# 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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### Can a neural network with ReLU activation represents exactly all $B$-bounded and $L$-Lipschitz $K$-max-affine functions?

A max-affine function is defined as the maximum over a set of affine functions, which is always convex. More specifically, we define a $K$-max-affine function $f:\mathbb{R}^d\to\mathbb{R}$ that can be ...

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### Weak lower semicontinuity of a sequence of Riemann sums

Let us have a sequence of functions $\{f^K\}_{K \in \mathbb{N}} \in C([0,1],\mathbb{R})$ which is uniformly bounded in $L^2((0,1))$. We observe a sequence of Riemann sums
$$R^K=\frac{1}{K} \sum_{k=0}^{...

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### Are Chebyshev polynomials a Schauder basis of $\mathrm{Lip}[-1,1]$?

It is known that every Lipschitz function $f \colon [-1,1] \to \mathbb R$ can be expressed as a series in the Chebyshev polynomials $$f = \sum_{n = 0}^\infty a_n T_n $$ which is absolutely convergent ...

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### Bound error in approximating $E_x [H(f(x))]$ with random $(1/n) \sum_{i=1}^n \Phi(f(x_i)/h)$ where $H$ is Heaviside function and $\Phi$ is normal CDF

Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently smooth" function. For simplicity, we may consider $f$ to be an affine function, i.e $f(x) \equiv b-x^\top w$, for some $(w,b) \in \mathbb ...

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### Best approximation of L1 function by Lipschitz function

Fix constant $L,C>0$ and $k\geq 1$ and let $f\in W^{1,k}(\mathbb{R}^d,\mathbb{R}^n)$ with $\|f\|_{W^{1,k}}\leq C$.
Is there a known estimate on the distance
$$
\|f - \operatorname{Lip}_L(\mathbb{R}^...

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### Relation between the local maxima and the local minima for approximating the generalized Laguerre polynomial

I have already asked my question in the link below:
Minima approximation for Laguerre polynomials
I have suggested to anyone to give me the approximations of the minima for the Laguerre polynomial, ...

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### Find weak approximation by smooth unit vector fields for Sobolev fields on manifold

I am considering the Sobolev space of unit tangent vector fields on a compact manifold:
$Γ_{W^{1,2}}(M, UTM)$.
I would like to approximate those weakly with smooth vector fields ($Γ_{C^∞}(M, UTM)$).
...

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### Series expansion for gaussian-like function

I need a series expansion to describe a general gaussian-like (bell shaped) function. I couldn't find a rigorous definition of "bell shaped" online but in essence the function should have ...

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### Distribution of quadratic polynomials mod $n$ and $n^2$

Suppose $n$ is odd, then both equations $x^2 = D \; mod \;n$ and $x^2 = D \; mod \;n^2$ have the same number of solutions for fixed $D$ coprime to $n$. What can be said about the relationship between ...

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### Fast decaying Fourier coefficients for indicator function

Let $0 \leq a < b \leq 1$. I wanted to compute the Fourier series expansion of the indicator function $f = \chi_{[a, b]}$ of the interval $[a, b]$, as $f(x) = \sum_{k\geq 0}a_k e(kx)$. My ...

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### Approximating exp(x) with a piecewise-linear function accurately

I am looking for the best way to approximate $\exp(x)$ on a finite domain $[0,M]$ with a piecewise-linear function. My initial approach is to take $K$ evenly-spaced segments from $0$ to $M$. For each ...

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### How to numerically compute $x \ln x$ and related functions near $0$?

I was recently trying to find a numerical solution to a thermodynamics problem and the expression $x\ln x$ appeared in one of the computations. I did not have to find its value very near $0$, so the ...

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### Computing $\int^{4b}_0 {e^{-tx}\biggl(\frac{\sqrt{4bx-x^2}}{(2b-2c)^2+4cx}\biggr) dx}$

I am a PhD student working on complex analysis. After integrating over a keyhole contour to obtain the inverse of a particular Laplace transform, I ended up with the following integral:
$$\frac{1}{\pi}...

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### Almost periodic functions in weak mixing extension

In Theorem 3.4.6 of the 'Lecture notes on ergodic theory' by Jesse Peterson, it is shown that in a weak mixing extension, every almost periodic function is trivial. I have a doubt in the proof of this ...

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### Almost periodicity and approximation in tracial von Neumann algebra

Let $N$ be a von Neumann algebra with a faithful normal tracial state $\tau$. For a countable group $G$, let $\sigma: G\rightarrow \text{Aut}(N)$ be a $G$- action on $N$ which preserves the tracial ...

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### Counting bounded residues congruent a fixed number modulo a range of numbers

Suppose we have a fixed "large number" $N$, and an interval of positive integers $[a, b]$ ($a$ and $b$ being in the same order; typically $b = 2a$) which are also "quite large" (i....

