Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
1answer
18 views

Name of a function space

For a real function $f$ on $\mathbb{R}$, define $e_n(f)$ to be the infimum of the $L_1$ distance between $f$ and piecewise constant functions on the subdivision of $\mathbb{R}$ into intervals of ...
1
vote
1answer
139 views

Approximation of a two-variable function by tensor products

Let $X$ and $Y$ be compact metric spaces and $f: X \times Y \to \mathbb{R}$ be a continuous function. We know that, for every $n \in \mathbb{N}$, by the Stone-Weierstrass theorem, there exist $k_n \...
0
votes
0answers
21 views

Maximizing the sum of piecewise linear functions using approximate, differentiable functions

Given an arbitrary (sorted) set of numbers $\{c_1,\dots,c_n\}$, define for each number the piecewise continuous linear function $$ f_i(x) =\begin{cases} x & 0\leq x\leq c_i \\ 0 & ...
-4
votes
0answers
125 views

Finding a paper [closed]

Could you please help me to find the paper "Rüssmann, Helmut Kleine Nenner. I. Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes. (German) Nachr. Akad. Wiss. Göttingen Math.-Phys. ...
4
votes
0answers
65 views

Orthonormal Basis of Multi-Dimensional Sobolev Space of Different Orders without Reproducing Kernel

Let $\Omega$ be an open subset of $\mathbb{R}^d$. Under regularity conditions, we know that the $s$-th order Sobolev space $H^s(\Omega)$ with $s \geq d/2$ is a reproducing kernel Hilbert space. In ...
2
votes
0answers
132 views

a Kernel free asymptotic for a sampling operator

Let $\Pi=\left\{ t_{k}\right\} _{k\in\mathbb{Z}}$ a sequence of real numbers such that $-\infty<t_{k}<t_{k+1}<+\infty$ for every $k\in\mathbb{Z}$, $\lim_{k\rightarrow\pm\infty}t_{k}=\pm\infty$...
1
vote
0answers
83 views

Final time maps of IVP's approximating functions $X\subseteq\mathbb{R}^n\to\mathbb{R}^n$

I originally posted this question on the Mathematics StackExchange and got told to consider putting it on here, on MathOverflow. I will word the question a bit differently: Let $X$ be a compact $k$-...
3
votes
1answer
106 views

Marsden's Identity and B-splines

Marsden's Identity states that for every $\tau$ in $\mathbb{R }$: $$(\cdot -\tau)^{k-1}=\sum_j\Psi_{j,k}(\tau)B_{j,k,t} \, ,$$ with $\Psi_{j,k}=(t_j-\tau)\times...\times(t_{j+k-1}-\tau)$. Following ...
0
votes
0answers
27 views

Approximating Uniform/Arbitrary distribution with Gaussian mixture model?

so Silverman in his 1986 book mentioned about approximating distributions with Gaussian mixture models but he didn't go much further into the topic...I'm just wondering, say I'm given a N-dimensional ...
2
votes
1answer
84 views

How far away is $\max_{x: x \in \{0, \ldots, N\}} |W(x/N)|$ from $\max_{0 \leq t \leq 1} |W(t)|$ ($W(t)$ a Wiener process)?

How far away is $$\max_{x: x \in \{0, \ldots, N\}} \left|W\left(\frac{x}{N}\right)\right|$$ from $$\max_{0 \leq t \leq 1} |W(t)|$$ In other words, if you simulate a Wiener process over a finite ...
2
votes
0answers
64 views

Universal approximation theorem for whole R^d

The well-known universal approximation theorem states that neural network with one hidden layer can approximate any continuous function on every compact subset of $\mathbb{R}^d$. My question is ...
0
votes
0answers
38 views

The fraction of vertices in the giant component as a function of $\lambda$

In the Erdos-Renyi random graph $G_{n,p}$ the fraction of vertices in the giant component, $v$, and the parameter $\lambda$, where $p = \lambda / n$, are well known to be related by $1-v = \exp(-\...
3
votes
1answer
103 views

Variation of steepest descent/Laplace methods for non-exponential integrands

I was wondering if versions of the Laplace/steepest descent methods exists for integrals of the type $$\int_C f(z) M(\lambda g(z)) dz$$ for $\lambda >>0$ functions $f(z), g(z): \mathbb C \...
2
votes
1answer
165 views

