# 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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### $L^1$ error between indicator function and smoothed out version

For a large parameter $r>0$, consider the indicator function $1_{[-r,r]}$ and its convolution with the (normalized) Gaussian $\frac{1}{\sqrt{\pi}}e^{-x^2}$, that is,
$$f_r(x) = \frac{1}{\sqrt{\pi}}\...

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### Practical calculation of Canterbury approximants

I'm looking for references on how to compute Canterbury approximants numerically from a practical point of view. The references on Canterbury approximants that I am aware of all appear rather abstract ...

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### Mean values of polynomial and holomorphic matrices

Lemma. Assume $H: \mathbb{R} \to \mathbb{R}^{d \times d}$ is a polynomial of degree $m$, such that for all $x \in \mathbb{R}$, $H(x)$ is a symmetric semidefinite matrix. For all $n \geq 0$ and real ...

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### Approximating a family of measurable functions

Let $X$ be a set of $N>0$ elements (with the counting measure) and consider a family of measurable functions $f_i:X\to [0,1]$, for $i\in \mathbb N$.
Any function $f_i$ can be seen as a point in the ...

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### Using programming to measure the uniformity of measurable subsets of the unit square?

This is a follow up to this post using this answer:
Let $S:=[0,1]^2$ be the unit square. "Partition" $S$ naturally into four congruent squares $S_{1,j}$ (with side length $1/2$ each), where ...

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### Solution or approximation to $\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx$

I'm looking for a solution or approximation to the following indefinite integral $$\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx.$$
I've tried Mathematica, but it does not converge to a solution....

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### Linear approximation of multivariate function of bounded second order partial derivatives

I have a question about linear approximation in the multivariate case.\
Let $f:B^d_r\to \mathbb{R}$ be a real-valued $C^2$-function defined on the $d$-dimensional ball of radius $r$ centered at the ...

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### Smooth approximation of nonnegative, nondecreasing, concave functions

Let $f\colon [0, \infty)\to\mathbb{R}$ be nonnegative, nondecreasing, and concave. Prove the following claim or give a counter example: There is a sequence of functions $f_n\colon [0, \infty)\to\...

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### Stone-Weierstrass theorem: coefficients of approximating sequence bounded?

Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a subalgebra of $C(X;\mathbb{R})$.
The Stone-Weierstrass theorem asserts that if $\mathcal{A}$ contains the constants and separates the points ...

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### Explicit bounds on derivatives of moments related to Bernstein polynomials

Background
This question relates to finding explicit bounds for the derivatives of moments related to Bernstein polynomials. Answering it will help me find explicit bounds for polynomials that ...

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### Approximation of function that has Lipschitz-continuous $n$-th derivative

Good afternoon. I'm trying to find in literature the solution for such a problem: for given function with $L_p$-Lipschitz continuous $p$-th derivative I need to find function $f_\varepsilon$ with $L_n$...

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### New method to compute square roots [closed]

In 2011 when I was in school I created a formula to calculate square roots... For $x\in\mathbb{R}$ with $x>0$ the following holds:
$$\sqrt{x} = \sum_{n=0}^{\infty}\frac{\left(\prod_{k=1}^{n}\left(\...

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### $\log$-classes of irrationals

Let $\mathbb{N}$ denote the set of non-negative integers. For $A\subseteq \mathbb{N}$ we let the (upper) density of $A$ be defined by $d^+(A) = \lim\sup_{n\to\infty} \frac {|A\cap \{0,\ldots, n\}|}{n+...

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### Approximation error of Chebyshev expansion of the second kind

Weierstrass' well known theorem states that every continuous function on $[-1,1]$ can be uniformly approximated to arbitrary precision by a polynomial function. Among these approximations it is known ...

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### "Almost rational" irrational

This is a follow-up to an older question.
Let $r\in \mathbb{R}\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\...

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### Smooth approximation of Hölder functions "from below"

We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1].$
I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\mathbb R)$ such ...

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### Approximation of pseudogeometric progression

Let $f_n(x)=1+x+x^{\sqrt{2}}+x^{\sqrt{3}}+x^{\sqrt{4}}+\cdots+x^{\sqrt{n}}$ be a sequence of functions on the interval $[0, 1]$. Is there a good closed form approximation for such a function ( ...

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### Approximation of Hölder continuous functions "from below"

We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1]$ with $f(0)=0$.
I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\...

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### How to prove approximation for fresnel integral converges

I was looking at the fresnel integral $S(x)=\int^x_0\sin(s^2)ds$. From reading I learned that this integral approaches $\frac{1}{2} \sqrt{\frac{\pi}{2}}$ as $x \rightarrow \infty$. Through messing ...

