# 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 we provide a good estimation for $(n!)!$?

I was thinking about this $$(n!)!$$ for $n\in\mathbb{N}$. I wanted to find a suitable approximation, or in any case a very good estimation for this. My first idea was to use Stirling ...
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### Converse of Taylor's Theorem

Let $n$ be a nonnegative integer and $a,b\in\mathbb{R}$ such that $a<b$. From Taylor's Theorem, we know that any $n$-time differentiable function $f:(a,b)\to \mathbb{R}$ satisfies the condition ...
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### Using the Saddle point method (or Laplace method) for a multiple integral over a large number of variables

I am trying to understand the saddle point method used in the large N limit of matrix models. First, for the case of the integral of a single variable I found this notes There they say that you can ...
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### Approximation by biholomorphisms

Assuming two domains $\Omega_1 \subset \Omega_2 \subset \mathbb{C}$ satisfy the criteria in Runge's theorem. So we know that any holomorphic function $f: \Omega_1\to \Omega_1$ can be approximated ...
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### Can you help me find a Fourier transform-able approximation function basis for compression?

I have four-dimensional, piece-wise smooth, discrete (4D voxel) data that I want to approximate/ compress using as few basis functions as possible. The data are discontinuous in three dimensions, ...
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### Uniform approximation on $[0,\infty)$

Let $f:[0,\infty) \to \mathbb R$ be a continuous function such that $f(x) \to 0$ as $x \to \infty$. If $\epsilon >0$ then there exists a polynomial $p$ such that $|f(x)-e^{-x}p(x)|<\epsilon$ for ...
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### Approximation of $\sin \pi x$ on $-1 \leq x \leq 1$

I am stuck in the following problem: I wish to give an approximation of $\sin(\pi x)$ on $-1 \leq x \leq 1$, when using the polynomial $$F_N(x)=\sum_{k=0}^{N}a_kx^{2k+1}$$ with the coefficients $a_k$ ...
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### Least squares regression (LSR)

I have the following Least square regression problem: $\underset{rank(X)\leq k}{\arg\min} \lVert DX - L \rVert_F^2$ Let's suppose I compute a QR decomposition of $D$ as $D = Q_DR_D$ and solve the ...
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### Stone Weierstrass on Banach algebras

Let $B$ be a complex Banach algebra. Let now $f\in \mathcal{C}(B)$ and $X$ be a compact subset in $B.$ Is there any version of the Stone Weierstrass theorem which asserts that we can approximate $f$ ...
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### Writing an integral in terms of the Hypergeometric Function

I have the following function defined as an integral: $G(x,k,s) = 1 - (k-1) \int_0^{x/s} (1-t)^{k-2} dt$ Or alternatively directly as: $G(x,k,p) = (1-\frac{x}{s})^{k-1}$ Is there any way (or ...
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### Approximation Theory for Deep Learning Models: Where to Start?

I am working as a novice-developer for company with Deep-learning(DL) Frameworks. DL is basically consists of several layers of combination of linear and non-linear(usually using ReLU) with millions ...
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### Approximate formula from data (7 inputs, 1 output)

I'd like to approximate a formula which 'fits' my data. The formula should take 7 inputs and produce a single ouput. The input variables are listed in columns A to G of my data (see image below). The ...
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### Approximation of dynamic systems

I was looking for a formal approach to simplify models of dynamic systems. Say we have a dynamic system given by $\frac{dx}{dt} = f(t,x,u), ~~~~x(t_0) = x_0$ $y = g(x)$ We know $f$ and $g$ but ...
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### Zolotarev number and commuting matrices

Recently in a post (link) upper bounds on the singular values $\sigma_j(X)$ of a matrix $X$ have been considered. To restate the central observation, it says that if $AX−XB=F$ for $A$ and $B$ normal ...
Let $F$ be a linear space with a norm $\lVert \cdot \rVert$ which is not strictly convex. Show that there is a function $f \in F$ and a subspace $S \subset F$, so that $f$ has different best ...