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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49 views

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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1answer
50 views

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 ...
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2answers
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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102 views

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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1answer
51 views

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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1answer
43 views

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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207 views

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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20 views

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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Approximation to continuous functions over an closed interval

Let $$f\in C[a,b]$$ A triangular system is a series of numbers \begin{matrix} x_{11}\\ x_{21}&x_{22}\\ x_{31}&x_{32}&x_{33}\\ \cdots \end{matrix} that $$a<x_{n1}<x_{n2}<\cdots<...
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79 views

Approximation of smooth function by harmonic function

Let $u(x,y)$ be a smooth function on the square $S:=[0, 1] \times [0, 1]$ (see, for example, Wiki for the definitions) and $\varepsilon > 0$. Is it possible to approximate $u(x,y)$ by a function $...
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68 views

Rate of convergence of Padé approximants

Let $f$ be an entire function of order $1$. Two questions: 1) Can one assert that the diagonal Padé approximants converge to $f$ (pointwise or uniformly over compacts of $\mathbb C$)? 2) if yes, can ...
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108 views

Stone–Weierstrass theorem for stronger topologies

The Stone–Weierstrass theorem gives an easy to check criterion on a (algebra) set of functions $D\subseteq C(X)$ which ensures that $D$ is dense. Are any similar results for density in $C_b(X)$ ...
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188 views

Function satisfying $f(x)^{f^{-1}(x)}=x^2$ with $ f^{-1}$ is a compositional inverse of $f$ and $f:\mathbb{R+}\to \mathbb{R+}$?

Let $f$ be a function such that :$f:\mathbb{R+}\to \mathbb{R+}$ and $f^{-1}$ is a compositional inverse of $f$ , I have tried to find solution of the following functional $f(x)^{f^{-1}(x)}=x^2$, I ...
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31 views

Higher-order inner products of an orthonormal basis

Let $\pi$ be a probability measure on some space $\mathcal{X}$, and let $\Phi = \{ \phi_k \}_{k \geqslant 0}$ be some (possibly complex-valued) orthonormal basis for $L^2 ( \pi )$, with $\phi_0 \equiv ...
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Approximating zero sets of real polynomials with “less complicated” polynomials

Let $K \subset \mathbb{R}^n$ be a compact subset, and let $P(x_1,\dots,x_n)$ be a real multivariable polynomial of degree $d$, whose vanishing set we denote by $Z_P$. Is it plausible to approximate $...
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63 views

Can this function be interpolated with a small power series

Does there exist a power series $\sum_i a_i x^i$ that is $1$ at $0$ and $0$ at integers from $1$ to $n$, and such that $\sum_i |a_i|$ is polynomial in $n$? I feel the answer might be no but I'm not ...
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48 views

if $\max_{z \in K} |\zeta(z+it)-f(z)|<\epsilon.$ then is this $\lim_{t\to \infty} \inf \frac {|\zeta'(z+it)|}{|f'(z)|} $ a finit limit?

Universality theorem of Riemann zeta function states that :Let $K$ be a compact subset with connected complement lying in the strip $\{1/2 < \operatorname{Re}(z)<1\}$, and let $f : K \rightarrow ...
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Sobolev convergence of Fourier series

Consider $f\in H^{\sigma}(S^1)=W^{\sigma, 2}$ (the usual Sobolev space on the circle) and let $S_Nf$ be its truncated Fourier series $S_Nf = \sum_{|n|\leq N} \hat{f}(n)e^{2\pi i n x}$. I am looking ...
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Inductive definition of Bernstein polynomials

For $n\in \mathbb{N}$ let $B_n$ be the linear operator taking a function $f$ on the unit interval $I=[0,1]$ to its $n$-th Bernstein polynomial $B_nf$, $$ B_nf(x):=\sum_{k=0}^n\binom{n}{k} f\Big(\...
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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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1answer
61 views

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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188 views

analytic approximations of the min and max operators

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 $\...
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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 ...
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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 ...
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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 ...
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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....
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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 $x \in \mathbb{R}^d$. ...
5
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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 $0\leq x < e^{-1}$, $g(x)= 1/x$ ...
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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(...
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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 ...
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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 ...
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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}.$...
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1answer
83 views

Terminology and approximation to logarithm of a sum of products of binomial coefficients

