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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What is the “right” way of approximating random variables with other random variables?

It is well-known that one can construct a Hilbert space of zero mean, finite variance random variables such that every random variable of this type $X$ can be represented as a linear combination of a ...
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55 views

Are of Polynomials of Continuous functions complete?

The Stone-Weierstrass Theorem guarantees that every continuous function defined on a closed interval [a,b] can be approximated as closely as desired by a polynomial. I'm curious in polynomials of ...
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Approximating the basis of a specific function

We are given a continuous function $g: A \to B $, where $A, B$ are compact subsets of $\mathbb{R}$. We define a function $f(x) := g(b_1x)+g(b_2x)+...+ g(b_mx)$, where $b_i < 1$ and $b_ix$ is a ...
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23 views

Number of simple functions needed to approximate another function

I have got a function $f$ which maps from one compact space to another. Function $f$ is smooth. I want to approximate it with some simple functions (e.g polynomials). Is there any theory that gives ...
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25 views

Optimal Fixed-Digit Rational Approximations of $\pi$

Is there a systematic way of finding optimal rational approximations to $\pi$ whose numerator and denominator have at most $n$ digits? More precisely: Let $D_n$ be the set of all positive integers ...
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Can we find a holomorphic function $g$ on an open disk such that $\sum_{i \in \mathbb{N}} |(f-g)(a_i)|^2 < +\infty$?

Let $f : \mathbb{C} \to \mathbb{C}$ be a continuous function with $f(0)=0$. Let $\{a_i\}_{i\in \mathbb{N}}$ be a set of scalars in $\mathbb{C}$ such that $$\exists C > 0 : \forall i\in \mathbb{N} ...
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37 views

Extension of function analytic on [-1,1]

Given that a function $f$ is analytic on $[-1,1]$, that is, for any $s ∈ [−1, 1]$, $f$ has a Taylor series about $s$ that converges to $f$ in a neighborhood of $s$. Can we conclude that $f$ is ...
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91 views

Truncation error with growing step size

When I read about finite difference methods (or really any approximation method), truncation error is often central to the discussion, and rightfully so. But it is also most often discussed in the ...
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19 views

What are some options for adaptive spline approximation of data in 1-D?

What are available options for adaptive spline approximation of data in 1-D? I've some data in a single dimension that I would like to approximate using some kind of spline, preferably a cubic. As ...
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Approximation of mean of a rational function of random variables

Let $\xi_i$ with $i\in\{1,\dots,n\}$ be iid random variables and let $Q(x,y)$ be a rational function. I need to compute one $x$ that satisfies $$\frac{1}{n}\sum_{i=1}^n Q(x,\xi_i)=0.$$ This is a ...
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$N$ birds are distributed on a telephone wire

$N$ birds are distributed on a telephone wire that can fit a maximum of $2N$ birds. The spacings between birds form a sequence $S$. The minimum space between birds is $1$ unit. The sequence is ordered ...
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The singular values of the best rank-$k$ approximation to a matrix

Let $A\in\mathbb{C}^{m\times n}$ be a complex matrix. Let $B_k$ be a best rank-$k$ approximation to $A$ such that \begin{equation*} B_k\in\arg\min\limits_{{\rm rank}(B)=k}||A-B||_F, \end{equation*} ...
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31 views

Distance of the element from the subspace of $l_{1}.$

Let $l_{1}$ be as follows $$l_{1}=\Big\{\{x_{n}\}_{n=0}^{\infty}\subset \mathbb C\: : \: \sum_{n=0}^{\infty}|x_{n}|<\infty\Big\}$$ and its subspace be $$V=\Big\{\{x_{n}\}_{n=0}^{\infty}\in l_{1}\: :...
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Expressions approximating Generalized Harmonic Number (truncated polynomials with shrinking error term preferred)

Specifically, $$H_m^{(2n)} \approx\ ?$$ and $$H_m^{(4n)} \approx\ ?$$ where $(m, n)$ $\in \mathbb N_{>1}$ I would not like to use special functions like the (Riemann zeta function) unless they ...
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Elementary approximations to $\zeta(s)?$

What are the best approximations in terms of elementary functions of one real variable for: $$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s},$$ for $Re(s)>1?$ There is not an elementary function that ...
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Uniform approximation of $L^2$ basis by smooth functions with bounded derivatives of all orders

Let $\mathcal{F}=\{f_i\}_{i\in\mathbb{N}}$ be an orthonormal Hilbert basis of $L^2[0,1]$. I am wondering whether it is possible to approximate the $f_i$ uniformly across $i$ in the $L^2$-norm by ...
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Can $x^n$ be uniformly approximated by the combination of $x^{k^2}$?

