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.
613 questions
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Rational all-pass interpolation of unitary-valued functions on the circle modulo torus–Weyl action
Let $\mathcal U(m)$ denote the unitary group, $T = \mathcal U(1)^m$ be the maximal torus of diagonal unitary matrices, and
$W = S_m$ be the Weyl group acting by column permutation.
Consider the right ...
2
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0
answers
89
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Asymptotic of dimensions of subvarieties of linear spaces that are nearly norm-dense in the unit balls
This is a follow-up to this question. I will start with the problem statement first (here, $B_n$ denotes the closed Euclidean unit ball of $\mathbb{R}^n$):
Fix sufficiently small $\varepsilon > 0$....
- 5,085
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81
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Carl's inequality on the $\ell_1$ ball in $\mathbb{R}^n$
I am having a hard time reconciling two facts about Kolmogorov widths and entropy numbers of a given convex body $K \subset \mathbb{R}^n$ with respect to the Euclidean distance. Let $K = B_{1}^n = \{\...
2
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2
answers
342
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Extension of the theorem of Stone-Weierstrass for vector lattices to the space $C_0(X)$ for locally compact Hausdorff spaces $X$
I am looking for a generalisation of the following version of the theorem of Stone-Weierstrass for vector lattices:
Theorem Let $X$ be a compact Hausdorff space and let $S \subset C(X, \mathbb R)$ ...
- 153
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1
answer
108
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Dealing with floor and ceil in Riemann-type approximations
I'm trying to prove the convergence of the following sum:
$$S_n := \sum_{k = \lceil a \sigma_n + \mu_n \rceil}^{\lfloor b \sigma_n + \mu_n \rfloor} \frac{1}{\sigma_n \sqrt{2\pi}} \exp\left( -\frac{(k -...
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Conjecture : $\left \lfloor{\displaystyle 1.5^n} \right \rfloor > \displaystyle \frac{3^n+1.5^n}{2^n}-1 $ , $\forall n \in \mathbb{N} > 1$
Motivation :
I want to prove $\forall n>1, d\mid 3^n-2^n \Rightarrow v_2(d+1)<n$ , but $\forall n>1$ , not just for a sufficiently large $n$ or $d$
My attempt :
We will first consider $3^n=2^...
- 211
5
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2
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392
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Sharp $L^{\infty}$ Bernstein inequality for bandlimited functions
I'm interested in finding a proof in the literature for the following result:
Let $f(x)$ let be a smooth function $\mathbb{R} \rightarrow \mathbb{R}$ such that $\hat{f}$ exists and is supported on $[-...
2
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0
answers
249
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Can we compute the Gaussian tail quickly with decent precision? (Former "Is this logistic approximation to the Gaussian integral valid?")
The original text is below. I suggest the following edit and reopening:
In some practical applications there is a need to evaluate the antiderivative for the Gaussian function on the line quickly ...
- 41
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357
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The linear span of the functions $φ(a+xt)$ is dense in $C([0,1])$, if and only if $\varphi$ is non-polynomial?
Original question: Let
$\varphi:[0,1]\!\to\!\mathbb R$ be continuous, strictly increasing, non-polynomial;
$\lambda$ be a finite signed Borel measure on $[0,1]$.
Assume that for every pair $(a,x)$ ...
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1
vote
1
answer
123
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Universal approximation theorem for nonnegative functions [closed]
The classical universal approximation theorem says that functions of the form $f(x)=\sum_{i=1}^N\alpha_i\sigma(w_i^Tx_i+b_i)$ can approximate any continuous function from a compact subset of $\mathbb{...
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162
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On the expected quality of a rank-$1$ approximation of a complex Gaussian noise covariance matrix
I am developing quality coefficients for a specific type of approximation of covariance matrices. I want to to specify meaningful lower bounds (or rather thresholds), for which I would like to use the ...
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$\ell_1$ norm of coefficients vector in polynomial interpolation (reference/history request)
A basic kind of multivariate polynomial interpolation formula for degree-$d$ polynomials has the following shape:
Theorem sketch. Let $z\in \mathbf{D}^n$ and $X=X_d\subset \mathbf{D}^n$ be the finite
...
- 151
3
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138
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Approximating a function on the positive reals by certain entire functions
Consider the following class of entire functions:
$$E=\left\{g\in H(\mathbb{C}) :\forall k\in \mathbb{N}, \sup_{\Re(z)\geq -k} |g(z)|e^{-|z|/k}<\infty \right\}.$$
Let $f(x)\in C^\infty([0,1])$. ...
- 323
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52
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Convergence Rate Truncates Besov-Type Sequence Norm
Let $x:=(x_{i,j})_{i\in \mathbb{N},\, j=0,\dots,2^i}$ be a real-sequence and consider the (small) Besov-type sequence spaces with quasi-norms for $0<q,p,\alpha< \infty$
$$
\|x\|_{\alpha,p,q}
:=
\...
