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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Expansions and Approximations of Functions

When looking at series expansions of functions, the main cases I've worked with have been Taylor series and Fourier series. Whilst the latter can arguably be viewed as a subset of the former (e.g. ...
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Runge's Theorem Application

Below is a question out of Gamelin's Complex Analysis which I cannot quite figure out. Any tips would help appreciated! "Let $(z_j)$ be a sequence of distinct points in a domain $D$ that accumulates ...
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If $f\in C[0,1]$ and $A\subset[0,1]$ is finite, can $f$ be approximated uniformly by polynomials that coincide with $f$ on $A$?

Let $f\colon[0,1]\to\mathbb{R}$ be continuous and $A$ a finite subset of $[0,1]$. Given $\epsilon>0$ is there a polynomial $p$ such that $$ |f(x)-p(x)|\le\epsilon\quad\forall ...
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+50

Prove $|P(0)|\leq 2n+1$

Let $P(x)$ be a polynomial with degree $\leq n$ and $|P(x)|\leq\frac{1}{\sqrt{x}}$ for $x\in(0,1]$. Prove that $|P(0)|\leq 2n+1$. The idea should be that if $|P(0)|$ is too large, then the polynomial ...
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Inapproximability of Combinatorial Optimization Problems

I've been reading the "Inapproximability of Combinatorial Optimization Problems" by Luca Trevisan (see: link). On pages 3-4 it mentions that a polynomial time algorithm for 3SAT would exist if there ...
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A proof in Hilbert & Courant vol 1 of Weierstrass theorem.

My question is regarding a derivation of an inequality on page 67 of Methods of Mathematical Physics. Here's a scan of the book: ...
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1answer
12 views

Approximation of function when variable tends to infinity

I'm reading these notes and on page 99 they approximate the function $$ q_2(x,t) = -12 \frac{3 + 4 \cosh(2x+24t) + \cosh(4x)}{\left( 3 \cosh(x-12t)+\cosh(3x+12t) \right)^2} $$ for when $t \rightarrow ...
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General Hermite interpolation problem

Good evening, i wonder if someone could help me with the proof of the general hermite interpolation error.
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Approximation theory in Lp spaces (Reference Needed)

I am looking for some reference on approximation theory in Lp spaces. I have found a number of papers like: paper1 , paper2 etc. I was wondering if there is a book or a monograph that will contain ...
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36 views

approximation with polynomials without stone weierstrass

This is from a course in real analysis. Let $f:G\rightarrow \mathbb{R}$ be a continuous bounded function and $G\subset\mathbb{R}^n$ an open bounded set. Prove that for every compact set $K\subset G$ ...
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Find maximum number of equal subsets of size 3 using each index only once.

Specifically, An array of numbers is being given(duplication allowed) and we have to find maximum numbers of subset(subset need not to be contiguous) each having size of 3 such that sum of each subset ...
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How to estimate ln(1.1) using quadratic approximation?

So the general idea for quadratic approximation is assuming there a function $Q(x)$ we want to estimate near $a$: $Q_a(a) = f(a)$ $Q_a'(a) = f '(a)$ $Q_a''(a) = f ''(a)$ But then how do you ...
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$4^\text{th}$ order polynomial approximation for: $f(x) = e^{-\cos(2\pi x)}$

I know how to find a polynomial approximation for a function $f(x)$ around a fixed number let's say $x=0$. However I do not have an idea about how to solve the following: Find a $4^\text{th}$ order ...
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1answer
49 views

Density In The Theorem/Proof of The Stone-Weierstrass Theorem

Yet again, I have a question that I could use some help with. Note that almost everything can be found in C. Pugh's, Real Mathematical Analysis (soft-cover, 2nd Edition, ISBN: 978-1-4419-2941-9); ...
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1answer
36 views

Proving Weierstrass Aproximation theorem with probability tools

While studying the Weak Law of Large Numbers on Allan Gut's Graduate Probability course I came across the proof of Weierstrass approximation theorem. Before the proof of uniform convergence he sets ...
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How to choose basis functions that contribute most efficiently per term?

GOAL: I would like to approximate some positive, scalar function, f(x,y) > 0, on a 2D field of finite size i.e. x=[a,b],y=[c,d] OBSTACLE: I am familiar with the set of basis functions used in the ...
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1answer
38 views

Construct a sequence of polynomials that converge uniformly on $[0,1]$, but whose derivatives fail to converge uniformly.

Can anyone help me with this? In what situation would the derivative of a polynomial not converge uniformly given the polynomial itself converges uniformly on a compact set. I don't even know where ...
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24 views

Representation of $f(x)$ with a superposition of functions standing wave functions $g(x,t)?$

I am trying to represent a continuous function, $f(x)$, defined on an interval of length L by a superposition of standing wave functions $g(x,t)$. Definitions for this problem are: $f(x)$ is in ...
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1answer
22 views

Can this approximation be made more formal?