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### Functions dense in $L^1[0,1]$ but not in $L^2[0,1]$

Is there a family of continuous functions $(f_n)_{n \in \mathbb{N}}$ on $[0,1]$ whose span is dense in $L^1[0,1]$ for the $L^1$-norm, but not dense in $L^2[0,1]$ for the $L^2$-norm?
Some preliminary ...

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### Density of $\mathrm{span} \{ f(\cdot - n) : n \in \Bbb Z \}$ in $C(\mathbb R)$

I recently came across a survey paper by Allan Pinkus. This paper contains the following result:
Suppose that $g \in L^1(\mathbb R)$ has support in an interval of length at most $2\pi$. If $f=\hat g$ ...

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### Smooth, non-analytic functions of non-normal matrices

My apologies if this isn't a well-enough-posed question, I think I'm partly unsure of what exact question to even ask.
There are many different ways in which we can take a function of a matrix.
We ...

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### Padé–Hermite approximants of the exponential of type II

I found the explicit expression of the Padé–Hermite approximants of the exponential function, of type I by complex integrals (see for instance Khémira - Approximants de Hermite–Padé, déterminants
d’...

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### Polynomial and rational approximation of continuous functions in $\mathbb{C}$

I am wondering what the state of the art is for polynomial and rational approximations to continuous/holomorphic functions in $\mathbb{C}$. The particular domains of interest are the closed unit ball $...

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### Distance to finite degree polynomials for BV functions

A result of Jackson establishes for lipschitz functions $f\in\text{W}^{1,\infty}(0,1)$ the bound $$\inf_{p\in\mathbf{R}_n[x]} \|f-p\|_\infty\lesssim \frac{1}{n}\|f'\|_\infty,$$
where $\mathbf{R}_n[x]$ ...

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### Weierstrass-type approximation of a system of the form $\{ x \mapsto f(x)^{p_n} : n \in \mathbb N\}$

Let $(p_n)_{n \in \mathbb N} \subset (0,\infty)$ be a sequence of distinct positive numbers such that their series of reciprocals diverges. By the theorem of Müntz-Szasz the closure of the span of the ...

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### Translating between interpolation vs approximation

tl;dr: Is there a way to translate interpolation error estimates into approximation error estimates for non-interpolating polynomials?
Given a function $f\in L^2(\Omega)$ and a basis $\{\phi_n\}$ for $...

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### Error estimates for orthogonal polynomial approximation

tl;dr: Are there explicit bounds for the approximation error by orthogonal polynomials?
There are various ways to formulate this question more precisely, so want I emphasize up front that this is a ...

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### Density of smooth function in Hilbert spaces

I am looking for a simple reference to the following fact:
If $f:\Omega\to\mathbb{R}$ is continuous, where $\Omega\subset H$ is an open subset of a separable Hilbert space $H$, then for any $\...

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### Concave functions: Series representation and converging polynomials

Background
We're given a coin that shows heads with an unknown probability, $\lambda$. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads ...

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### P1-finite element as convolution of P0-finite element

For a vector $u\in\mathbf{R}^N$ let's denote $\pi_N(u)$ the unique piecwise linear and $1$-periodic function matching the components of $u$ on the discretization $x_k = \frac{k}{N}$ of the unit ...

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### Convexification of difference of convex functions

I am looking for a reference/a hint to the following problem:
We are given $f_1(x),f_2(x)$ convex functions (say, on $\mathbb R^d$) such that $f_1(x) \to\infty$ for $\|x\|\to\infty$. Also there is an $...

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### Minimum norm polynomial subject to equality constraints

I had asked this question in the math stackexchange about a year ago. I did not get any response, so I am asking it here.
Given distinct points $x_1,\dots,x_m \in [0,1]^n$ and real numbers $y_1,\dots,...

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### Using the Holtz method to build polynomials that converge to a continuous function

Background
We're given a coin that shows heads with an unknown probability, $\lambda$. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads ...

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### On $L^2$ spaces which have an orthogonal basis of characters (complex exponentials)

Suppose $\Omega \subset \mathbb{R}^n$. What conditions on $\Omega$ make it so there exists a countable set $\Lambda$ such that $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ form an orthogonal basis ...

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### Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$

Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials?
In this post, $\phi$ is said to be polyharmonic in $\mathbb{...

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### Determining polynomial approximations of piecewise constant functions

Let $t_1 < t_2 < \cdots <t_m$ be real, and $X = \cup_{i=1}^{m-1} (t_i, t_{i+1})$ be a union of real open intervals. Let $f:X \rightarrow \{-1, 1\}$ be any piecewise constant function of form
...

