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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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Representability of smooth invertible Lipschitz functions by a finite composition of near-identity functions

Theorem 1 of this paper shows that For every positve integer $K$ and for every nonempty bounded domain $\mathcal X \subseteq \mathbb R^d$, the restriction on $\mathcal X$ of any function $h: \...
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23 views

On different norms of the interpolating operator

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
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28 views

Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
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1answer
77 views

Minimax Approximation to Sine Function on interval [-K, K]

Let $p_{n,K}(x)$ be the polynomial of degree $n$ that minimizes $\epsilon_{n,K} = ||p_{n,K}(x) - sin(x)||_\infty$ on the interval $[-K, K]$. Question: what is the asymptotic behavior of $\epsilon_{n,K}...
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92 views

Does log-concave approximable distribution satisfy transportation-cost inequality?

Definition: Recall that a distribution $\mu$ on $\mathbb R^d$ is said to be log-convave with constant $c > 0$, if density $d\nu \propto e^{-V}dvol$ satisfying the curvature condition $$ \...
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1answer
133 views
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Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
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22 views

Approximate a function by smooth functions with bounded below second-order derivative

For $i=1,\ldots,N$, let $f_i:(a,b)\to \mathbb{R}$ be some given functions, and define $f=\min_{i}f_i$ for each $x\in (a,b)$. I was wondering, under what kinds of conditions of $f_i$, we can find ...
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1answer
91 views

Given a local metric which is $C^1$-close to another, can we extend it globally while preserving the approximation?

Let $M$ be a smooth closed manifold, and let $g_0$ be a Riemannian metric on $M$. Let $U$ be a neighbourhood of $p \in M$, and suppose that we are given a metric $g$ on $U$, which satisfies $\| g-...
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1answer
135 views

Is the kernel of a Fredholm operator stable under perturbation?

This is a follow-up of this question. In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator? Let $(X,\| \cdot \|)$ be an infinite-dimensional normed vector space. ...
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Approximate in $W_1$ sense, an empirical distribution with restriction of true distribution on a set

Let $\mu$ be a probability distribution on a metric space $X=(X,d)$ (to avoid unnecessary complications, assume the full filtration $2^X$) and let $x_1,x_2,\ldots,x_N$ be a sample of size $N$ drawn i....
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120 views

Can we stay invertible while approximating linear maps in Sobolev spaces?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$. Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ ...
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1answer
105 views

Giving Uniform Bound on Differences of Sums of Converging Polynomials

The title does not quite capture the essence of the difficulty, please allow me to be more explicit here. I thought of this question when I was trying out an open problem by Ovidiu Furdui(See problem ...
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108 views

Degree of 2-variable real polynomial that’s large on a square and small on a nearby “L”

I’d like to know what’s the minimum possible total degree of a real polynomial $p(x,y)$ such that: (1) $p(x,y)\in [0,1]$ whenever $x,y\in [0,1]$ with either $x$ or $y$ at most $\epsilon$, and (2) $p(...
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1answer
144 views

Can a continuous map on a Hilbert manifold be approximated by a map which has infinitely many critical points?

It is well-known that a continuous map $f:M\to\mathbb{R}^n$ from a Hilbert manifold can be closely approximated by a smooth map $g:M\to\mathbb{R}^n$ which has no critical points. But, can such a ...
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159 views

On effective constructions in the functional analysis of Volterra's integration operator

Let $V: L^2(0,1) \to L^2(0,1)$ be the Volterra integration operator: $V(f)(x) := \int_0^x f(t) \, dt$. Is there a universal function $C(L,\varepsilon) < \infty$ such that the following uniform ...
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44 views

Approximating norms using numerical integration? [closed]

I have a sequence of functions $u_m$ in $H^1(\Omega)$, where $\Omega$ is Lipschitz such that $u_m(x)=\int_{|y|\le \epsilon} \, f_m(x,y) \, dy$, but the integral cannot be expressed in terms of ...
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45 views

What are the compact elements in the domain of functions on the Interval Domain, [D,D]?

Take the interval domain $D = \{[a,b]| a \le b, \in \mathbb{R}\}$, ordered by reverse inclusion. Next, take the domain of functions $D' = \{f| f:D \rightarrow D \}$, ordered as follows $f \le g$ if $...
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53 views

Continuity of solution to 2nd Order PDE w.r.t. the coefficients

I am considering the following 2nd order PDE : On a domain $R$, for some $r > 0$, \begin{equation*} \frac{1}{2}\sum_{i, j = 1}^{K}\gamma_{i}\gamma_{j}U_{x_{i}x_{j}}(x) + \sum_{i = 1}^{K}\frac{\...
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2answers
50 views

dense subalgebra in measurable functions set

Take $\mathcal M_D$ the space of measurable functions from a compact set $D\subseteq \mathbb R_n$ to ℂ. I'm wondering if a Stone-Weierstrass-like theorem holds in this space, with the convergence in ...
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126 views

Intersection of Sobolev space with the space of continuous functions

While doing some problems, I came across the space $H=H^1(\Omega) \cap C(\Omega)$, where $\Omega$ is subset of $\mathbb{R^n}$. So far, by definition of these subspaces, We know that none of these are ...
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1answer
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automorphisms of a measurable space can be approximated by continuous measure preserving maps?

