Approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby.

learn more… | top users | synonyms

0
votes
0answers
22 views

Runge's Approximation Theorem

This is a homework problem, so my apologies for a seeming lack of motivation. I want to understand what's going on in this problem more than I want someone to just write down the solution, so if ...
0
votes
0answers
18 views

Approximation of a continuous bounded function on $\mathcal{P}_2(\mathbb{R}^d)$ by Lipschitz functions

Hello everyone I am currently struggling with the following problem. Consider a bounded, measurable and continuous function $f: \mathcal{P}_2(\mathbb{R}^d) \rightarrow \mathbb{R}$ where ...
0
votes
1answer
21 views

How to solve this question using approximation theory?

I am asked to find the first three terms in the taylor series of the function $$ f(x)=(x-1)\ln x $$ around $x_0=0$. Then to find the maximum error in my approximation in the interval $[0.5,1.5]$. ...
0
votes
2answers
45 views

Prove that $|x|$ can be uniformly approximated in [-1,1] by polynomials?

I know how to aproximate it, is it enough to actually aproximate it to prove it can be aproximated? I just noticed there is a suggestion here at the end. Sorry for lateness Do Taylor arround the ...
1
vote
1answer
28 views

Bound for the max value of a Lagrange polynomial

Given some Lagrange polynomial $L$(x) that interpolates over the points $x$0, $x$1,..., $x$n with values in the set $A$ = {$a$0, $a$1,..., $a$n } on some interval [a,b], show that the max value that ...
0
votes
1answer
12 views

Does a closed subalgebra of $C(X)$ which separates points and vanishes nowhere contain 1?

Let $X$ be a compact Hausdorff space and $C(X)$ be the real valued continuous functions on $X$. Suppose that $\mathcal A$ is a (topologically) closed subalgebra of $C(X)$. Is it true that if $\mathcal ...
0
votes
2answers
69 views

Why Taylor Series or any other approximation method give us approximation of function? Why not give exact equivalent of function?

Lately I am started studying approximation of functions by polynomials and the need for approximation of functions? But what I failed to understand and books did not explain me is that why finding ...
1
vote
1answer
78 views

If a function is odd/even, then its best polynomial approximation is also odd/even. [duplicate]

If $f$ ∈ $C([a,b])$ is an even/odd function, then show that the best approximation among the polynomials of degree n is also even/odd. I'm almost certain that to show this directly, you should ...
0
votes
0answers
19 views

Effect of location of nodes for interpolation

I've been doing some numerical experiments to see how the location of the interpolating nodes affects the performance of the interpolator. I am just curious about this because it seems like the ...
0
votes
0answers
33 views

Given n points, show there is a unique polynomial through all of them

if x0, x1,..., xn are distinct points in [a,b] and $A$ = { a0, a1,..., an } ∈ Rn+1, show there is a unique polynomial $p$A of degree at most "n" such that $p$A(xj) = aj for each "j". Then, show ...
0
votes
0answers
28 views

Slope of linear approximation to a function?

Assume $f$ ∈ $C([a, b])$ is twice continuously differentiable and $f$ ''(x) > 0 on [a, b]. Show that the best linear approximation (polynomial of degree one) $p$ to $f$ has the slope $p$'(x) ...
0
votes
0answers
21 views

How well are certain functions approximated by simple functions?

Let us restrict ourselves to functions on the unit circle. It is well known that any function on $L^2(S^1)$ can be approximated arbitrarily well using simple functions. Now suppose $f \in L^2(S^1)$ ...
2
votes
0answers
46 views

A single analytic function that can approximate all others

The problem in it's entirety is this: Given some simply connected domain $U$ then show that there is a function $g\in\mathcal{O}(\mathbb{C})$ such that for any given $f$ there exists a sequence ...
0
votes
1answer
27 views

Richardson extrapolation - deriving methods for forward difference

I am rather stuck on a question from a textbook that I am practicing from which goes as follows: The forward difference formula can be expressed as $$f'(x_{0}) = \frac{1}{h}[f(x_{0} + h) - f(x_{0})] ...
0
votes
0answers
17 views

How to obtain an analytic approximation of this wavelet-looking function?

I would like the approximating function to be infinitely differentiable. I have tried fitting (1) a polynomial and (2) a combination of sin, cos, and exp functions to approximate the function shown ...
4
votes
1answer
52 views

Approximating polynomials over $\mathbb{C}$ with an entire function

Given a series of polynomials $p_{n}$ and a series of non-intersecting balls $B_{n} \subset \mathbb{C}$ show that there exists some function $f \in \mathcal{O}(\mathbb{C})$ such that $lim_{n ...
0
votes
2answers
36 views

Interpolation by splines: how to set up the equation system for finding the coefficients of the spline (in a B-spline basis)

Problem. I want to interpolate a function $f$ in some equidistant points $x_0<x_1<x_2<x_3<x_4$ using a quadratic spline. My attempt. I assume that we can use the interpolation points as ...
0
votes
2answers
31 views

How could I solve this equation: $ n-ne^{x\ln(2)}+xe^{x\ln(2)}\ln(2)=ax^{n-1} $ for $x$?

