# 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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### Error analysis of approximating Fourier transforms

Consider the problem of computing the Fourier transform of a function, $f(x).$ $$\hat{f}(k) = \int_{-\infty}^{\infty} dx~ f(x)~ e^{i k x} .$$ Suppose I want to approximate this transform by a ...
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### Approximation of conditional expectation of unknown function

I am given a multidimensional markovian stochastic process $X_1,X_2,...X_n$ with continuous state space and unknown to me function $V$. I want to approximate expectation $E(V(X_k)|X_{k-1} = x)$ ...
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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 ...
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### How does Weierstrass' theorem follow from Mergelyan's theorem?

According to Theorems 1 and 3 in this review article we have Weierstrass: Suppose $f$ is a continuous function on a closed bounded interval $[a,b] \subset\mathbb{R}$. For every $\epsilon > 0$ ...
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### Applying Bishop’s Theorem to $\langle xy;x^2y\rangle$

I am using Bishop’s Theorem in the version given by Wikipedia¹: Let $\mathfrak{A}$ be a closed subalgebra of the Banach space $C(X,ℂ)$ of continuous complex-valued functions on a compact Hausdorff ...
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### Closed form of this product or approximate?

What is the closed form of this product: $$\prod_{i=1}^{k-1}\left(1-e^{-a(b- ic)^2}\right)$$ where $a,b,c$ are constants?
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### tan nx is not as popular as Chebyshev polyomials?

I noticed that $\tan(nx)$ can be written as the ratio $P(\tan x) / Q(\tan x)$, where $P$ and $Q$ are polynomials. Indeed, I found a short proof here Expansion of $\tan(nx)$ in powers of $\tan(x)$ ...