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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111 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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121 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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1answer
398 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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1answer
176 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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56 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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54 views

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
55 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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267 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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36 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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1answer
55 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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36 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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165 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
328 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
50 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
93 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
170 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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1answer
163 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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153 views

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
28 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
101 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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2answers
64 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
47 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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29 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
115 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
247 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
53 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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2answers
60 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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30 views

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
87 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
37 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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2answers
643 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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0answers
25 views

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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1answer
86 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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119 views

Zolotarev number and commuting matrices

Recently in a post (link) upper bounds on the singular values $\sigma_j(X)$ of a matrix $X$ have been considered. To restate the central observation, it says that if $AX−XB=F$ for $A$ and $B$ normal ...
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1answer
71 views

Show that there exists 2 different best approximations

Let $F$ be a linear space with a norm $\lVert \cdot \rVert$ which is not strictly convex. Show that there is a function $f \in F$ and a subspace $S \subset F$, so that $f$ has different best ...
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119 views

Uniform approximation of second derivative via Bernstein polynomial

Let $f:[0,1]\mapsto(0, +\infty)$ be a continuous function. Define by $$ B_k(f,t):=\sum_{j=0}^k f(j/k) {k \choose j} t^{j} (1-t)^{k-j}, \quad t \in [0,1] $$ the associated Bernstein polynomial. For $...
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2answers
171 views

Prove that a set is not strictly convex

I want to prove that the set $\{ f:||f||_{\infty}=1 \}$ where $f$ belongs to the space of continuous functions on $[a,b]$ is not a strictly convex set. As a counterexample, I'm asked to use $f(x)=x$ ...
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0answers
53 views

Approximating $\frac{\frac{N}{2}!\frac{N}{2}!}{(\frac{N}{2}-m)!(\frac{N}{2}+m)!}$ without using logs

This question came up in a recent problem. It basically states: In the limit of $N\gt\gt m \gt\gt 1$ show that $$\frac{\frac{N}{2}!\frac{N}{2}!}{(\frac{N}{2}-m)!(\frac{N}{2}+m)!} \approx \exp\left({-\...
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0answers
44 views

approximate bijective function such that the inverses are bijective and “easily” computable

I have a infinitely differentiable, bijective function $f:[0,1]\to[0,1]$, and I would like to approximate this function by a series of other functions $T_i$ (think Taylor) – with the conditions that ...
2
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1answer
68 views

Approximation for the following integral needed

I have following integral $$\int_0^{\infty}e^{-ax-bx^m}dx$$ where $a>0, b>0, m>1$. I can get an approximation for the above integral when $b$ is small. However, I want to get an expression ...
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1answer
71 views

Best approximation of a function out of a closed subset

I'm studying approximation theory and I saw this exercise on Rivlin book an introduction to the approximation of functions: Prove that if $V$ is a normed linear space, $W$ a finite-dimensional ...
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27 views

Clarification for definition of admissible: $\Delta\in (K)$

I am reading through the following book: E.M. Nikishin, V.N. Sorokin: Rational Approximations and Orthogonality, Translations of Mathematical Monographs, vol. 92, Amer. Math. Soc., Provindence RI, ...
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1answer
91 views

Almost locality of cubic spline interpolation

The natural cubic interpolating spline is the unique $C^2$, interpolating cubic spline, endowed with two extra boundary conditions. Obtaining this spline, denoted by $s(x)$, involves the inversion of ...
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1answer
90 views

construction of entire function by using Runge's Approximation theorem

I have been trying the some exercises given in the book of "Robert E green and Krantz".But I am unable to do it,the exercise is following Construct a sequence of entire function $f_j$ such that $f_j$ ...
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1answer
85 views

Linear programming or mixed integer linear programming approximation

This may well be a stupid question. Given a one-dimensional non-convex/concave piecewise linear $\mathbf R\to\mathbf R$ function, is there a way to translate its minimization problem into a linear ...
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0answers
184 views

Curve fitting N points with n(fixed) quadratic curves

I essentially have a constrained curve fitting problem that I need to solve efficiently. The following problem arises when performing practical calibration of RSSI (signal strength), providing ...
4
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1answer
56 views

Relationship between the Prime and Triangular numbers ${T_n\over P_n}\sim {n\ln{\pi}\over 2\ln{P_n}}$

I was observing $T_n:=1,3,6,10,15,...$ and $P_n=2,3,5,7,...$ of these sequences. $T_n={n(n+1)\over 2}$ is the triangular numbers and $P_n$ is the prime numbers We came acrossed this relation ...
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1answer
33 views

calculate the derivative of a function containing factorial

I got an expression for free energy F in a book as: $$F(N_1) = \frac{N_1N_2}{N}W-kT(\log N!-\log N_1!-\log N_2!),\tag1\label1$$ where $N=N_1+N_2$. As what the book later introduced, the derivative of ...
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1answer
151 views

Fejer monotone with respect to the convex hull of $C$

Let $C$ be a nonempty subset of a real Hilbert space $H$. Let $\lbrace z_n \rbrace_{n=1}^{\infty} \subset H$ be Fejer monotone with respect to $C$. Show that $\lbrace z_n \rbrace_{n=1}^{\infty} $ is ...