Simple but entangled inequalities

Do there exist functions $F,G$ on $[0,1]$ with $0\le F,G< 1$, such that for all $x, y\in [0,1]$ with $x+y\le 1$, the following hold? 1) $G(x)\le x$, 2) $G(1)<1$, 3) $F(x)>0$ if $x>0$, ...
1
vote
1answer
75 views

Extended Global approximation theorem

In Evans, $\textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $\partial U$ is $C^1$. Suppose as well that $u \in W^{k,p}(U)$ for some $1\leq p < \infty$. Then, there ...
5
votes
2answers
208 views

Probability of at least two of $n$ independent events occurring subject to some conditions

Given a set of independent Bernoulli random variables $\{x_1, \dots, x_n\}$, let $p = \sum_{0<i\leq n}\Pr[x_i = 1]$ and $X=\sum_{0<i\leq n} x_i$. We know that for any $i$, we have $\Pr[x_i = 1]\...
1
vote
1answer
132 views

For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often

Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that ...
1
vote
1answer
111 views

For which $n$, can we find a sequence of $n+1$ distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve

Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the ...
23
votes
2answers
392 views

Does every positive continuous function have a non-negative interpolating polynomial of every degree?

Let $f:[a,b] \to (0,\infty)$ be a continuous function. Then is it necessarily true that for every $n\ge 1$, we can find $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that the ...
0
votes
0answers
23 views

Approximate sum of log terms from sum of terms?

Suppose I have a program that processes time series data and calculates the discounted cumulative sum of future reward, like so $$ X_{t} = r_{t} + \gamma r_{t+1} + \gamma^2 r_{t+2} + ... + \gamma^N ...
1
vote
0answers
130 views

Property of Fixed Point Function

Given an operator $\mathcal{T}$ that maps from a function $f: \mathbb{R}^d\rightarrow \mathbb{R}$ to another function $f': \mathbb{R}^d\rightarrow \mathbb{R}$, we are interested in the fixed point $f^*...
1
vote
1answer
86 views

Accurate estimate/lower bound for the l2 approximation error of trigonometric polynomial approximation

This is a restated version of my original very broad question. Let $P$ be probability a measure on an interval $[a,b]$ ($-\infty<a<b<\infty$) that's dominated by Lebesgue measure. Let $\...
1
vote
1answer
133 views

Approximate the following series on the euclidean grid

I had trouble in finding a closed form solution for the following series, so now I am trying to find a good approximation for it. The $\sqrt{i^2 + j^2}$ in the exponent comes from distances on the ...
1
vote
2answers
163 views

A min-max approximation

Let $n\ge 1$ be an integer, $\mathcal P_n$ be the vector space of all polynomial functions over $[a,b]$, of degree at most $n$. My question is : Is it true that $$\inf_{x_0,x_1,...,x_n\in[a,b], x_0&...
0
votes
1answer
83 views

Chebyshev interpolation [closed]

Let's define the n-th degree Chebyshev polynomials by $$ T_{n} (x)=\cos(n\arccos(x)).$$ Find a polynomial $P$ such that $$\mid y- P (x) \mid$$ is minimal, using the first three Chebyshev ...
1
vote
0answers
38 views

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: \...
2
votes
0answers
39 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 $...
4
votes
1answer
67 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 $...
2
votes
1answer
99 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}...
0
votes
1answer
115 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 $$ \...
4
votes
1answer
171 views

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 $...
3
votes
1answer
100 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-...
3
votes
1answer
159 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. ...
0
votes
0answers
22 views

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....
5
votes
2answers
131 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 $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ ...
1
vote
1answer
106 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 ...
9
votes
0answers
132 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(...
1
vote
1answer
148 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 ...
5
votes
0answers
171 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 ...
1
vote
0answers
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 ...
0
votes
0answers
48 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 $...
0
votes
0answers
57 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{\...
0
votes
2answers
53 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 ...
2
votes
0answers
174 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 ...
3
votes
1answer
78 views

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 ...
0
votes
0answers
79 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 $|P(\mathbf{x})| \leq 1$....
10
votes
1answer
362 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(\...
5
votes
1answer
101 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 $P_m(z)e^z-Q_n(z)$ ...
0
votes
1answer
163 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 $C_{\infty}$ ...
1
vote
0answers
45 views

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 ...