I want to have a solution for $x$ in this equation. $$ n-ne^{x\ln(2)}+xe^{x\ln(2)}\ln(2)=ax^{n-1}$$ Thanks !
2
votes
0answers
53 views

Is the Preimage of an Open Disk a Continuous Action

Let the kernel-like operator, $K$, be defined as $K(f,R) = \text{Closure}({\{x; \lvert f(x)\rvert < \frac{1}{R}\})}$ where $f:\mathbb{R} \to \mathbb{R}$ is a continuous function on the complex ...
2
votes
3answers
49 views

How to show that $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$

How can we show that: $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$ for large N.
0
votes
0answers
12 views

Finite element approximation in semi-norm $|\dot|_{H^1}$ and $L^2$

Let $\Omega$ be a polyhedral domain in $\mathbb{R}^n$. Consider the Poisson equation $-\Delta u=f$ in $\Omega$ subject to the following non-homogeneous boundary condition $u= g_D$ on $\Gamma_D$; ...
0
votes
0answers
34 views

Integral involving double exponentials

I am trying to estimate the growth in time of a certain quantity. In order to do this, I have to evaluate the following integral: $\int_0^{+\infty} \exp(-b x)\exp(-c\,t \exp(-a x \log x)) dx$ with ...
3
votes
2answers
39 views

Validity of an Approximation

I am considering the approximation of the following integral: $$\int_{-W/2}^{W/2}\exp[-p( x + iq)^2 ] dx \approx \int_{-\infty}^{\infty}\exp[-p( x + iq)^2 ] dx = \sqrt{\frac{\pi}{p}}$$ For large ...
0
votes
1answer
14 views

Higher Order Polynomial Interpolation

I am trying to approximate some log and exp functions in my code. I have implemented linear and cubic splines, but I want more accuracy. I am thinking about biquadratic splines (4th order, quartic), ...
1
vote
2answers
40 views

Uniform approximation by even polynomial

Proposition Let $\mathcal{P_e}$ be the set of functions $p_e(x) = a_o + a_2x^2 + \cdots + a_{2n}x^{2n}$, $p_e : \mathbb{R} \to \mathbb{R}$ Show that all $f:[0,1]\to\mathbb{R}$ can be ...
0
votes
0answers
14 views

Does this operator produce an interpolating spline of it's argument?

I'm studying a bit of spline functions theory: In this book, chapter 6, a lot of error bounds are given when spline functions are used to approximate functions belonging to a specific function spaces. ...
1
vote
0answers
28 views

Geometric Meaning of Modulus of Smoothness

Could you explain me the "geometric meaning" of the following definition (it's taken from a book on spline functions theory)? Definition: Given $1 \leq p \leq \infty$ a positive integer $r$, and ...
0
votes
0answers
19 views

Show that the Bernstein operator is not a projection

I'm currently trying to show that the Bernstein operator is not a projection, but I can't find a good counter-example to show that it's not a projection. I was thinking about starting with some ...
1
vote
0answers
13 views

are there any other properties like sub-modularity which can help compute the maximization problem of a set function

Now I have a set function $f$ (it's monotone), which inputs a node set $S$, and output a real value $r=f(S)$. And I want to compute $$ \max_{S,|S|\leq k}f(S). $$ where k is given number. And I know ...
3
votes
0answers
63 views

Calculate integral curve through a discretely sampled vector field

given a finite sample of a smooth vector field, i.e. a set $$S=\{(p_i,v_i)\mid i\in \{1,\dots, n\}, p_i,v_i\in\mathbb{R}^2\}$$ where the $p_i$ are the points at which we have a vector $v_i$, how do I ...
2
votes
1answer
42 views

Stronger Whitney approximation for smooth manifolds

If $M$ and $N$ are smooth manifolds, then any continuous map $f:M\rightarrow N$ is homotopic to a smooth map $g:M\rightarrow N$. If $f$ is smooth on a closed set $K\subseteq M$, the homotopy can be ...
4
votes
1answer
32 views

Approximation of $L^1$ function with compactly supported smooth function with same mass and same uniform bounds

Recently, I have asked this question. Now, I even want to make this better. Given $f\in L^1(\mathbb{R})$ with $0\leq f\leq 1$, I can find for any $\epsilon>0$ a $g\in C_c^\infty(\mathbb{R})$ such ...
1
vote
0answers
20 views

Sign of approximation error and remainder (residual)

In the Wikipedia article Taylor series it is said that: The error incurred in approximating a function by its $n$th-degree Taylor polynomial is called the remainder or residual and is denoted by ...
1
vote
1answer
25 views

Inverse problems with Graphical Approximation and Graphs

Suppose an inverse problem with graphical approximation for the system where only a small subset of system features are known hence undetermined scenario. The system can be represented by a graph. ...
1
vote
0answers
23 views

References on Inverse Problems, Approximation theory and Algebraic geometry

For example, you approximate structure functions of finite simple graphs in cases where only cut sets of the systems are known. The inverse problem means to build possible scenarios in underdetermined ...
1
vote
0answers
17 views

approximation of weakly differentriable bochner functions

Given a function $u\in L^2(0,T;H^1(\Omega))$ with $u_t\in L^2(0,T;(H^1(\Omega))^*)$. Can we approximate $u$ by functions $u^k$ with $$u^k=\sum\limits_{i=1}^{n(k)}c_i^k\phi_i^k,\text{ where } c_i^k\in ...
0
votes
0answers
14 views

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. ...
5
votes
2answers
49 views

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 ...
26
votes
2answers
411 views

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 ...
1
vote
0answers
25 views

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 ...
0
votes
1answer
28 views

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: ...
0
votes
1answer
16 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 ...
0
votes
0answers
23 views

General Hermite interpolation problem

Good evening, i wonder if someone could help me with the proof of the general hermite interpolation error.
1
vote
0answers
26 views

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 ...
1
vote
0answers
43 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 $K\subset G$ ...
1
vote
1answer
39 views

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 ...
1
vote
2answers
57 views

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 ...
2
votes
4answers
47 views

$4^\text{th}$ 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 ...
2
votes
1answer
73 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); ...
1
vote
1answer
40 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 ...