
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Is the best degree $n$ polynomial approximation an interpolation on $L^2[0,1]$?
The Question:
Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous. Let $p:[0,1] \rightarrow \mathbb{R}$ be the $L^2$-closest degree $n$ polynomial to $f$. That is, $p$ minimizes $\int_0^1|f(x) - p(x)|^...
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Is Kolmogorov-Arnold (representation) neural network dense?
The Kolmogorov-Arnold neural networks (KAN), Ziming Liu et al, KAN: Kolmogorov-Arnold Networks draws inspiration from the Kolmogorov-Arnold representation theorem(KA theorem). However, the former, as ...
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Polynomial Approximation of Piecewise Continuous Functions
I'm looking for results about constructing polynomial approximations of piecewise continuous functions. Specifically, I'm wondering about whether there is a straightforward approach to the following ...
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Approximating integral of product of gaussian and cosines
I am trying to evaluate the following integral
\begin{equation}
I=\int_{-\infty}^{\infty}\exp\left(-\frac{x^2}{b}\right)\prod_i^N\cos{\left(a_ix\right)}dx,
\end{equation}
where $b>0$ and $a_i$ are ...
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$f$ convex $\Rightarrow$ $L(f)\geq f$
Assume that $L:C[a,b]\to C[a,b]$ is a sequence of linear operator acting on the set $C[a,b]$ of continuous functions over [a,b]. Besides, $L(1)=1$ and $L(x)=x$. I recently found that from this is ...
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Approximating the function 1 in Sobolev energy
I would like to find a sequence of smooth compactly supported functions $ u_n \in C^\infty_c([0,+\infty))$ with the following two properties:
$u_n \to 1$ a.e. on $[0,+\infty)$ as $n \to + \infty$;
$\...
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Good approximation of $\sin(x)^5$ to use for ODE?
I need to find an approximate solution of $x'(t)= - \sin(x)^5$ with $x \in [0, \pi]$. I know there's no explicit solution, but I wonder if there are good approximations (whatever that means, let's say ...
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Approximating self-maps of $[0,1]$
Let $\mathrm{Inc}([0,1])$ denote the space of continuous, increasing functions $f:[0,1]\rightarrow [0,1]$ such that $f(0)=0$ and $f(1)=1$. I want to find a countable family of functions $f_n\in \...
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How to approximate any line segment within a circular region using the minimum number of connected rotating axes
This problem arises from my personal experience in developing a game mod. At that time, I wanted to create a drone system for vehicles, but due to the limitations of the game itself, I could only ...
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Explain the proof of Kolmogorov Arnold representation theorem
Can someone explain the outline of proof strategy of Kolmogorov Arnold representation theorem? Any proof of any variant (eg. George Lorentz's variant) would suffice. I would be grateful if you could ...
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Closed subspaces of $L^1(\mathbb{R})$ that is not isomorphic to a subspace of a space with an unconditional basis
It is known that $L^1([0,1])$ does not admit any unconditional basis. Even more, $L^1([0,1])$ is not isomorphic to a subspace of a space with an unconditional basis.
My question is the following ``...
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