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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55 views

Does this “inverse Taylor series” exist in literature?

So next in my list of (my) useless ideas here's the idea of an "inverse Taylor series"...enjoy! Let $f$ be an analytic function, so that \begin{equation} f(x)=\sum_{n=0}^\infty a_n x^n \end{equation} ...
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Approximating integral term in integro-differential equations

I am trying to find an approximate solution to the following integro-differential equation for the $n$-dimensional vector $\mathbf{x}(t)$ in some interval $t\in[t_0, t_1]$: $$ \frac{d\mathbf{x}(t)}{dt}...
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1answer
33 views

Uniform approximation of an analytic function and its derivatives

The Lavrentiev's theorem is stated as follows Let $K \subset \mathbb{C}$ be a compact set. Then every continuous function $f: K\to \mathbb{C}$ can be approximated uniformly by polynomials if and ...
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Approximating numbers with powers of two other numbers

Given $0 \lt a,b\in\mathbb{R}, a\ne b$, for all $x, \epsilon$ do there exist integral $n,m$ such that $|\dfrac{a^n}{b^m} - x| < \epsilon$?
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55 views

Relation between error and coefficients in polynomial approximation

Let $f$ be a function and $(p_i)_{i=0}^\infty$ a sequence of polynomials such that the first $n$ span the space $\mathcal{P}_n$ of polynomials of degree $\leq n$ for all $n$. Furthermore, let $$ \...
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628 views

Prove that $\left\{x^{\lambda_i}\right\}_{i=0}^n$ is a Chebyshev system on $(0,\infty)$

The definition of Chebyshev system is as follows. A linearly independent system of $n$ basis functions $\varphi_1,\ldots,\varphi_n$ on $[a,b]$ is a Chebyshev system on $[a,b]$ if every non-trivial ...
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How to approximate $f(x)=a\cdot e^{x}+b$?

There is common way in which one can approximate $f(x) = a\cdot e^{bx}$ . Just use $ln$ for both formula's sides and make it linear. What about $f(x) = a\cdot e^{bx}+c$ ? How to determine $a$, $b$ and ...
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1answer
43 views

Functional iteration to find the root of a linear equation

I'll try to explain what I've understood about the subject and I want you to please tell me whether or not my arguments are correct. Given the equation $$(a-1)x+b=0$$ The obvious solution is $$\bar{x}...
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24 views

Does convergence of Bernstein polynomials imply convergence of its coefficients?

Assume a sequence of Bernstein polynomials $\{p_N(t)\}_{N=1}^\infty$ uniformly converges to a continuous function $f(t)$ on $[0,t_f]$. Is it true that the Bernstein coefficients of $p_N(t)$ (e.g., $p_{...
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Approximation of functions of 2 variables by multiplication of unidimensional functions

Let a function $f:\mathbb R^2\rightarrow \mathbb R$, such that $f\in C^{1, 2}_b(\mathbb R^2)$ (that is $\partial_x f$, $\partial_y f$ and $\partial_{yy} f$ are continuous and bounded). Is it possible ...
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36 views

Stone-Weierstrass theorem for two dimensional functions.

Let $K\subset \mathbb{R}$ and $\mathcal{A}$ be a sub-algebra of $C(K, \mathbb{R}^2)$. Is there any Stone-Weierstrass-type theorem that I can claim $\mathcal{A}$ is dense in $C(K, \mathbb{R}^2)$? Note:...
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1answer
299 views

Why are Cartoon like functions named like this?

I just saw an interesting talk about shearlets on PyData Berlin. One point when defining shearlets is the use of so called cartoon like functions, which are ...
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1answer
20 views

Asymptotic approximate solution of the parabolic cylinder differential equation

In chapter 3 (example 4) of the book "Advanced Mathematical Methods for Scientists and Engineers", by Bender and Orszag, I want to get the approximate solution for $+\inf$ for the parabolic cylinder ...
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3answers
46 views

Can $\sin(1/x)$ be approximated pointwise by polynomials over $(0,\infty)$

Can the function $f(x)=\sin(1/x)$ on $(0,\infty)$ be approximated by a sequence of polynomials pointwise on the domain?I am sure that uniform approximation is not possible because $\lim_{x\to 0+}\sin(...
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Density results of the span of two-dimensional polynomials

This is a question related to Multi-dimensional uniform approximation results and Asymptotic properties of a power series. I'm wondering whether $$V := span \left\{ \begin{bmatrix}\text{Re}((\beta^p +...
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Error bounds for the simple function approximation theorem in the number of level curves

The simple function approximation theorem says that any bounded, measurable function on a compact subset of $\mathbb{R}$ is the point wise limit (almost everywhere) of a sequence of simple functions. ...
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1answer
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Multi-dimensional uniform approximation results

Recently I've been investigating results from approximation theory, especially the uniform approximation by polynomials. I find most of the interesting results are for one-dimensional, uni-variate ...
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2answers
475 views

