# 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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### Convergence speed of discrete approximation

Here I asked the question about approximating the function $g(x) := \mathbb{E}(f(x,Y))$, where $x \in R$ and $Y$ is a random variable. If you follow the link you will see that $g(x)$ can be ...
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### Truncation error with growing step size

When I read about finite difference methods (or really any approximation method), truncation error is often central to the discussion, and rightfully so. But it is also most often discussed in the ...
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### What are some options for adaptive spline approximation of data in 1-D?

What are available options for adaptive spline approximation of data in 1-D? I've some data in a single dimension that I would like to approximate using some kind of spline, preferably a cubic. As ...
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### Approximation of mean of a rational function of random variables

Let $\xi_i$ with $i\in\{1,\dots,n\}$ be iid random variables and let $Q(x,y)$ be a rational function. I need to compute one $x$ that satisfies $$\frac{1}{n}\sum_{i=1}^n Q(x,\xi_i)=0.$$ This is a ...
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### $N$ birds are distributed on a telephone wire

$N$ birds are distributed on a telephone wire that can fit a maximum of $2N$ birds. The spacings between birds form a sequence $S$. The minimum space between birds is $1$ unit. The sequence is ordered ...
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### Tight upper bounds for a monotonically increasing non-linear recurrence

I have the following non-linear recurrence: $$y_{n+1} = \sqrt{\frac{2}{1+y_n}}y_n,\quad y_0 \in[0,1]$$ Some basic thought shows that $0$ and $1$ are fixed points of this, and that $0$ is repelling ...
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### Sign of approximation error and remainder (residual)

In the Wikipedia article Taylor series it is said that: The error incurred in approximating a function by its $n$th-degree Taylor polynomial is called the remainder or residual and is denoted by ...
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### How could I obtain this approximation of the May-Wigner theorem?

I'm trying to understand the complete proof of the May-Wigner theorem. We have a real random $n\times n$ matrix $B$ with its non-zero elements $B_{ij}$ are chosen independiently from a fixed ...
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### Can we approximate continuous functions arbitrarily well with polynomials? (beyond Weierstrass )

Let $f:(0,1) \to \mathbb{R}$ be continuous, and let $\delta:(0,1) \to \mathbb{R}$ be continuous and positive. Does there always exist a polynomial $p(x)$ satisfying $|f(x)-p(x)| < \delta(x)$ for ...
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### Landau inequality for several variables

For $f \in C^n(\mathbb{R})$ and $0 < \alpha < n$, Landau-Kolmogorov inequlity is geven by $$\|f^{(\alpha)}\| \leq K(n,\alpha)\| f\|^{1-\alpha/n}\|f^{(n)}\|^{\alpha/n}, 0 < \alpha < n,$$ ...
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### Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?

Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
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### Approximating the lateral derivatives

Let $f : [0,2] \to \mathbb{R}$ be a continuous function with continuous derivatives of all orders in every point except at $t = 1,$ where the lateral derivatives exist. We know that one can ...
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### Examples of transcendental functions giving almost integers

Informally speaking, an "almost integer" is a real number very close to an integer. There are some known ways to construct such examples in a systematic way. One is through the use of certain ...
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### Optimize a fixed size susbset

So I'm trying to solve this problem: There are many people who apply for jobs at a company. Each applicant has some technical skills required for jobs. The skills possessed by different ...
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### Advanced Methods for Approximating Surfaces based only on partial derivative estimates

I'm looking for information on interpolating a surface function p(x,y) based only on estimates of the partial derivatives at points on a grid. Obviously, any such approximation is subject to a ...