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^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 $m$.
...
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1
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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 ...
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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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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 ...
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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{...
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1
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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 ...
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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 $X$ ...
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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 ...
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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
...
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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 ...
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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
$$
S_N(f):=\sum_{...
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2
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Approximation for complex variables
Approximation theory, which aims to provide the optimal polynomial function approximating the target function in a given domain such as $x\in[-1,1]$, has been well-developed for real variables. In ...
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Solution or approximation to $\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx$
I'm looking for a solution or approximation to the following indefinite integral $$\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx.$$
I've tried Mathematica, but it does not converge to a solution....
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Sum of terms after partial fraction decomposition
I am facing the following problem (all $a_i$ being positive and all different)
$$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+\...
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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 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 \...
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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' \...
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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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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/...
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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 ...
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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 ...
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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 ...
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1
answer
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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
$$
\|f - \operatorname{Lip}_L(\mathbb{R}^...
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Is polynomial interpolation with RKHS in some way more advantageous than simple Lagrange interpolation?
[Question originally posted here but maybe it is more suitable for this site.]
The reproducing kernel Hilbert space associated with the polynomial kernel $K(x,z)=(1+xz)^{d-1}$ (or other similar ...
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2
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How expressive is $e^A$ in the sense of universal approximation?
For any real matrix $B\in\mathbb{R}^{n\times n},n\ge 2$ and precision $\varepsilon$, is there a real matrix $A\in\mathbb{R}^{n\times n}$ such that $\|e^A-B\|_F<\varepsilon$? ($F$ refers to ...
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Fast decaying Fourier coefficients for indicator function
Let $0 \leq a < b \leq 1$. I wanted to compute the Fourier series expansion of the indicator function $f = \chi_{[a, b]}$ of the interval $[a, b]$, as
$$
f(x) = \sum_{k\geq 0}a_k e(kx).
$$ My ...
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1
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Rational approximation for continuous function on curve $\Gamma$
Let $\Gamma \in C^{1,\lambda}$ be an oriented Jordan curve in complex plane $\mathbb{C} $, $\mathrm{R}(\Gamma)$ the set of all rational functions without poles on $\Gamma $. "$\mathrm{R}(\Gamma)$...
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1
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$L^1$ error between indicator function and smoothed out version
For a large parameter $r>0$, consider the indicator function $1_{[-r,r]}$ and its convolution with the (normalized) Gaussian $\frac{1}{\sqrt{\pi}}e^{-x^2}$, that is,
$$f_r(x) = \frac{1}{\sqrt{\pi}}\...
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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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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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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 $b$?
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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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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 ...
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1
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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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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,...
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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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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 ...
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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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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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$\log$-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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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:
$$\sqrt{x} = \sum_{n=0}^{\infty}\frac{\left(\prod_{k=1}^{n}\left(\...
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When we use Bernstein polynomials in application
When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", &...
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Approximation error of Chebyshev expansion of the second kind
Weierstrass' well known theorem states that every continuous function on $[-1,1]$ can be uniformly approximated to arbitrary precision by a polynomial function. Among these approximations it is known ...
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"Almost rational" irrational
This is a follow-up to an older question.
Let $r\in \mathbb{R}\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\...
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A conjecture on consistent monotone sequences of polynomials in Bernstein form
A Conjecture
In the following, a polynomial $P(x)$ is written in Bernstein form of degree $n$ if it is written as— $$P(x)=\sum_{k=0}^n a_k {n \choose k} x^k (1-x)^{n-k},$$ where $a_0, ..., a_n$ are ...
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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): ...