2
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1
answer
148
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Decaying rate of minimax of an inner product
For $n = 1, 2, \dots,$ let $f_n (x) = x^n - \frac{1}{n+1}$ for $x \in [0, 1].$
Consider the quantity
$$
R (N)
=
\inf_{c \in \mathbb{R}^N, \|c\| = 1}
\sup_{x \in [0, 1]}
\sum_{n = 1}^N c_n f_n (x)
,
$$
...
- 105
2
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166
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Approximation by Blaschke products in Carathéodory's theorem
A classical theorem of Carathéodory says that if $f$ is a holomorphic function in the unit disc such that $|f(z)| \leq 1$, $z \in \mathbb{D}$, then there exist finite Blaschke products $B_n$ which ...
4
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1
answer
333
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Bounds on Sobolev norms of mollified functions - interpolation
Let $a,b\in \mathbb{R}$ with $a< b$, $n\in \mathbb{N}$ with $n>0$, $1\le p\le \infty$, let $W^{a,p}(\mathbb{R}^n)$ be the corresponding Sobolev space, and let $(m_{\delta})_{\delta>0}$ be a ...
- 57
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0
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119
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Approximation subspaces in Hilbert spaces
Let $H$ be a Hilbert space and $(v_n)_{n = 1}^N$ be as sequence of normalized vectors in $H$ and let $\varepsilon > 0$. The following seems like a problem some people have already thought about: ...
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65
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Question about the number of correct digits of $e$ generated by a self-iterated quadratic map
For a fixed $n$, the sequence is defined as $S_n(k+1) = S_n(k) \times S_n(k)$, with $S_n(0) = 2^n + 1$. It's quite obvious that the first $n$ binary digits of $S_n(n)$ match those of $e$, give or take....
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0
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106
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Does sequential rank-one approximation (Eckart–Young Theorem) yield a global minimum?
Problem Formulation
Given real values $ z_{i,j}^{(k)} $ for indices
$
i = 1,\ldots, n, \quad j = 1,\ldots, m, \quad k = 1,\ldots, p,
$
our goal is to estimate the parameters $\alpha_{i}^{(k)}$, $\...
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1
answer
84
views
Does the Weighted Average of $f$ with $b_{n,K}$ Equal $f(n/K)$?
Let’s suppose that for an affine function
$$
f(u) = au + b,
$$
the Bernstein operator reproduces $f$ exactly; that is, the Bernstein operator $B_K(f)$ satisfies
$$
B_K(f)(x) = \sum_{n=0}^{K} f\left(\...
- 105
1
vote
1
answer
246
views
Riemann sum error bounds without using derivatives or the Mean Value Theorem?
I know that if a function $f$ is continuously differentiable on $[a,b]$, one can bound the difference between its integral and a Riemann sum using either the Mean Value Theorem or Taylor expansions. ...
- 37
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151
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Proof of a more general statement for Weierstrass approximation theorem
It is written in a wikipedia article that
Let $f$ be a continuous function on the interval [0, 1]. Consider the Bernstein polynomial
$$B_n(f)(x) = \sum_{\nu = 0}^n f\left( \frac{\nu}{n} \right) b_{\...
user553626
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2
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Why exactly is Simpson's rule better than the Trapezoidal rule? [closed]
I am reading up on numerical integration and have trouble to really understand why or rather in what sense Simpson's rule is better than the Trapezoidal rule in general. There is a lot of stuff ...
5
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0
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107
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What is the maximal advantage of randomized over deterministic algorithms for approximation in the worst-case?
Let $X\subset Y$ be Banach spaces and $B_X:=\{x\in X: \|x\|_X\le1\}$ be the unit ball of $X$.
The goal is to find an approximation of every element from $B_X$ with error measured in $Y$ by using at ...
-1
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1
answer
199
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On an equality obtained from analysis of an algorithm
I came across the following equality $$\Big(\frac{c n^b}{x}\Big)^{dx}=n^{n/x}$$ where $b,c,d\geq0$ are fixed and $n$ increases when I was analyzing an algorithm.
What is a good approximation for $O(n^{...
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0
answers
62
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Approximation of the function $f(z)=z^2/|z|$ by $C^1$ immersions
Let $D$ denote the unit disk in $\mathbb C=\mathbb R^2$. We consider the function $f:D\rightarrow\mathbb C $ defined by $$f(z):=\frac{z^2}{|z|}.$$ Then as proved in Global invertibility (p324 Remark 4)...
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163
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On mollifiers acting between $L^2$ and Sobolev spaces
(I'm reposting here this question from MSE as it didn't receive any answer for two weeks.)
Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by
$$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{...
- 617
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0
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Solve coupled ODEs analytically in the limit of a small parameter
I have the following set of coupled second order non-linear ODEs :
$$ x^2 a''(x) + x a'(x) - \Big(\frac{1}{\epsilon^2}\Big)b^2(x) a(x) = 0 \\
x b''(x) - b'(x) - 2x b(x)a^2(x) = 0$$
with boundary ...
