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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8 votes
0 answers
145 views
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L^2 polynomial approximation in higher dimensions

Let $\mu$ be a measure on $\mathbb{R}^n$ I'm looking for known upper-bounds on $$\| f-P_m \|_{L^2(\mu)} $$ where $P_n$ is the orthogonal projection of $f$ onto the polynomials of degree less than $m$. ...
0 votes
1 answer
87 views

Approximating a sequence of tempered distributions "uniformly" by Schwartz functions

This question has been motivated by the post making sense of distributions on the diagonal. Let $T$ be a tempered distribution on $\mathbb{R}^2$ and $\eta$ be a given mollifier on $\mathbb{R}$. For $f ...
2 votes
2 answers
426 views

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 ...
6 votes
5 answers
3k views

Approximation to the ratio of a Gaussian CDF to PDF

Johnstone and Silverman (2005) claimed that for large x $\frac{1-\Phi(x)}{\phi(x)} \approx \frac{1}{x}$ where $\Phi(x)$ and $\phi(x)$ are the CDF and PDF for a normal random variable. I was able ...
0 votes
0 answers
22 views

Metric entropy of mixed norm spaces with exponent-free bounds

Suppose $\mathcal{F}\subset L^p([0,1]^d)$ is a subset with the following property: The $L^q$-covering number of $\mathcal{F}$ is independent of $q$, for all $1\le q\le\infty$. An example of $\mathcal{...
2 votes
1 answer
135 views

Upper bound of a product of sines

Consider the function $$ f_n(t)= \prod_{1 \leq k \leq n-1,\\ \gcd(k,n)=1} \sin\Big(t-\frac{k \pi}{n}\Big),\quad t \in [0,\pi].$$ I wonder whether it is possible to compute some nontrivial upper ...
10 votes
2 answers
5k views

Approximate a probability distribution by moment matching

Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ ...
0 votes
0 answers
60 views

Multivariate Jackson inequality for Chebyshev approximation

There is an approximation of a multivariate function by a Chebyshev polynomial of degree n. One needs to understand how the approximation error behaves depending on the degree of the polynomial or ...
1 vote
0 answers
30 views

Error bounds for a Romberg-style improvement of a non-linear approximation

I have a (possibly non-linear) functional $F$, which I want to numerically approximate by a (typically non-linear) $\widehat{F}_h$. For a suitable class of functions, I have asymptotic error behaviour ...
-1 votes
1 answer
162 views

Best approximation of the modulus function

While there is extensive study regarding the best approximation of function with polynomial functions in the real domain, the study of approximation of complex variables becomes much sparse. See this ...
1 vote
0 answers
46 views

Best approximation rates of various classes of functions by truncated Fourier series

Let $f\in C([-1,1]^d)$ have periodic boundary, $N$ be a positive integer, and let $S_N(f)$ be the best approximation of $f$ by its truncated Fourier expansion truncated approximation $$ S_N(f):=\sum_{...
3 votes
2 answers
475 views

Approximation for complex variables

Approximation theory, which aims to provide the optimal polynomial function approximating the target function in a given domain such as $x\in[-1,1]$, has been well-developed for real variables. In ...
0 votes
1 answer
123 views

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....
1 vote
1 answer
67 views

Sum of terms after partial fraction decomposition

I am facing the following problem (all $a_i$ being positive and all different) $$F=\int_0^1 \,dx\prod_{i=1}^n \frac 1{x+a_i}=\int_0^1\sum_{i=1}^n \frac {A_i}{x+a_i}\,dx=\sum_{i=1}^n A_i \log \left(1+\...
3 votes
1 answer
196 views

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 ...
7 votes
1 answer
296 views

Approximating functions on the real line

While it is not possible to approximate any function with polynomials on the entire real line, I am wondering if there are modified conditions under which the approximation is possible. Consider $f \...
1 vote
1 answer
540 views

Strong convergence of differential quotient in $L^2(0,T;V^*)$

I have got a problem regarding the weak differentiability of Bochner-integrable functions. Let $(V,H,V^*)$ be a Gelfand-triple and \begin{align*} w \in W(0,T) := \{w\in L^2(0,T;V) ~\vert~ \exists w' \...
16 votes
1 answer
562 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 $...
10 votes
3 answers
20k views

Approximation of a normal distribution function

I am reviewing and documenting a software application (part of a supply chain system) which implements an approximation of a normal distribution function; the original documentation mentions the same/...
1 vote
0 answers
64 views