For each $n\in\mathbb{N}$ , can $x^n$ be uniformly approximated by the linear combination of $\left(x^{k^2}\right)_{k\in\mathbb{N}}$ ? In order to facilitate a solution, we might as well try to ...
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Approximate solution for high order polynomial (order 12) [closed]

I'm trying to get an approximation value of y by x from the following equation $\ x = $$\sum_{i=1}^{12} y^i$ The current suggestion is to take y=1+z and z tending to zero or y tending to 1 any ...
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40 views

Tight upper bounds for a monotonically increasing non-linear recurrence

I have the following non-linear recurrence: $$y_{n+1} = \sqrt{\frac{2}{1+y_n}}y_n,\quad y_0 \in[0,1]$$ Some basic thought shows that $0$ and $1$ are fixed points of this, and that $0$ is repelling ...
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$(N+1)^{\text{th}}$-order terms in $N^{\text{th}}$-order approximation

I am working with a formula for which I'd like an $N^{\text{th}}$-order approximation. After some simplification, I am able to solve for a form of the expression with some but not all $(N+1)^{\text{th}...
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35 views

Unbounded approximation ratio

Suppose that there is a specific instance of a graph for which the approximation ratio of an algorithm polynomially increases with the number of nodes of the graph, say the approximation ratio is $n^2$...
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55 views

What is the complete Partial Derivatives approximation formula to calculate the value of a function?

We know the approximation formula using partial derivatives to calculate the value of a function with some variables $x$ and $y$, which can be extended to any number of variables to be \begin{...
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A question on the existence and uniqueness of a cubic Hermite interpolant

I have been trying to solve a particular problem that establishes both the existence and uniqueness of a cubic hermite interpolant on some generic interval $[a,b]$. Briefly, for a function $f$ we ...
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28 views

How could I obtain this approximation of the May-Wigner theorem?

I'm trying to understand the complete proof of the May-Wigner theorem. We have a real random $n\times n$ matrix $B$ with its non-zero elements $B_{ij}$ are chosen independiently from a fixed ...
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116 views

Can we approximate continuous functions arbitrarily well with polynomials? (beyond Weierstrass )

Let $f:(0,1) \to \mathbb{R}$ be continuous, and let $\delta:(0,1) \to \mathbb{R}$ be continuous and positive. Does there always exist a polynomial $p(x)$ satisfying $|f(x)-p(x)| < \delta(x)$ for ...
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45 views

Do all convergent sub-sequences of a sequence in a compact set $\mathcal K$ converge to the same element in $\mathcal K$?

I came upon this when trying to understand the proof to "Theorem on Existence of Best Approximations in a Metric Space" as given by Cheney (1981). Let $\mathcal K$ denote a compact set in a metric ...
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44 views

Landau inequality for several variables

For $f \in C^n(\mathbb{R})$ and $0 < \alpha < n$, Landau-Kolmogorov inequlity is geven by $$ \|f^{(\alpha)}\| \leq K(n,\alpha)\| f\|^{1-\alpha/n}\|f^{(n)}\|^{\alpha/n}, 0 < \alpha < n,$$ ...
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Smooth approximation (under supremum norm) of distance to algebraic set in $\mathbb{R}^n$.

Given a set $S$ which is the zeroes of a finite number of homogenous polynomials in $x\in\mathbb{R}^n$, I want a constant $\alpha$ and a $C^2$ approximation, denoted $d$, to the function $d(x,S)=\inf_{...
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19 views

Approximating the lateral derivatives

Let $f : [0,2] \to \mathbb{R}$ be a continuous function with continuous derivatives of all orders in every point except at $t = 1,$ where the lateral derivatives exist. We know that one can ...
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Optimize a fixed size susbset

So I'm trying to solve this problem: There are many people who apply for jobs at a company. Each applicant has some technical skills required for jobs. The skills possessed by different ...
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280 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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Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?

Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
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838 views

Examples of transcendental functions giving almost integers

Informally speaking, an "almost integer" is a real number very close to an integer. There are some known ways to construct such examples in a systematic way. One is through the use of certain ...
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Advanced Methods for Approximating Surfaces based only on partial derivative estimates

I'm looking for information on interpolating a surface function p(x,y) based only on estimates of the partial derivatives at points on a grid. Obviously, any such approximation is subject to a ...
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21 views

Generalization of Power Series

Given a smooth function $f(x):$ Does there always exist an expansion of x around a point $x_{0}$ of the form $$\sum_{n=0}^\infty \frac{h(f^{(n)}(x_{0}))}{n!}g(x-x0,n)$$ for some functions $h(c),g(a,b)...
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Regression through linear Fourier coefficient fitting?

