# 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 “right” way of approximating random variables with other random variables?

It is well-known that one can construct a Hilbert space of zero mean, finite variance random variables such that every random variable of this type $X$ can be represented as a linear combination of a ...
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### Are of Polynomials of Continuous functions complete?

The Stone-Weierstrass Theorem guarantees that every continuous function defined on a closed interval [a,b] can be approximated as closely as desired by a polynomial. I'm curious in polynomials of ...
1answer
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### Approximating the basis of a specific function

We are given a continuous function $g: A \to B$, where $A, B$ are compact subsets of $\mathbb{R}$. We define a function $f(x) := g(b_1x)+g(b_2x)+...+ g(b_mx)$, where $b_i < 1$ and $b_ix$ is a ...
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### Number of simple functions needed to approximate another function

I have got a function $f$ which maps from one compact space to another. Function $f$ is smooth. I want to approximate it with some simple functions (e.g polynomials). Is there any theory that gives ...
2answers
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### Optimal Fixed-Digit Rational Approximations of $\pi$

Is there a systematic way of finding optimal rational approximations to $\pi$ whose numerator and denominator have at most $n$ digits? More precisely: Let $D_n$ be the set of all positive integers ...
2answers
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### Expressions approximating Generalized Harmonic Number (truncated polynomials with shrinking error term preferred)

Specifically, $$H_m^{(2n)} \approx\ ?$$ and $$H_m^{(4n)} \approx\ ?$$ where $(m, n)$ $\in \mathbb N_{>1}$ I would not like to use special functions like the (Riemann zeta function) unless they ...
0answers
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### Elementary approximations to $\zeta(s)?$

What are the best approximations in terms of elementary functions of one real variable for: $$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s},$$ for $Re(s)>1?$ There is not an elementary function that ...
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### Uniform approximation of $L^2$ basis by smooth functions with bounded derivatives of all orders

Let $\mathcal{F}=\{f_i\}_{i\in\mathbb{N}}$ be an orthonormal Hilbert basis of $L^2[0,1]$. I am wondering whether it is possible to approximate the $f_i$ uniformly across $i$ in the $L^2$-norm by ...
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### Can $x^n$ be uniformly approximated by the combination of $x^{k^2}$?

For each $n\in\mathbb{N}$ , can $x^n$ be uniformly approximated by the linear combination of $\left(x^{k^2}\right)_{k\in\mathbb{N}}$ ? In order to facilitate a solution, we might as well try to ...
1answer
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### Relating the Nonlinear and Linear operators in the Homotopy Analysis Method

The question refers to chapter two of the book Liao, Shijun. Homotopy analysis method in nonlinear differential equations. Beijing: Higher Education Press, 2012. Link to book pdf from Chinese .edu ...