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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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Rigorous error bounds for polynomial regression

Consider a set of $N$ points $(x_i , y_i)$. I want to find a $d$ degree polynomial $P_d(x)$ that will minimize the error, $$ e_d = max_{i \in [N]} ~|P_d(x_i) - y_i| $$ The question I have is about ...
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34 views

Polynomial approximation in Sobolev spaces

Let $U \subseteq \mathbb{R}^n$ be an open connected set. Let $p>1$, and let $f \in W^{1,p}(U)$. I have recently heard that $f$ can be locally approximated in $W^{1,p}$ by polynomials. That is, ...
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13 views

Simultaneous asymptotic expansion in multiple points

Let $\Omega\subset\mathbb R$ be open and connected. Assume I have some non-linear, smooth function $g:\Omega \to\mathbb R$. Given disjoint base points $a_1,\ldots,a_n\in \Omega$ (and possible $\pm\...
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30 views

Boundary value problem results in system of three non-linear sine equations

I have the following equation which I am trying to find an exact solution for if possible, if not at least some approximation. The equation in general is a simple sine function, with an unknown ...
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17 views

Analytical approximations of infinite tetration?

What are some analytical approximations of infinite tetration within its convergence radius? https://en.wikipedia.org/wiki/Tetration#Extensions_of_the_domain_for_(iteration)_%22heights%22
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1answer
86 views

Representation of $\pi$ using algebra and exp/log.

Can $\pi$ be represented exactly using a mixture of algebraic as well as exp/log functions, all real valued? I know it can't be done using only algebra since its transcendental, but what if we ...
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16 views

randomized approximate matrix inverse or adjoint of a square matrix

I have been reading about some random matrix theory, JL, and related topics and am wondering if there are any methods to calculate an approximate inverse of a SPD matrix $\mathbf{A}$, or possibly even ...
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1answer
19 views

Closed linear span of translations of simple step functions

This paper utilizes Wiener's tauberian theorem to indicate that the closed linear span of translations of any simple step function is equal to $L^p[a,b]$, where $1< p \leq \infty$ and $[a,b]$ are ...
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56 views

Barycentric subdivision of planar graph approximating a (top.) path

First of all: I started reading about simplicial complexes only recently (I'll do my best to get the terminology right). For the problem: Suppose we are given a planar drawing of a cycle (i.e. Jordan ...
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102 views

Can we provide a good estimation for $(n!)!$?

I was thinking about this $$(n!)!$$ for $n\in\mathbb{N}$. I wanted to find a suitable approximation, or in any case a very good estimation for this. My first idea was to use Stirling ...
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66 views

Converse of Taylor's Theorem

Let $n$ be a nonnegative integer and $a,b\in\mathbb{R}$ such that $a<b$. From Taylor's Theorem, we know that any $n$-time differentiable function $f:(a,b)\to \mathbb{R}$ satisfies the condition ...
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29 views

Approximation of a continuous function by piecewise constant function

Let $f:(0,1) \to \mathbb{R}$ be continuous and increasing. Define $$f_n(t) := \sum_{i=0}^{n-1} f(T^n_i)\chi_{(T^n_i, T^n_{i+1})}(t)$$ where $\{T^n_0, T^n_1, ..., T^n_n\}$ is a uniform partition of $(0,...
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33 views

proof of chebychev polynomial

Can someone explain the Intuition or the reasoning of the following Chebychev Theorem (page 72 Theorem 2.7.2): In order that the ordinary polynomial $P(x)$ among all polynomials of degree < $n$ ...
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1answer
47 views

Advantage of Bernstein polynomial basis

The well-known "Bernstein polynomials" on the interval [0,1] are defined as $$ B_{N,i}(x)=\binom{n}{i}x^{i}(1-x)^{n-i}, \ \ i=0,...,N. $$ My question is about advantage of these polynomials in ...
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25 views

Approximating Log(Gamma(z)) for small z as Log(Gamma(z + 1)) - Log(z)

I'd like to implement a numerical approximation to the log Gamma function, and I found Gergő Nemes' approximation described here: https://en.wikipedia.org/wiki/Stirling%27s_approximation. This seems ...
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How far apart can L1 and L2 lines fit to the same data be?

Given $n$ points $(x_i, y_i)$ in the unit square with $x_i = {{i - 1} \over {n - 1}}$ uniformly spaced and $0 \leq y_i \leq 1$, consider the best-fit L1 line and the best-fit L2 line: $ \qquad$ L1 ...
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1answer
45 views

Proving this condition for convergence in a Banach space

I have difficulty proving the following claim from a paper (a free version is here, see Lemma 2.4 on page 9): Let in a Banach space $X$ a sequence $\{x_n\}_{n=1}^\infty$ be given. Assume that for ...
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56 views

Simpson's Rule in Matlab [closed]

I have made the following code based on Simpson's expansion: function I = simprule(f, a, b, n) h = (b-a) / n; x = a:h:b; S = 0; L = 0; for l = 1:2:n %generates the odd number array S = S + 4*...
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27 views

Convergence rate of multivariate polynomials.