When considering oscillating systems in physics, we end up with some response function like $$F(\omega) = \frac{\omega^2}{(\omega_0^2 - \omega^2)^2 + (\omega/\tau)^2},$$ where $\omega_0$ and $\tau$ ...
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How to find coefficients in a multivariate Chebyshev polynomial approximation

How do perform a multivariate Chebyshev approximation? Let \begin{align} \vec{x} & = x_{0}, x_{1}, ... , x_{n},\\ \vec{a} & = a_{0}, a_{1}, ... , a_{n},\\ \vec{b} & = b_{0}, b_{1}, ... , ...
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Can a smooth vector-valued function approximated by functions taking value in a dense subspace?

Question Suppose $E$ is a Banach space, and $F\subseteq E$ is a dense vector subspace. Suppose $f\in C^k([0,1],E)$, the space of $k$-times continuously differentiable $E$-valued functions, endowed ...
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1answer
23 views

Finding a polynomial sequence approximating certain function on complex plane

I'm trying to find a polynomial sequence on complex plane, which converges to $1$ on the upper-half plane, $0$ on the real line, and $-1$ on the lower-half plane...but just don't have a clue. Thanks ...
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is it true that $ \frac{1}{\epsilon}\ln (1- \epsilon x) \approx - \frac{1}{1-\epsilon}x $?

Target is to approximate $\frac{1}{\epsilon}\ln (1- \epsilon x) $ ($\epsilon, x \in (0,1) $). Here is one using $\ln (1+y) \approx y $: $$ \frac{1}{\epsilon}\ln (1- \epsilon x) \approx -x $$ I ...
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asymptotic expansion/approximation

Find the small solution of $$y''-y\left ( 1-y^{2} \right )=0 \text{ with } y\left ( 0 \right )=\epsilon \ll 1$$ Making a pun, I decided that $$y^{3}\left ( 0 \right )\ll y\left ( 0 \right )$$ so ...
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Is there a way to approximate a polynomial as another, binary-coefficient polynomial?

Let's say I have a polynomial: $$p(x) = \sum_{n=0}^N a_n x^n$$ where $x \in \mathbb C$. Does there exist theory and/or methods on approximating $p$ as: $$p(x) \approx \hat p(x) = \sum_{m=0}^M b_n ...
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102 views

Does an extension operator in Sobolev spaces commute with derivative operators?

Assume that $\Omega\subseteq \mathbb R^d$ is open and has a Lipschitz boundary. Let $\tau\geq0$. Then we know that there exists a linear operator $E:H^\tau(\Omega)\to H^\tau(\mathbb R^d)$ such that ...
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1answer
62 views

Richardson's extrapolation of composite trapezoidal rule

I have applied Richardson's formula to the composite trapezoidal rule, $I_h(f)=\frac{h}{2}(f(a)+\sum_{k=1}^{n-1}f(a+kh)+f(b))$, in an attempt to better approximate the integral $I(f)=\int_0^1 ...
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The derivatives of Bernstein polynomials for a $C^1$ function uniformly converge

Let $f:[0,1]\rightarrow \mathbb{R}$ be continuous, and suppose that $f'$ is also continuous. Show that $$\frac{d}{dx}B_{n}(.;f)\rightarrow f'$$ uniformly. Here $B_{n}(.;f)$ represents the $n$th ...
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Approximation of Sets of Finite Perimeter

Fix an open set $\Omega \subset \mathbb{R}^n$. If $E$ is a measurable subset of $\Omega$, we may define the perimeter of $E$ in $\Omega$, denoted by $P(E;\Omega)$, to be $$P(E;\Omega) = ...
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Limit of this function as $x \rightarrow \infty$?

I was interested in approximating $\log x$ without using the $\log$ function, specifically for values of $x \in [1, e]$, and I got this monster of a function: $$\left( ...
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Polynomial approximation with multiplicative error bounds

Problem: Given an monotonically increasing function $f(x):[0,a]\rightarrow \mathrm{R}^{+}$, approximate $f(x)$ with polynomial function $g_k(x)=\sum_{i=0}^k a_ix^i$ such that $|f(x)-g_k(x)|\leq ...
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How to prove that problem is NP-hard by making a reduction?

Convering by triples Data: A set Y of cardinality 3n and a family C = ($C_{1},...C_{m}$) of triples of elements of Y: for all i, $C_{i}$ $\subset Y$ and |$C_{i}$| = 3. We admit that COVERING BY ...
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Approximation algorithm for ratio is $2$?