Suppose first that $D=[0,1]$ is equipped with the usual Lebesgue measure, and that $\varphi$ is a measure-preserving transformation $\varphi:D\to D$ that is bijective and whose inverse is also measure ...
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61 views

Bounded polynomial having coefficients that are bounded linearly in degree and number of variables

Let $P(\mathbf{x})$ be a bounded multivariate polynomial of degree at most $d$ (for my purposes it can be either coordinate degree or total degree) over $[-1,1]^n$, and assume $|P(\mathbf{x})| \leq 1$....
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309 views

Real polynomial bounded at inverse-integer points

Let $p$ be a real polynomial and $N$ be a positive integer. Suppose I tell you that $|p(\frac{1}{k})| \le 1$ for all $k\in\{1,\ldots,N\}$, and also that $p(\frac{1}{N})\le -\frac{1}{2}$ while $p(\...
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1answer
94 views

Padé multipoint approximants of the exponential function

One says that a pair of polynomials $(P_m,Q_n)$ over $\mathbb C[z]$, with $\text{deg }P_m=m$, $\text{deg }Q_n=n$, is a "multipoint Padé approximant of the exponential function" if $P_m(z)e^z-Q_n(z)$ ...
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1answer
98 views

Approximation of a continuous function by a smooth one on an open set

I'm interested in the following kind of theorems : Let $M$ be a real analytic manifold and $U$ an open set of $M$. Let $f : U \to \mathbb{R}$ a continuous function. Then, there is a $C_{\infty}$ ...
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Bounding quantiles of the noncentral chi distribution

I need to bound the empirical quantiles for a noncentral chi distribution (not chi-squared) $\chi_\nu(\lambda)$, where $\nu$ is the number of degrees of freedom and $\lambda$ the non-centrality ...
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168 views

What are the possible $L^{\infty}$ closures of an integration-invariant linear subspace of $C([0,1],\mathbb{R})$?

Let $S \subset C([0,1],\mathbb{R})$ be an $\mathbb{R}$-linear subspace that is invariant under the $T := \int_0^x$ integration operation: if $g \in S$ then the function $f = Tg$ defined pointwise by $...
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Role of polyhedral domain in convergence of finite element method

I am reading a paper by Diening and Kreuzer where they consider the convergence of finite element approximations for $p$-Laplace equation when using a certain algorithm. In the paper, they assume ...
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257 views

approximating the $|x|$ function

I'm familiar with Newman's rational approximation of the absolute value function via rational functions. Are there other explicit functions that approximate $|x|$ with exponential error? I was under ...
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217 views

Normal approximation to the pointwise/Hadamard/Schur product of two multivariate Gaussian/normal random variables

Let $X \sim \mathcal{N}\left( {{\mu _x},\sigma _x^2} \right)$ and $Y \sim \mathcal{N}\left( {{\mu _y},\sigma _y^2} \right)$ be two univariate and independent Gaussian/normal random variables and let $...
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1answer
74 views

Optimal $L^2$ bounds of cubic spline interpolation

Let $s(x)$ be the natural cubic spline interpolant of a function $f\in C^4$. There are known bounds on the $L^{\infty}$ error, $\|f^{(r)}(x) - s^{(r)} (x) \|_{\infty} $ for $r=0,1,2,3$. See Hall & ...
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145 views

Closure of polynomials of a function in $L^2$

Suppose $f \colon I \to \mathbb{R}$ is a function in, say, $L^\infty$, and $I \subset \mathbb{R}$ is a bounded interval. We may assume further regularity on $f$, such as Lipschitz continuity or strict ...
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1answer
119 views

Holes of a compact set in $\mathbb{R}^n$ that do not contain holes of a larger open set

Let $K$ be a compact subset of an arbitrary open set $\Omega\subset \mathbb{R}^n.$ It is said that a connected component $W$ of $\Omega\setminus K$ is $\Omega$-bounded if $\overline{W}$ is a compact ...
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Finding a tight upper bound of $\int_0^\infty e^{-a\sqrt{1-e^{-x}}-x^2/2} dx$ as a function of $a$, $a>0$

The integral converges as it is easily seen to be upper bounded by $\sqrt{\pi/2}$. However, Laplace's method does not seem to work out as the maxima of the function $S(x) = -a\sqrt{1-e^{-x}}-x^2/2$ ...
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1answer
45 views