Approximation of continuous function by multilayer Relu neural network

For continuous/holder function $f$ defined on a compact set K, a fix $L$ and $m_1,m_2,\dots,m_L$, can we find a multilayer Relu fully connected network g with depth $L$ and each $i$-th layer has width ...
0 votes
1 answer
103 views

Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems? (time-varying case)

Because flowmaps are homeomorphic maps on a compact domain $\Omega$, I was wondering if there is any literature that proves that diffeomorphism $\Phi(x)$ can be expressed as a flowmap of a certain ...
4 votes
2 answers
275 views

Approximating a fraction with a given denominator

Let $M$, $N$ be large natural numbers (say ~200 bits). Let $L$ be a smaller number, (say ~100 bits). I want to approximate the fraction: $$\frac{M}{N} \sim \frac{k}{L+r}$$ where $r$ is at most $L$. In ...
6 votes
1 answer
300 views

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}^...
0 votes
0 answers
49 views

Is polynomial interpolation with RKHS in some way more advantageous than simple Lagrange interpolation?

[Question originally posted here but maybe it is more suitable for this site.] The reproducing kernel Hilbert space associated with the polynomial kernel $K(x,z)=(1+xz)^{d-1}$ (or other similar ...
5 votes
2 answers
342 views

How expressive is $e^A$ in the sense of universal approximation?

For any real matrix $B\in\mathbb{R}^{n\times n},n\ge 2$ and precision $\varepsilon$, is there a real matrix $A\in\mathbb{R}^{n\times n}$ such that $\|e^A-B\|_F<\varepsilon$? ($F$ refers to ...
0 votes
1 answer
611 views

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 ...
0 votes
1 answer
124 views

Rational approximation for continuous function on curve $\Gamma$

Let $\Gamma \in C^{1,\lambda}$ be an oriented Jordan curve in complex plane $\mathbb{C} $, $\mathrm{R}(\Gamma)$ the set of all rational functions without poles on $\Gamma $. "$\mathrm{R}(\Gamma)$...
1 vote
1 answer
104 views

$L^1$ 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}}\...
3 votes
1 answer
82 views

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 ...
1 vote
1 answer
113 views

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 ...
1 vote
1 answer
87 views

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 ...
1 vote
1 answer
81 views

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 ...
0 votes
0 answers
68 views

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 ...
8 votes
6 answers
7k views

How to approximate a solution to a matrix equation? [closed]

Suppose a matrix equation $Ax = b$ has no solution ($b$ is not in the column space of $A$) How can I find a vector $x^\prime$ so that $Ax^\prime$ is the closest possible vector to $b$?
0 votes
0 answers
59 views

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 ...
4 votes
0 answers
542 views

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 ...
2 votes
1 answer
109 views

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\...
6 votes
0 answers
400 views

Using the Lorentz operators to build polynomials that converge to a continuous function

Questions Let $f(\lambda):[0,1]\to (0,1)$ have a $\beta-\lfloor\beta\rfloor$)-Hölder continuous $\lfloor\beta\rfloor$-th derivative, where $\beta>0$. Find explicit bounds, with no hidden constants,...
2 votes
1 answer
408 views

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 ...
0 votes
0 answers
118 views

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 ...
8 votes
0 answers
513 views

Concave and other bounded functions: Series representation and converging polynomials

Main Question Suppose $f:[0,1]\to[0,1]$ is continuous, polynomially bounded, and belongs to a large class of functions (for example, the $k$-th derivative, $k\ge 0$, is continuous, Lipschitz ...
4 votes
1 answer
575 views

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-...
0 votes
0 answers
48 views

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 $L_n$...
0 votes
0 answers
107 views

$\log$-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+...
11 votes
1 answer
1k views

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(\...
4 votes
4 answers
2k views

When we use Bernstein polynomials in application

When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", &...
0 votes
0 answers
86 views

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 ...
2 votes
0 answers
134 views

"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\...
3 votes
0 answers
355 views

A conjecture on consistent monotone sequences of polynomials in Bernstein form

A Conjecture In the following, a polynomial $P(x)$ is written in Bernstein form of degree $n$ if it is written as— $$P(x)=\sum_{k=0}^n a_k {n \choose k} x^k (1-x)^{n-k},$$ where $a_0, ..., a_n$ are ...
17 votes
2 answers
1k views

Explicit and fast error bounds for polynomial approximation

Main Question This question is about finding explicit, calculable, and fast error bounds when approximating continuous functions with polynomials to a user-specified error tolerance. EDIT (Apr. 23): ...

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