Questions over a specific case of the Muntz-Szasz theorem proof

On page 157 of this site: http://arxiv.org/pdf/0710.3570.pdf the author is proving a specific case of one direction of the Muntz-Szasz theorem. I do not understand the following 3 claims: 1) For $...
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1answer
24 views

Orthogonal Polynomials approximation and $L^2(\mathbb(R))$

I have another basic question, this time about approximation of functions. Given $(p)_{i\in \mathbb{N}}\in \mathcal{A}=\{$all the families of orthogonal polynomials $\}$ and $f\in L^2(\mathbb{R})$, ...
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How to apply the Stone-Weierstrass-Theorem to nonlinear systems? How to integrate Volterra kernels for TI systems?

I'm currently studying Volterra series as an input-ouput-representation of nonlinear systems. For this, I found a variety of interesting papers. For instance, Lesiak & Krener (1978), Brockett (...
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1answer
28 views

Taylor series to approximate derivative with difference quotient function

Consider the following example on Taylor series. Let's consider the difference quotient function of center $x_0$: $$f'_h(x_0)=\frac{f(x_0+h)-f(x_0)}{h}$$ For $h$ sufficiently small, the difference ...
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1answer
68 views

How to expand $b\int_0^\infty \operatorname{sech}^2\big(b\cdot f(x)\big)\,dx$ for large $b$?

Suppose $f(x)$ has a single zero in $(0,\infty)$ at $x=c$ and has a Taylor expansion about this point with some nonzero radius of convergence $0<R\leq\infty$. For concreteness, I'm working with the ...
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Necessary condition for the linear span of a set of functions to be dense in $C([-1,1])$

Let $P=\{x^{k_i}\}_i$ be a set of monomials defined on $[-1, 1]$. By Stone-Weierstrass theorem, if $k_i= i$, then $\text{span}\: (P)$ is dense in $C[-1, 1]$ under sup-norm topology. However the ...
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Instantaneous smoothing effect of sphere-valued maps

The instantaneous smoothing effect of the heat equation is the property that the solution to $$\begin{cases} \partial_t u= \Delta u, & t>0 \\ u(0, x)=f(x), & x\in \mathbb R^d,\end{cases}$$ ...
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23 views

On the invertibility of basis function matrix for interpolating functions

Problem: Let $f: \mathbb{R}^2 \to \mathbb{R}$ and $g: \mathbb{R}^2 \to \mathbb{R}$ satisfies $$ f(y(x)) = g(x) \ \forall x \in \mathbb{R}^2 $$ where $y : \mathbb{R}^2 \to \mathbb{R}^2$ is a map that ...
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3answers
28 views

An example for a seminorm on $\mathbb{R}^n$

Can any one come up with an example of a seminorm that is not a norm on $\mathbb{R}^n$ ? A seminorm on a real vector space $V$ is a function $N:V\rightarrow \mathbb{R}$ that satisfies that 1) $N(x)\...
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38 views

Does every vector space have a Hamel basis ? And is every linear comination representation finite?

If $V$ is a linear space, then a set $B$ of linearly independent vectors in $V$ that span $V$ is called a Hamel basis for $V$. Does every infinite dimensional vector space have a Hamel basis ? My ...
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function parameterization with known sums!

I want to find a function, let's say $y= a x + b$ but I don't have sample $(x,y)$ pairs but what I have is samples of following form $((x_1, x_2, ..., x_n), \sum_{i=1}^n y_i)$ where n is also a known ...
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Robustness of a model to learnt parameters

There is a recent push to study how sensitive a model is to small changes in its input. This has also been studied from an adversarial point of view: e.g what is the smallest input perturbation that ...
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28 views

Wavelet expansions

I'm looking into wavelets to approximate a known square-integrable function $$ f(x) = \sum_{j,k} a_{j,k} \times 2^{j/2}\psi(2^j x - k), \qquad a_{j,k}=2^{j/2}\int f(x) \psi(2^jx-k)dx$$ and I'm happy ...
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1answer
39 views

Polynomials can approximate the identity (proof)

From Terence Tao's Analysis II: I'm having problems with 14.8.2.(c), since it seems like the choice of $N$ depends on $c$ (and vice versa). The only proof I have relies on the convergence of the ...
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20 views

Approximation theorems and sketches of their proofs

I would like to collect (with support of users here) approximation theorems and sketches of their proofs. Each answer would give one approximation theorem and sketch of a proof. Any other proof of ...
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32 views

Is there a class of functions that have a cyclical or constant derivative?

I'm working on approximating functions in A.I., and I noticed that everyday functions seem to, at some point, have either a cyclical, or constant, derivative. For example, a straight line has a ...
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1answer
60 views

Approximating function of the integral $\dfrac{\sin(x)^k}{x^k}$

I needed a formula giving the value of the following integral; $$J(k)=\int_0^\pi dx\dfrac{\sin(x)^k}{x^k}$$ I didn't find any analytic expresion for it. I only found this interpolating function: $$J(k)...
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Approximation subject to derivative constraints, textbook reference?