Given a continuous function $f:[a,b]\rightarrow\mathbb{R}$, it is a well known result that, for each $n\geq0$, there exists a polynomial $p_n$ that best approximates $f$ in $\|\cdot\|_\infty$ among ...
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35 views

Chebyshev coefficients of $e^{-x}$

Im trying to derive, given $n$, the Chebyshev coefficients $c_k$ of $e^{-x}$ on [-1,1]. That is $c_k=\frac{(e^{-x},T_k)_w}{(T_k,T_k)_w}$, $k=0,1,...,n.$ I have problems computing $(e^{-x},T_k)_w= \...
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21 views

Spline Approximation Results in $L^2(\mathbb{R})$ norm

I have seen in several papers that one can approximate a function $f \in C^{(k-1)}$ via splines in $S_\pi^k$ of order $k$ with extended knot sequence $\pi$ using a local approximation operator $Q: C^{(...
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1answer
41 views

approximating irrational roots of algebraic equations with the Pierce expansion

Let $ p (x) = 1 - x \lfloor \frac{1}{x} \rfloor $ then the Pierce expansion of a real number $x \in {R}$ is expressed by \begin{equation} x_1 = \sum_{n = 1}^{\infty} (- 1)^{n + 1} \prod_{m = 1}^n ...
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27 views

Approximating a piece-wise function

I would like to approximate a piece-wise function. The aim is to get a function as $f(x) \approx ...$ without piece-wise definition (only one expression, not depending of $x \leq 1$ or $x \geq 1$), ...
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1answer
14 views

Question about $O$

Consider two non-negative sequences $f(n)$ and $g(n)$ and suppose that $\exists C>0, \ \forall \varepsilon >0, \exists N \in \mathbb{N}$ s.t. $$\forall n \geq N, \ \ f(n)<C \cdot g(n)+n^{\...
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114 views

What is the optimal Fourier series convergence rate estimate for $|x|$?

What is the known best estimate of the rate of convergence in $\|\cdot\|_\infty$ (or maximal absolute value) of the Fourier series of $|x|,\, x\in[-1,1]$? If I look at the coefficients of the Fourier ...
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1answer
71 views

Smooth floor function

I want a monotonic function on the positive real numbers that behaves like floor but in smooth way, like smoothstep but for all integers. It should follow this simple rule. slope is zero at ...
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1answer
39 views

Smart tricks used to prove formulas by approximation

I've recently seen a proof of the inversion formula for the Fourier transform for $f,\hat{f}\in L^1(\mathbb{R}^d)$. The main idea of the proof is this We have $$\int_{\mathbb{R}^d}e^{2\pi i x\...
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1answer
50 views

Show the best linear approximation $p$ to $f$ has slope $p'(x)=(f(b)-f(a))/(b-a)$.

Suppose $f\in C([a,b])$ is twice continuously differentiable and $f''(x)>0$ on the interval. Show that the best linear approximation $p$ to $f$ has the slope $p'(x)=(f(b)-f(a))/(b-a)$. To my ...
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1answer
68 views

Vector-valued Weierstrass theorem

I'm looking for a version of Weierstrass's approximation theorem that works for a continuous function $f:D \to \mathbb{R}^d$. Versions that I know of Multivariate Weierstrass theorem? uses a ...
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146 views

recurrence relation in approximating theory

I am stuck in a little part of a problem: I wish to give an approximation of $\sin(\pi x)$ on $-1 \leq x \leq 1$, when using the polynomial $$F_N(x)=\sum_{k=0}^{N}a_kx^{2k+1}$$ with the coefficients $...
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38 views

Looking for a way to move from $y = f(x)$ to $ln(y) = F(ln(x))$

Greeting. I have to find the value of P: $P = \sum_{0}^{r}{p(k)} $ where $p(k) = \binom{N}{n_1}^{-1}\binom{N}{n_2}^{-1}\binom{N}{k, n_1-k, n_2-k}$ for given $n_1, n_2, N$ The problem is for some $...
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Using the Saddle point method (or Laplace method) for a multiple integral over a large number of variables

I am trying to understand the saddle point method used in the large N limit of matrix models. First, for the case of the integral of a single variable I found this notes There they say that you can ...
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1answer
22 views

Approximation by biholomorphisms

Assuming two domains $\Omega_1 \subset \Omega_2 \subset \mathbb{C}$ satisfy the criteria in Runge's theorem. So we know that any holomorphic function $f: \Omega_1\to \Omega_1$ can be approximated ...
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1answer
84 views

Simple L^2 bound for bivariate Sobolev function on a square

I have a rather basic question about Sobolev functions. I would need a reference or proof for the following inequality which seems to be well-known in approximation theory. Question: Let $\Omega=[x,x+...
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57 views

Suppose $f$ is in $C([0,1])$ and $f(0)=f(1)=0$. Show that $f$ is the uniform limit of a sequence of polynomials with $p_n(0) =p_n(1) = 0$ for all $n$.