Basically suppose on was given an unknown function/data and expected to write a function so that $Y=f(X)$, this can be done by linear regression in simple cases very easily. However, suppose that the ...
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195 views

Smoothing of a step function using smoothstep. (Curve fitting)

I was trying to smoothen the step function (zero when $x$ is less than $2/3$ and equal to $1$ when $x$ is greater then $5/6$) as in the picture below. Trying to fit $f$ in between $2/3$ and $5/6$ ...
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Finding optimal knots for function approximations

I would like to approximate a continuous (complex) function $f(x)$ in the interval $[a,b]$ $ (x\in\mathbb{R})$ by local polynomial functions of order $3$ (cubic Hermite spline or cubic C2 spline). Is ...
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25 views

Reference request: Bounded function can be approximated by continuous functions in $L_1$ with bounded $L_\infty$-norm

I think that it is well-known that a real valued function $f\in L_\infty[a,b]$ can be approximated by continuous functions $f_n$ with respect to the $L_1$-norm, i.e. $||f_n-f||_{L^1}\to0$, where the $...
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References for a proof of a Jackson's inequality?

Let $g:[0,2\pi]\to \mathbb{C}$ which is $\mathcal{C}^k([0,2\pi],\mathbb{R})$ and periodic. If $\mid f^{(k)}(x) \mid\le 1$ then for each $n\in \mathbb{N}^*$, there exists a trigonometric polynomial $T_{...
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Approximating a bounded measurable function from below by a sequence of smooth functions

Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a bounded measurable function. Is it possible to find a sequence of functions $\{f_n \}_n: \mathbb{R} \to \mathbb{R}$ in $C^{\infty}_c( \mathbb{R})$ ...
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Metric projection from space of bounded functions to finite-dimensional linear space

Apologies if the answer is obvious or should be easy to find, but so far I've had no luck. Let $X$ be a subspace of $\mathbb{R^k}$ for a finite $k$ and let $\mathcal{B}(X)$ be the Banach space of ...
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Does the Sobolev space $W^{1,p}(\Omega), p>2$ has a monotone basis?

A Shauder basis in a Banach space is monotone if $\|P_{n}f\|\leq\|f\|,$ where $P_{n}$ is the projection to the sum of the first n elements of the basis. For Hilbert spaces this is always the case if ...
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Is there a theory of “almost symmetry” generalizing group theory?

Apologies for the inescapably soft question. Does there exist a theory that aims to develop tools analogous to those of group theory, except for the study of objects that are merely almost ...
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36 views

approximating a decreasing function with hyperbolic functions

Let $y = f(x)$, where $x, y \in \mathbb{R}_1$ and $f \in \mathcal{C}^1$ with $f'(x) \le 0$. Partition the range using $t$ points ${y_1, \ldots, y_t}$ and - on each interval $[y_m, y_{m+1})$ - ...
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Mitigating the Runge Phenomenon with Constrained Norm Minimization

I am interested in polynomial interpolation of a set of points in $\{(x_1, y_1), \ldots, (x_n, y_n)\} \subset \mathbb{R}^2$. On the wikipedia page for Runge's phenomenon, the Constrained ...
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Do two close functions share some local minima?

Let $f,g:\mathbb{R}^n \to \mathbb{R}$ be two differentiable functions. Assume that $\| f -g \|_{\infty} \leq \epsilon$. On what conditions on $f$, for every local minima $x$ of $g$, there is a local ...
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101 views

Error analysis of approximating Fourier transforms

Consider the problem of computing the Fourier transform of a function, $f(x).$ $$ \hat{f}(k) = \int_{-\infty}^{\infty} dx~ f(x)~ e^{i k x} .$$ Suppose I want to approximate this transform by a ...
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159 views

Approximation of conditional expectation of unknown function

I am given a multidimensional markovian stochastic process $X_1,X_2,...X_n$ with continuous state space and unknown to me function $V$. I want to approximate expectation $E(V(X_k)|X_{k-1} = x)$ ...
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66 views

Relating the Nonlinear and Linear operators in the Homotopy Analysis Method

The question refers to chapter two of the book Liao, Shijun. Homotopy analysis method in nonlinear differential equations. Beijing: Higher Education Press, 2012. Link to book pdf from Chinese .edu ...