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 $\ln T(m)$ ...
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1answer
150 views

Logarithm of an integral involving generalized real binomial coefficients

I could not find a closed form for this integral although I think it should have been studied. What is a good approximation to $I$ in $$I=\ln\Bigg(\int_{0}^y\binom{2m}{m(1+x)}dx\Bigg)$$ where $0\...
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1answer
104 views

Criteria for $\epsilon$-Density

Let $Y$ be a compact, separable metric space and $X=C(Y)$ Banach space. There are many criteria for a linear subspace $Z\subseteq X$ to be dense; notably the Stone-Weierstraß theorem. Are there ...
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1answer
152 views

Can we approximate a vector field on the plane with non-vanishing vector fields in $W^{1,2}$?

Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated. Does there exist a sequence of ...
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131 views

An integral involving many exponential terms with quadratic exponents in the denominator

Given $k$ points $\{p_1,\cdots, p_k\}$ in $\mathbb{R}^n$ and positive constants $r_1, ..., r_k$ and another positive constant $\alpha>0$. Is there a way to compute/approximate the following (...
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1answer
566 views

Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have distinct singular values?

$\newcommand{\SO}[1]{\text{SO}(#1)}$ $\newcommand{\dist}{\operatorname{dist}}$ Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth. Set $$...
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192 views

bounds on derivatives of mollifiers/mollified functions

Consider the standard mollifier $$ \phi(x) = C\exp\left(-\frac{1}{1-x^2}\right), \quad -1<x<1. $$ such that $\int\phi(x) = 1$. Let $f(x) = |x|$ and consider the convolution $f\ast \phi$. I am ...
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1answer
125 views

Approximation of the indicator function of an interval by polynomials

Suppose $f:[-1, 1]\rightarrow\mathbb{R}$ is a polynomial. I am curious what the minimal degree of $f$ can be such that for $0<a<b<1$, $f$ satisfies the following two properties: 1) $\forall ...
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265 views

Best constant approximation in $L^p(\Omega)$

For $\Omega$ a bounded open set of $\mathbf{R}^d$ and $f\in L^p(\Omega)$ the infimum \begin{align*} \inf_{C\in\mathbf{R}} \|f-C\|_p \end{align*} is reached (by compactness). For $1<p<\infty$ ...
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1answer
99 views

Upper bound on Lp distance of functions before and after change of variables

Setup I am trying to upper-bound the difference between two functions: one before the change of variables and the other after. For example, let $r \in \mathbb{N} \cup \{\infty\}, 1 \leq p < \...
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1answer
27 views

Family of funcitons that approximates uniform density on an ellipsoid

Given an nondegerate ellipsoid $E$ in $\mathbb{R}^d$, described as $E = \{x\in\mathbb{R}^d: (x-x_0)^TQ_0(x-x_0)\leq 1\}$ and let $\chi_E$ be the characteristic function supported on $E$. I am thinking ...
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39 views

Error bounds for spline interpolation. Hall and Meyer's conjeture

Hall & Meyer, 1976, J. Approx. Theory, show for $f \in C^4[a,b]$ and a mesh $a = x_1, \ldots x_n = b$ with $h = \max x_{j+1} - x_j$ then for $\pi f$ a cubic spline interporlant over the mesh ...
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74 views

The $L_\infty$ norm of the derivative of the $L_2$ spline projector

A. Shadrin Acta Mathematica, 2001, shows that the $L_\infty$ norm of the $L_2$ projector $P_\Delta$ onto the spline space $S_k(\Delta$) is bounded independently of the knot-sequence. I.e. for a ...
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42 views

Approximating a subexponential function with a submultiplicative function

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire holomorphic function of growth no higher than minimal exponential type. Does there exist a measurable non-negative function $g:\mathbb{R}\to\mathbb{R}_+$ ...
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57 views

Approximation of measured-valued function by continuous functions

For each $x\in R^d$, let $\nu(x,dz)$ be a L\'evy measure, i.e., $$ \int_{R^d}(|z|^2\wedge1)\nu(x,dz)<\infty. $$ Let $\mu$ be a probability measure on $R^d$ such that $$ \int_{R^d}\int_{R^d}(|z|^2\...
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1answer
93 views

Approximation of function in general measure space

Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with $$ \int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...

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