# 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.

421
questions

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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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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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vote

**2**answers

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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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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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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26 views

### 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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37 views

### 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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113 views

### 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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72 views

### 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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292 views

### 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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87 views

### 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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59 views

### 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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**1**answer

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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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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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 $x \in \mathbb{R}^d$. ...

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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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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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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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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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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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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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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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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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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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**1**answer

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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**1**answer

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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**1**answer

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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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 \...