I'm looking for a reference (preferably a textbook treatment) that can help me answer the following question. Suppose $\mathcal{F}$ is a space of functions on a subset of $\mathbb{R}^d$ with ...
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1answer
36 views

Stone-Weierstrass Theorem (Lattices)

I am struggling with a portion of a proof concerning the lattice version of the Stone-Weierstrass theorem. In particular, there is a subset $\mathcal{A}$ of the set of all real-valued continuous ...
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1answer
23 views

Are metric continuous measures setwise sequentially dense in finite Borel measures?

Let $(\mathcal{X},d)$ be a complete separable metric space. Say that a Borel measure $\sigma$ of $(\mathcal{X},d)$ is a metric continuous measure if for each $x\in\mathcal{X}$ the function $$(0,+\...
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1answer
50 views

approximation of 'any' bounded continuous function using bounded continuous functions with compact support

Suppose that $\phi$ is a bounded, continuous function with compact support $I$ (i.e. a bounded interval), then given any $\epsilon > 0$, there exists a simple function $\phi_{\epsilon}$ s.t. $$ \...
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Interpolation error of $C^k$ function derivatives

Consider the interval $I=[-1,1]$ and the set of points $A=\{r_1, ..., r_n\}\subset I$. Suppose that there exists a $k$-times continuously differentiable $f\in C^k(I, \mathbb{R})$ of which we know its ...
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3answers
100 views

Linear Fit: why do we minimize the variance and not the sum of all deviations? [duplicate]

my question is about the linear fit and the least squares method. Why do we decide to minimize the quantity $$ S = \sum_{i=1}^n r_i^2 $$ instead of this one: $$ r_i = y_i - f(x_i, \beta) $$ ? ...
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Choosing basis functions for function approximation

Suppose that $Y$ follows multivariate normal $\mathbb{N}(\mu, \Sigma)$. And we know that $f(Y)$ follows $\mathbb{U}(0,1)$, where $f: \mathbb{R}^n \to \mathbb{R}$ Can we find exact form of $f$ given ...
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1answer
33 views

Is there any example demonstrating nonlinearity of best polynomial approximation operator?

For any $f\in C[0,1]$, it is well known that there exists an unique $p^{*}\in P_n[0,1]$ such that $||f-p^{*}||_{\infty}=\inf\limits_{p\in P_n[0,1]}||f-p||_{\infty}$. In this fashion, one can define an ...
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How do I find a Bezier curve that goes through a series of points?

When someone has the 4 control points P0, P1, P2, P3 of a 2D cubic Bézier curve, that person can calculate a series of hundreds of points along the curve that start from P0 at t=0 and end at P3 at t=1 ...
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2answers
64 views

How to prove that $(1+x)^r$ behaves like $1+rx$ for small x without calculus?

Of course the fact that, in the neighborhood of $x=0$, $$ (1+x)^r=1+rx+o(x) $$ can be easily proven for integer $r$. For positive values, it's a trivial consequence of the binomial formula. For ...
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2answers
129 views

Approximating a clamp function using only addition, multiplication, division and subtraction

I'm trying to construct a function which satisfies the following: $$ \begin{align} f(x) = 0 & \qquad x \leq 0\\ f(x) = x & \qquad 0 \lt x \lt 1\\ f(x) = 1 & \qquad x \geq 1\\ \end{align} $...
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312 views

Why do deep neural networks work well?

The universal approximation theorem, as I understand it, states that for any continuous bounded function $f: X \rightarrow \mathbb{R}$ with compact domain $X$ and any threshold $\varepsilon$ there is ...
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1answer
51 views

How small must $x$ be for the error of $\cos(x) \approx 1$ to be below a certain threshold

I might be missing some background knowledge on this subject, but nevertheless I am interested. In some cases like this, the answers talk about finding the taylor series for $\cos(x)$ and then ...
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63 views

even and odd cubic polynomials

Book says approximating even function $P_3$ must be of form $ax^2+b\ $so they neglected $x,x^3\ $. This function approximate even function $|x|^3$ very nicely in the book after some tweaking. I am ...
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1answer
37 views

Approximation formula for a simple counting problem

Let $a,b,c$ be positive integers with $\gcd(a,b)=\gcd(a,c)=\gcd(b,c)=1$ and let $n =$ the number of positive integers $\leq N$ not divided by $a,b,c$. Set, $$m = N\cdot (1-1/a)(1-1/b)(1-1/c).$$ I ...
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1answer
23 views

Is an infinite sequence of orthogonal functions in $H$ closed in $H$?

Consider some countably infinite sequence of elements $f_n$, each belonging to an infinite dimensional Hilbert space $H$, that are all orthogonal to every other member of the sequence. Is this set ...