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### Integrating a B-Spline basis function with respect to the standard normal PDF

I am looking for ways to evaluate exactly (i.e. analytically or semi-analytically) integrals of the type:
$$
\int_{-\infty}^{+\infty}B_{i}^k(u)e^{-\frac{(u-\mu)^2}{2\sigma^2}}du,
$$
where $B_i^k$ is a ...

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### Smooth cut-off in homogeneous Besov space

Given a Littlewood-Paley decomposition
$$1 = \chi(\xi) + \sum_{j \geq 0}\varphi(2^{-j} \xi), \quad \xi \in \mathbb R^n$$
where $\chi$ is smooth, supported on a ball, and $\varphi$ is smooth, supported ...

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### Approximating linear bounded operator on $L^2([a,b])$

I have the following problem: I'm given a linear bounded operator $P\in \mathcal{L}(L^2([a,b]))$, $a,b\in \mathbb{R}$ and I want to find a sequence of approximating linear bounded operators $(P_n)_{n\...

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### Is there any known relationship between logarithm of modified Bessel function of second type and its ratio?

Is there a known relationship between $\log(K_s(x))$ and $\frac{K_{s-1}(x)}{K_s(x)}$ for $s < 0$ and $x > 0$?

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### Best way to introduce B-splines?

I have the option of mentoring pure math undergrads in a topic lying within Approximation Theory and I really want to do $B$-splines. Mostly because I have recently found applications of them in my ...

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### Approximating the lateral surface area of a generalized cylinder with arc lengths of cross section curves

Given a generalized cylinder1 whose axis is defined by $r(s) = (x(s), y(s), z(s))$, we want to derive a lower bound of the lateral surface area between the cross sections of $r(a)$ and $r(b)$, where $...

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### Approximation of $C^1$-smooth equivariant maps by infinitely smooth ones

Let $M,N$ be smooth closed manifolds acted by a finite group $G$. Let $f\colon M\to N$ be a $C^1$-smooth $G$-equivariant map.
Is it true that for any $\varepsilon>0$ there exists a $C^\infty$-...

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### Bounds for the beta CDF

This question is closely related to a previous question that I asked here:
An inequality involving the beta distribution
Let $a,b$ be strictly positive integers, and let $F_{a,b}(x)$ denote the CDF ...

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### Polynomial approximation for square root function with fast convergence and bounded coefficients

Let $\delta, \varepsilon \in (0,1)$. I am interested in a sequence $\{f_n\}$ of polynomial approximations of the square root function $x \to x^{1/2}$ on $[\delta,1]$, of the form
$$
f_n(x) = \sum_{i=0}...

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### Upper bound an integral with exponential function

I am working on my research about approximation a function. I come up with the following integral. I run some simulations and saw that the integral would converge to zero as n goes to infinty. Here is ...

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### Standard function spaces with the approximation property

A Banach space $\mathcal{X}$ is said to have the approximation property (AP) if, for every compact set $K \subset \mathcal{X}$, there is a sequence of finite rank operators $\{U_n : \mathcal{X} \to \...

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### Density and Fourier approximation

Let $\mathbb{T}$ denote the 1-d torus and $H^s(\mathbb{T})$ the Sobolev space of order $s\geq0$ of complex-valued functions on $\mathbb{T}$ with the identification $H^0 (\mathbb{T}) = L^2 (\mathbb{T})$...

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### Chebyshev Equioscillation Theorem in presence of extra conditions

Let $P_\ell$ be polynomials of degree $\ell$. For $f \in C[0,1]$, define the minimax error $E_\ell(f) = \min_{p \in P_\ell} \max_{x \in [0,1]} |f(x) - p(x)|$. We know that for the above scenario the ...

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### Approximating a subclass of $L^2(\mathbb{R})$ by Schwartz functions within similar subclass

It is well-known that real valued Schwartz functions on $\mathbb{R}_+$ $\mathcal{S}(\mathbb{R}_+)$ are dense in the set of square integrable functions on $\mathbb{R}_+$ $L^2(\mathbb{R}_+)$. We can ...

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### Asymptotic approximation of a convex body by a high-dimensional parallelepiped

Consider a continuous random variable $X$ with support $[0,1]$ and log-concave density $f(x)$. Define an $n$-dimensional convex body in Euclidean space as
$$
\mathscr{X}_n=\left\{ \mathbf{x}=\left(x_1,...

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### Do higher-order splines with Lipschitz derivatives exist on finite sets?

Fix $k\in \mathbb{N}^+$ and let $E=(e_i,f_i)_{i=1}^I\subset \mathbb{R}^n\times \mathbb{R}^m$ be a non-empty finite set with $e_i\neq e_j$ whenever $i\neq j$.
If $n=m=1$ then it's easy to see that:
$$
...