- 11
3
votes
1
answer
238
views
How can discrete Fourier transform approximation prove the completeness of complex exponentials in $L^2(T)$?
I have a question about the completeness of complex exponentials in function spaces.
For the discrete set $ S = \{1, 2, \ldots, n\} $, it is clear and intuitive that $ e^{2\pi ikx/n} $ for $ k = 0, 1, ...
- 1,017
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2
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433
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Approximation of smooth real functions with an integer integral by a sum of $\delta$-functions with coefficients $\pm 1$
This question arose from a quantum physics research (my article Entropy 2022, 24(2), 261, where I considered approximation of a smooth charge density (built from a wave function) by a collection of ...
- 183
1
vote
0
answers
209
views
Difference of two completely monotonic functions
We know by the Hausdorff-Bernstein-Widder theorem that any completely monotonic function on the positive half line $[0, \infty)$ is given by the Laplace transform of a positive Borel measure on $[0, \...
1
vote
3
answers
310
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The approximated function of $\mathbb{E}\left\{ \ln\left(1+X\right)\right\}$, where $X\sim\operatorname{Gamma}\left(\kappa\ge1,\theta>0\right)$
Given $X \sim \operatorname{Gamma}(\kappa, \theta)$ with CDF $F_X(\kappa, \theta)$, where $\kappa \geq 1$ and $\theta > 0$, the expected value of $\mathbb{E} \left\{ \ln(1+X) \right\}$ is ...
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605
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The curse of dimensionality of the Kolmogorov–Arnold neural network
The Kolmogorov–Arnold neural networks (KAN), Ziming Liu et al., KAN: Kolmogorov–Arnold Networks is inspired by the Kolmogorov–Arnold representation theorem (KA theorem). Though it is not proved in the ...
- 2,363
1
vote
1
answer
189
views
Sufficient condition for uniform convergence of the Stieltjes transform
Let $\mu$ be a probability measure and $\mu_N$ be a sequence of probability measures. For simplicity we may assume them to have compact supports contained in $[-1,1]$. Define
$$G_\mu(z):=\int\frac{\mu(...
- 400
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votes
1
answer
162
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Approximation on $H^1_0(B)$ and cut-off functions
Let $u \in H^1_0(B)$, where $B$ is the unit ball in $\mathbb{R}^N$. Given $\epsilon > 0$, I need to show there exists a function $\chi_\epsilon \in C^\infty_0(\mathbb{R}^N)$ such that
$$
\| u - \...
- 239
0
votes
0
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153
views
Hilbert spaces that include algebraic polynomials
This question is motivated by a phrase I found in several books/papers about approximation theory, for example, M.J.D.Powell's Approximation Theory and Methods: ''Let $\mathcal{H}$ be a Hilbert space ...
- 53
1
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0
answers
157
views
Multivariate polynomial approximation
Let $f$ be a function on $[-1,1]^d$ with some smoothness property, for example, it is in the Sobolev space $W^{k,p}$.
Let $P_n$ is a space of polynomials with degree $n$. My question is what is the ...
- 53
3
votes
1
answer
260
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Density beyond Stone–Weierstrass
$\DeclareMathOperator\tr{tr}$I need density assertions for spaces of polynomials which are not (that I know of) algebras. One goes like this: Fix $n\in\mathbb N$ let $S$ denote the set of self-adjoint ...
- 761
4
votes
1
answer
263
views
Multivariate polynomial approximation of functions in Sobolev space
I found a result of the estimation error of polynomial approximation in page
6 of https://scg.ece.ucsb.edu/publications/theses/ARajagopal_2019_Thesis.pdf
The statement is for $f \in W^{k, p}\left([-1,...
- 53
1
vote
1
answer
330
views
approximating differentiable functions with double trigonometric polynomials
Let $Q = [0,1]^2$. For sake of notation, let
$$
f^{(i,j)}(x,\xi) = \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x,\xi).
$$
Fix some non-negative integer $k$. Moreover let $f\in C^k(Q)$ if
$$
\|...
- 171
0
votes
1
answer
254
views
Numerical integration with integrable singularity
Suppose I have a numerical estimation of discrete samples of a smooth function $C(t)$ at $t = a, \dots, T = Na$ and I want to (numerically) compute the integral of $f(t) = \frac{C(t)}{\sqrt{t}}$. In ...
- 33
3
votes
0
answers
515
views
Interchange limit and supremum of functionals over a bounded convex set
Let $(H, \langle\cdot,\cdot\rangle)$ be a separable real Hilbert space and $B\subset H$ be (nonempty) convex and bounded, and suppose that $(\alpha_k)\subset H$ is a sequence for which the limit $\...
- 453
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votes
1
answer
258
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 ...
- 3,745
2
votes
0
answers
199
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 ...
- 828
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votes
0
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260
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
2
votes
0
answers
66
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
...
- 3,788
-1
votes
1
answer
311
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 ...
- 149
4
votes
3
answers
859
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 ...
- 149
1
vote
1
answer
98
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+\...
- 1,502