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### Carleman approximation for functions from $\mathbb R$ to (closed convex subset of) a Lie algebra

I am looking for an approximation result dealing with continuous functions of a real parameter with values in (some subset of) the unitary algebra. However, I wouldn't be surprised if the following ...

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### Explicit and fast error bounds for approximating continuous functions

Main Question
This question is about finding explicit, calculable, and fast error bounds (no hidden constants) when approximating continuous functions with polynomials or simpler functions to a user-...

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### Decomposing convex functions as (simple, useful, convex) + (convex)

Let $f$ be a nice convex function, let $X = \{ x_i : i \in [N] \}$ be a collection of points in the domain of $f$, and define the 'bundle approximation'
$$
f(x; X) = \max \{ f(x_i) + \langle \nabla f (...

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### Zeros in $[0,1]$ of functions $f \in \mathrm{span} \{ p(x - \lambda_k)e^{\lambda_k x} : k=1,\dots, n \}$

Let $n \in \mathbb N$, let $p:\mathbb R \to \mathbb R$ be a real polynomial, and let $\lambda_1< \lambda_2 <\dots < \lambda_n$. Now let
$$
f \in \mathrm{span} \left \{ p(x - \lambda_k)e^{\...

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### Measuring how "close" $\alpha\in[0,1]\setminus\mathbb{Q}$ is to being rational

Let $\mathbb{N}_+$ denote the set of positive integers and let $\mathbb{N}_0 = \mathbb{N}_+\cup\{0\}$. Fix $\alpha\in[0,1]\setminus \mathbb{Q}$. For $n\in\mathbb{N}_+$ we let the approximation radius ...

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### Measurability of Brjuno numbers

A positive irrational number $\alpha\in{\mathbb R}\setminus {\mathbb Q}$ is said to be a Brjuno number if $$\sum_{i=1}^\infty\frac{\log q_{i+1}}{q_i} < \infty$$ where $q_i>0$ is the denominator ...

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### Optimal polynomial approximation of rational function $\frac{1}{1-x}$

I've been working on the following polynomial approximation problem. I want to find the optimal Chebyshev approximation of the rational function $\frac{1}{1-x}$ on the real interval $x\in[-\rho, \rho]$...

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### A robust version of "a holomorphic function is determined by its modulus"

It is well known that if $f(z)$ and $g(z)$ are both holomorphic on a (path-)connected open set $C$ and $\lvert f(z)\rvert=\lvert g(z)\rvert$ on $C$ then $f(z)=cg(z)$ on $C$ for some constant $c$. Do ...

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### Is there an approximate formula for this summation function?

Consider the function $$\sum_{n=1}^\infty \frac{\cos(nx)}{n^r},$$ where $r\in\mathbb{N}$. Is there any approximate formula (closed form possibly avoiding this type of summation) for this function? I ...

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### The relative error of approximating a binomial

Are there any good approximations for a binomial CDF that work well in terms of the relative error, as opposed to absolute? For the usual normal approximation, the absolute error is very well-studied ...

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### Interpolation on Sobolev space on $[0, 1]^d$ over finite meshes

Let $\Omega = [0, 1]^d$ and suppose that $f \colon \Omega \to \mathbb{R}$ lies in order $m > d/2$ Sobolev space; i.e.,
$$
\|f\|_{H^m(\Omega)}^2 = \sum_{|\alpha| \leq m} \|D^\alpha f\|_{L^2(\Omega)}^...

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### Error bounds for Sobolev space norm approximation on a finite grid

Suppose that $f : [0, 1] \to \mathbb{R}$ is an element of the order-$k$ Sobolev space,
\begin{multline}
f^{(k - 1)}~\text{is absolutely continuous},\quad \|f\|_{W^k}^2 := \int_0^1 f^{(k)}(x)^2 \, dx &...

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### Finding the set of best approximation

Given $X$=$l^1$ and its dual space $X^*=l^\infty$. Now take $f=(1, 1/2, 2/3, 3/4,...) \in X^*$. Then clearly $\|f\|_\infty = 1$. I have found that $H=\ker f$ is a proximinal hyperplane in $X$.
Note: A ...

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### Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm

Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c,$ $|f(b)| = c$ and $\varepsilon > 0.$
It is easy to see that if $\|f\|_{\infty}< \frac{\varepsilon}{2} =: \...

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### Construction of the Lipschitz function with a given Lipschitz constant and given two values

Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c$ and $|f(b)| = c$. Is there a Lipschitz function $g$ such that $|g| \geq c,$ $g(a)=f(a),$ $ g(b)=f(b)$ and Lipschitz ...

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### Bounding the discrete $l^p$ norm by the continuous $L^p$ norm for trigonometric polynomials

Let $ X_N = \text{span} \{\cos(2\pi lx): l=0, \cdots, N-1 \} $ with $ x \in [0, 1] $ and $ Y_N = \{v =(v_0, \cdots, v_{N-1}): v_j \in \mathbb{C}\} = \mathbb{C}^N $. Then $ X_N $ is the space of ...

