# 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 construct a computable sequence of trigonometric polynomials that converges pointwise to a given continuous function defined on the Torus?

Consider any continuous function $f$ on an $m$-dimensional Torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the ...
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### Can we construct a sequence of trigonometric polynomials that converges pointwise to a given continuous function on the Torus?

Consider any continuous function $f$ on an $m$-dimensional Torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the ...
225 views

### Chebyshev rational approximation of $e^{x}, x >0$: does it exist?

It's well known that the scalar function $e^x$, for $x \in (-\infty,0]$ can be approximated by Chebyshev rational approximation. In practice, one wants to use a partial fraction decomposition form ...
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### Expected value of a truncated binomial

Let $X\sim B(n,p)$ be a binomial random variable and fix $0<k<n$. Are there any well-known bounds for $\mathbb{E} (X-k)^+$, where $(X-k)^+ =\max\{0,X-k\}$? I am particularly interested in ...
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### A uniform mixture of order statistics

Let $0<k<n$ be integers, and let $X$ be a random variable obtained as follows: sample $n$ points independently and uniformly at random in the unit interval, and select (uniformly) one of the $k$...
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### Uniform Lipschitz function approximation by shallow neural networks

Fix $d\in \mathbb{N}$. Let $F_1$ be the set of all 1-Lipschitz functions mapping $[0, 1]^d$ to $\mathbb{R}$. For $\varphi: \mathbb{R} \rightarrow \mathbb{R}$ and $m \in \mathbb{N}$, let $N_\varphi^m$ ...
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### Finding the upper bound and uniform convergence

For every positive integer $m$, let $x_0^{(m)}$ < $x_1^{(m)}$ < ... < $x_m^{(m)}$ be $(m+1)$ distinct points in $[0, \pi]$ and let $p_m$ $\in$ $P_{m+2}$ be the Hermite interpolant of the ...
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### Polynomial satisfying a functional equation [closed]

I am currently stuck with the following question: Let $q$ be a polynomial of degree $n+1$ with distinct positive zeros $x_0, ... , x_n$. Find a polynomial $p \in P_n$ that satisfies the functional ...
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### Upper bound for eigenvalue of symmetric kernel

Let $V \in L^2(D \times D)$ be symmetric kernel defining the compact and nonnegative integral operator \begin{equation}\mathcal{V}: L^{2}(D) \rightarrow L^{2}(D), \quad(\mathcal{V} u)(x)=\int_{D} V\...
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### Hardness results for approximating Hölder continuous functions

Let $f \in \mathrm{Lip}^{L,\alpha}[a,b]$, and let $f_{h} \in C^{L}$ be a spline which interpolates $f$ at $a + ih$. Then standard theorems show that \begin{align*} \left\| f - f_{h} \right\|_{\infty} \...
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### Reference request: Projection operators in metric spaces

Given a metric space $(X,d)$ and a subset $S\subset X$, the projection $P_S$ onto $S$ is well-defined as a set valued function. I am interested in learning more about properties of these projections ...
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### Example of a nonconvex Chebyshev set in a metric space with continuous projection?

Question: Is there an example of a nonconvex Chebyshev set $S$ in a metric space $(X,d)$ whose projection map is continuous? For convexity to be well-defined, we need to assume that $X$ is a vector ...
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Question: What is the state of the art on analytic approximations of $\min$ and $\max$? My hunch is that numerical analysts probably have a better solution than the one I propose here. For any $\... 0answers 47 views ### Approximation of a Sobolev map with fixed singular values by smooth maps with the same singular values Let$0<\sigma_1<\sigma_2$, and let$D \subseteq \mathbb{R}^2$be the closed unit disk. Let$f \in W^{1,\infty}(D,\mathbb{R}^2)$, and suppose that the singular values of$df$are a.e. equal to ... 1answer 302 views ### Lower-bound for$\Pr[X \geq m]$subject to$E[X]>m$where$X$is a binomial random variable Given an integer number$m>0$and a real number$\alpha\in [1, 2]$, I am interested in finding a lower-bound for$\Pr[X\geq m]$subject to$X \sim \text{Binomial}(n, m\alpha/n)$. For large values ... 0answers 75 views ###$L^p$-convergence of truncated Legendre series approximation to inverse error function$\text{erfinv}$Bounding the$L^p$-error for an$n$-th order Legendre series approximation I have function$f\,\colon (-1, 1) \to \mathbb{R}$where$f \in L^q(-1, 1)$for any$q \geq 1$. I approximate$f$using a ... 2answers 72 views ### Interpolation nodes for linear spline (piecewise-linear) interpolation of$x \ln x$I need to approximate$x \ln x$on$[0,1]$as a piecewise-linear function. If$P(x)$is a piecewise-linear approximation, I want to minimize $$\max_{0 \le x \le 1} |P(x) - x \ln x| \rightarrow \min_P.... 0answers 38 views ### Probabilistic Approximation of non-linear Dynamical System by Diffusion Process Setting Suppose I have a discrete dynamical system given by:$$ X^{n+1} = f(X^{n}) \qquad X^0 =x , $$where f is some diffeomorphism from \mathbb{R}^{d} to itself, and some x \in \mathbb{R}^d. ... 1answer 201 views ### Polynomial approximation (in L^1 norm) at minimal cost Define the cost of a polynomial \sum_{i=0}^N a_i x^n to be \sum_{i=0}^N |a_i|. Let g:[0,1]\to \mathbb{R} be a function to be approximated — say, g(x)=0 if 0\leq x < e^{-1}, g(x)= 1/x ... 1answer 70 views ### Convergence of Chebyshev interpolation in L^1 Let f\in C^0([-1,1]) and P_n(f) its interpolation polynomial at the Chebyshev nodes. I would be interested to know about any existing results (positive or negative) about the convergence of P_n(... 0answers 49 views ### Any good references on the decay rate of Legendre coefficient? Let P_n:[-1,1]\rightarrow\mathbb{R} be the n-th Legendre Polynomial. and, let$$a_n:=\int_{-1}^1 f(t) P_n(t)\, dt$$for some f:[-1,1]\rightarrow\mathbb{R}. Are there any good references on the ... 1answer 126 views ### On the set of good approximators in the sense of Dirichlet's theorem This question came up when thinking about an older question that hasn't been answered as of now. Let \mathbb{N} be the set of positive integers. If \alpha\in\mathbb{R}, we say q\in\mathbb{N} is ... 1answer 100 views ### Density of the set of numbers that are “good approximators” to a given real in the sense of Dirichlet's approximation theorem Let \mathbb{N} be the set of positive integers. Given a set A\subseteq \mathbb{N} we let the (upper) density of A be defined by$$\mu^+(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$...
Denote $$T(m)=\sum_{1\leq n_m\leq n_{m-1}\leq\dots\leq n_2\leq n_1\leq m}\prod_{i=1}^{m}\binom{n_i}{n_{i+1}}.$$ Is there a name for this kind of summation and is there a good estimate for $\ln T(m)$ ...