Let consider a graph $G$ and a subset of its vertices. Let $G_s$ be the complete graph on $S$ with the following weights on its edges: the weight of $(x,y)$ is equal to the length of the shortest path ...
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Does Stone-Weierstrass apply in $\mathbb{R}$?

Does the Stone-Weierstrass theorem (Weierstrass approximation theorem) apply in $\mathbb{R}$. The Stone-Weierstrass theorem in $\mathbb{R}$ would be: Given continuous ...
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Is the set $\Sigma_n^0$ closed with respect to the norm $\sup_{[0,1)}(\cdot)$

Take $\Omega=[0,1)$ and define $$T_n(\Omega):=\{\qquad\Pi=\{I_j\}_{j=1}^n ~~|~~I_j:=[t_{j-1},t_j),\\ \qquad\qquad\qquad~~t_{j-1}<t_j~~\text{with }j=1,\dots,n,\\ ...
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Why polynomial interpolation is considered as better than others?

Why polynomial interpolation is considered as better than others? In case of interpolation, the function $\phi(x)$ to approximate the unknown function $f(x)$ may be polynomial, exponential, ...
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Approximation of non-analytic function

I have a function which is of the form \begin{equation} f(x) = \frac{1 - x^{1/2} + x - x^{3/2} + \ldots}{1+x^{1/2} - x + x^{3/2} - \ldots}. \end{equation} Intuitively, I would assume that for small ...
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Rigorous rationale for the Pade Approximant?

I recently asked a Question for which the only Answer I got was a recommendation to punt and use the Pade Approximant. This is the first time I recalling seeing this, and intuitively it seems like ...
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1answer
51 views

Existence of minimum for $\inf_{k \in \mathbb{R}} E[|X-k|^p]$

Suppose $E[|X|^p ]< \infty$ for the given $p \in \mathbb{R}^{+}$. How to show that the following expression has a minimum \begin{align} \inf_{k \in \mathbb{R}} E[|X-k|^p] \end{align} That is ...
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Approximating basic trigonometric functions to a integrable form

I wondered if there is a way of approximating trigonometric functions in terms of basic functions (possibly trigonometric functions) so that one can derive the indefinite integral of said function. ...
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Floquet exponent for Mathieu equation solution by using WKB method

Let's have Mathieu equation $$ \tag 1 y''(t) + (A - 2qcos(t))y(t) = 0 $$ Let's assume case $q << 1$. Then Eq. $(1)$ can be rewritten in a form $$ \tag 2 \frac{1}{2\epsilon }y''(t) + (b - ...
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Hash Collision Probability Approximation

If an item is chosen at random $k$ times from a set of $n$ items, the probability the chosen items are all different is exactly $\dfrac{n^\underline{k}}{n^k}=\dfrac{n!}{(n-k)!n^k}$. For large $n$, the ...
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Schwartz class dense in $L^2$, reference

Does anyone know a good reference where it is shown that the Schwartz class $\mathcal{S}(\mathbb R)$ is a dense subset of $L^2(\mathbb R)$? Many thanks
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Pertubative Solution To Vector Recurrence Relation?

Suppose we have the recurrence relation $$\mathbf{v}_n = F_n \mathbf{v}_{n+1},$$ where $\mathbf{v}_n$ is a sequence of $2\times1$ vectors, and $F_n$ is a sequence of $2\times2$ matrices. If $F_n=F$ ...
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Series expansion of rect function.

It is well-known that so-called rect function, $$\mathrm{rect}(t) = \begin{cases} 0 & \mbox{if } |t| > \frac{1}{2} \\ \frac{1}{2} & \mbox{if } |t| = \frac{1}{2} \\ 1 & \mbox{if } |t| ...
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Boundedness of smooth functions approximating an Lp function

We all know that the space of smooth functions on Euclidean space with compact support is dense in the Lp spaces, for p strictly less than infinity. Now my question is: suppose there is a function f ...
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Nonnegative coefficients in Chebyshev expansion

I am considering the Chebyshev expansion of analytic functions, and notice that when all the successive derivatives of my function are nonnegative, then the Chebyshev coefficients are themselves ...
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1answer
47 views

Approximation of characteristic function by mollifiers

I have been asked to show that the Heaviside function $H := \chi_{[0,+ \infty)}$ does not admit weak derivative in $L^1_{loc}(\mathbb{R})$. Here's my reasoning: By definition the weak derivative of ...
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27 views

Interpolation with RBF

I have a function that is continuous and differentiable over $\mathbb{R}$ and its support is the whole real line. I want to approximate it through a linear combination of Gaussian functions. I know ...
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A Question from Murray - Asymptotic Analysis

I'm stuck on one of the questions related to the method of stationary phase in Murray's book on Asymptotic analysis. The question is as follows; If $h(t)$ has a single stationary point at $t_0$, ...