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### Generalization of the min-entropy that looks at the top $n$ probabilities

The min-entropy of a random variable $X$ can often be much easier to compute than the Shannon entropy. This is because the min-entropy is simply a function of the most probable value, and sometimes, ...

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### Approximating a strictly increasing non-negative function on a non-negative domain by polynomials with non-negative coefficients

Let $f:[0,2]\rightarrow [0,\infty)$ be a strictly increasing smooth function. The Weierstrass approximation theorem says that we can uniformly approximate $f$ by polynomials. But my concern is
...

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### Can the second-order difference control the first-order difference for nowhere differentiable functions?

Suppose that $f$ is a continuous, nonconstant function on $[0,1]$. Fix some $0<a<1$. Is it possible to establish the following inequality
$$ |f(x+h)-f(x)| \leq C \left[ |h|^a + |2f(x)-f(x+h)-f(x-...

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### Average as a constant approximation in $L^p$

Let $I=[0,1]$. For $p\in[1,\infty]$ define $C_p$ as the best constant such that for all $f\in L^p(I)$
$$
\left\|f-\int_If\,\right\|_{L^p(I)}\leq C_p\inf_{c\in\mathbb{R}}\left\|f-c\,\right\|_{L^p(I)}.
$...

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### Rate of uniform approximation by piecewise constant functions

Definitions and Notation:
Fix a positive constant $M>0$ with positive integers $m,n$ and the standard orthonormal basis $e_1,\dots,e_n$ of $\mathbb{R}^n$.
For every positive integer $N$, define the ...

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### Uniform norm bounds for linear approximation of 1-Lipschitz functions

This problem seems like it should be quite easy/standard, but I've not found a solution written down anywhere.
Consider the set of 1-Lipschitz functions on the $[0,d]$ interval. Define the linear ...

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### Show that $\frac{1}{n} \sum_{i=1}^n a_i \operatorname{erf} \left( \frac{b_i-x}{\sqrt{2}} \right) \to x$ for some sequence $\{a_n\}$ and $\{b_n\}$

Consider the following function
\begin{align}
f_n(x)=\frac{1}{n} \sum_{i=1}^n a_i \operatorname{erf} \left( \frac{b_i-x}{\sqrt{2}} \right)
\end{align}
where $\operatorname{erf} $ is the error ...

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### A function is of bounded variation if and only if the errors of its best approximation by trigonometric polynomials satisfy $\sum\frac{e_n}n<\infty$?

Let $\mathcal P_n$ be the set of trigonometric polynomials of degree less than or equal to $n$ and let $\lVert\cdot\rVert_\infty$ be the supremum norm. The error of the best approximation of $f$ of ...

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### Under what conditions is the least-squares approximation bounded with the same Lipschitz gradient constants?

Let $f(x):\mathbb{R}^K\Longrightarrow \mathbb{R}^L$ denote a multivariate continuously differentiable function. All the partial derivatives of $f$ (all its Jacobian elements) are bounded from above ...

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### Approximation in Bochner spaces

Is there any result like the Bramble-Hilbert lemma for Bochner spaces?
More specifically: let $H$ be a (e.g.) Hilbert space, $I\subset \mathbb R$ a bounded interval, and $L \in \mathcal L(H^k(I;H), Y)...

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### Smooth approximation of the $\max\{0,x\}$ function with controlled derivatives

Motivation/Hand-Wavy Question:
In this post, it was asked what the best local approximation of $f(x):=\max\{0,x\}$ is by a polynomial of a given degree; with the answer provided by Chebyshev's ...

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### Chebyshev-like polynomials [closed]

In some approximation problems I'm working on, the errors turned out to be polynomials of various degrees whose graphs on the interval $[-1,1]$ look like this:
As you can see, these things look a bit ...

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### Does a matrix product have an upper bound on the largest coefficient?

Let $A$ and $B$ be two $n\times n$ random matrices. Matrix $A$ has coefficients taken from a normal distribution $ \mathcal{N}(\mu_A,\sigma_A)$, and matrix $B$ has coefficients taken from $ \mathcal{N}...

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### Numerically expanding a function in a rational-power "basis"

I have some scientific code which interfaces with a library which accepts real functions specified as any number of additive terms with exponential powers. For instance, it is capable of accepting ...

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### Bounds on the expectation of a function of a hypergeometric random variable: A "Jensen gap"

Main Question
Let $f:[0,1]\to [0,1]$ be continuous, let $B_n(f)$ be the $n$-th degree Bernstein polynomial of $f$, and let $r\ge 3$.
Given certain assumptions on $f$, what is an explicit and tight ...