Approximation error of 1-Lipschitz function on cubical mesh

Let $\Omega = [0,1]^d$ and consider $f : \Omega \to R$ Lipschitz continuous with constant 1. Consider the regular decomposition of $\Omega$ into $d$-dimensional cubes $\Omega_i$, $i=1 ... k^d$ with $...
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2answers
174 views

Finding a tight upper bound of $\int_0^\infty e^{-x^2/2-a(1-e^{-x})}dx,\ a>0$, as a function of $a$

Is there a method to find a tight upper bound on the given integral? Note that the integral is upper bounded by $\sqrt{\pi/2}$, and thus converges. I first thought about applying Laplace's method. ...
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1answer
177 views

A good starting position for maximizing a function with Newton-Raphson / Halley's method

I'm attempting to find the maximum of this function: \begin{align*} h(\mathbf{t}) = -\left\{\sum_{i=1}^{n}\lambda_i e^{\boldsymbol{\theta}_i^\intercal \mathbf{t}}\right\} + \boldsymbol{\alpha}^\...
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What are good ways to 'relax' a uniform approximation into independent saddle-point expressions once the uniform approach is no longer needed?

I am doing long-running project that involves asymptotic saddle-point estimation of integrals (for flavour, it's this sort of stuff) and I would like to ask if there are established ways in the ...
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2answers
111 views

Approximately complemented subspaces

Definition: Suppose $E$ is a subspace of normed space $X$. Then $E$ is approximately complemented in $X$ if for any compact subset $K$ of $E$ and any $\epsilon>0$ there is a continuous linear ...
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1answer
150 views

Approximating a compact $C^1$ hypersurface without boundary

Can we approximate (arbitrarily closely) a compact $C^1$ hypersurface in Euclidean space without boundary with a polygonal hypersurface, such as a simplicial complex? To clarify, I want to have the $\...
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1answer
137 views

Literature Request: Finite Dimensional “Approximations” of Linear Operators

I am interested in finding literature pertaining the problem posed by this question, which is the degree to which an operator $A$ on an infinite dimensional (separable) Hilbert $X$ space can be "...
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79 views

Integral of exponential of quadratics + exponentials

Eq 2 of this paper states this integral: \begin{align*} r^{-\beta} = \frac{1}{\Gamma(\beta)}\int_{-\infty}^{\infty} e^{-re^t + \beta t} dt \end{align*} Is there is name for this identity, or the class ...
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122 views

Can Carlsons's iterative algorithm for $\arctan x$ be inverted to get one for $\tan x$?

In the article An algorithm for computing logarithms and arctangents, by B. C. Carlson, the following iterative algorithm for arctangents is given: The algorithm uses that $2^n\tan(2^{-n}\arctan(x))=\...
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28 views

Truncation error of product and composition of functions?

I'm trying to solve the following problem numerically $$\int_0^a\alpha(x)\frac{dw(x)}{dx}\frac{d v(x)}{dx}dx.$$ For any $\alpha$,w and v functions of x. The problem is that I don´t know what the ...
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61 views

Approximating partial $p$-series

Inspired by this question, could help to obtain further numerical values. Suppose we want to obtain a good approximation of $S_p(n) = \sum_{k = 1}^n \frac{1}{n^p}$ for an integer $p \geq 1$. It is ...
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1answer
136 views

Tight L2 bound on moments approximation and reference

Consider $f\in L^2(I)$, where $I$ is the unit interval and $L^2$ is w.r.t. Lebesgue measure, and consider an approximation of $f$ denoted by $\tilde{f}\in L^2$. The error in approximated the moments ...
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153 views

Is it possible to estimate number of the tuples?

There is the problem. It seems very easy but I struggle with it for a few hours. We fix $n$ positive integer $a_i\in N^*,\forall i\neq j,(a_i,a_j)=1$,and fix a number $0\leq \delta\leq \frac{1}{2}$. $...
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2answers
93 views

Cubic interpolating Spline - Number of extremum points

Question: Given $f\in C^2 [a,b]$, and $s$ its "natural cubic spline" interpolant on some grid/knots $a= t_0 < t_1<t_2 < \ldots < t_n = b$, is there a bound on the number of extremum point ...
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1answer
119 views

Pade approximation of a rational function

I believe I have a naive or hard question because I couldn't find any results in the Internet yet. Any help is greatly appreciated. So suppose I have two rational functions $R_1(x)$ and $R_2(x)$, i....
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183 views

A Lemma on convex domain which is a Lipschitz domain

I am reading the following paper: https://docs.wixstatic.com/ugd/1de1d9_cd82cb002eaa4eefa9af574eb5efdff2.pdf I am stuck on the proof of lemma 2.3 on page 6. I don't see why does the property (i) of ...