I am stuck on how to start. My idea is the following but I am not sure how it will lead to anything: We know that because $f$ is continuous function on real interval $[a,b]$, then for all $\epsilon$, ...
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2answers
25 views

How can one measure how well is a function approximated over an interval?

I am currently studying Taylor polynomials. I was wondering if the fact that a (continuous) function approximates another over an interval can be quantified? At a point it is easy, you just compute $|...
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23 views

Can you help me find a Fourier transform-able approximation function basis for compression?

I have four-dimensional, piece-wise smooth, discrete (4D voxel) data that I want to approximate/ compress using as few basis functions as possible. The data are discontinuous in three dimensions, ...
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1answer
44 views

Uniform approximation on $[0,\infty)$

Let $f:[0,\infty) \to \mathbb R$ be a continuous function such that $f(x) \to 0$ as $x \to \infty$. If $\epsilon >0$ then there exists a polynomial $p$ such that $|f(x)-e^{-x}p(x)|<\epsilon$ for ...
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1answer
154 views

Approximation of $\sin \pi x$ on $-1 \leq x \leq 1$

I am stuck in the following problem: I wish to give an approximation of $\sin(\pi x)$ on $-1 \leq x \leq 1$, when using the polynomial $$F_N(x)=\sum_{k=0}^{N}a_kx^{2k+1}$$ with the coefficients $a_k$ ...
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0answers
42 views

Least squares regression (LSR)

I have the following Least square regression problem: $\underset{rank(X)\leq k}{\arg\min} \lVert DX - L \rVert_F^2$ Let's suppose I compute a QR decomposition of $D$ as $D = Q_DR_D$ and solve the ...
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59 views

Sobolev approximation lifts to $L^p$ convergence of the exterior powers

I am reading the book "Geometric Function Theory and Non-linear Analysis", where the following claim is used: Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set. Let $f \in W^{1,s}(\Omega,\...
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Show that for any open subset of $\Bbb R$, there is a fraction with prime terms that belongs to it [duplicate]

Be $\Bbb P\Bbb Q$ the set of all fractions $f_{m,n}=\frac{p_m}{p_n}$ whose numerador and denominator are both prime numbers. i) Show that for any open set $A\subset \Bbb R^+$, there is at least one $...
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1answer
47 views

Stone Weierstrass on Banach algebras

Let $B$ be a complex Banach algebra. Let now $f\in \mathcal{C}(B)$ and $X$ be a compact subset in $B.$ Is there any version of the Stone Weierstrass theorem which asserts that we can approximate $f$ ...
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1answer
30 views

Writing an integral in terms of the Hypergeometric Function

I have the following function defined as an integral: $G(x,k,s) = 1 - (k-1) \int_0^{x/s} (1-t)^{k-2} dt$ Or alternatively directly as: $G(x,k,p) = (1-\frac{x}{s})^{k-1}$ Is there any way (or ...
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199 views

Approximation Theory for Deep Learning Models: Where to Start?

I am working as a novice-developer for company with Deep-learning(DL) Frameworks. DL is basically consists of several layers of combination of linear and non-linear(usually using ReLU) with millions ...
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Is there any bound/approximation of Stirling numbers of the second kind of the form $S(ax+b, x+1)$?

I'm looking for an approximation or upper bound for Stirling numbers of the second kind that look like this: $S(ax+b, x+1)$, where a and b are fixed constants. Any tips are appreciated.
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Approximate formula from data (7 inputs, 1 output)

I'd like to approximate a formula which 'fits' my data. The formula should take 7 inputs and produce a single ouput. The input variables are listed in columns A to G of my data (see image below). The ...
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83 views

Normal approximation to the pointwise/Hadamard/Schur product of two multivariate Gaussian/normal random variables

Let $X \sim \mathcal{N}\left( {{\mu _x},\sigma _x^2} \right)$ and $Y \sim \mathcal{N}\left( {{\mu _y},\sigma _y^2} \right)$ be two univariate and independent Gaussian/normal random variables and let $...
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1answer
47 views

Approximation of dynamic systems

I was looking for a formal approach to simplify models of dynamic systems. Say we have a dynamic system given by $\frac{dx}{dt} = f(t,x,u), ~~~~x(t_0) = x_0$ $y = g(x)$ We know $f$ and $g$ but ...
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18 views

Probability of Getting Several Nice Sequences of Coin Tosses/Approximating Erf^{-1}

Given $\epsilon_1>0$ and a natural number $m$, I want a lower bound on $N$ that guarantees that with probability, say, at least $.995$, the number of heads in each of $m$ different sequences of $N$ ...