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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34
votes
5answers
2k views

How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...
17
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5answers
5k views

Polynomial approximation of circle or ellipse

Trying again, with a somewhat simpler sounding question, since my previous one (Generalizations of equi-oscillation criterion) got zero response: Let $F:[0,1] \to R^2$ be a parametric polynomial ...
10
votes
1answer
471 views

$\epsilon>0$ there is a polynomial $p$ such that $|f(x)-e^{-x}p|<\epsilon\forall x\in[0,\infty)$

Could any one tell me how to solve this one? Given $f\in C[0,\infty)$ such that $f(x)\to 0$ as $x\to\infty$ we need to show that for any $\epsilon>0$ there is a polynomial $p$ such that $|f(x)-e^{-...
10
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4answers
2k views

Is Ramanujan's approximation for the factorial optimal, or can it be tweaked? (answer below)

Ramanujan's famous factorial approximation, $$n!\approx\sqrt{\pi}\left(\frac{n}{e}\right)^n\root\LARGE{6}\of{8n^3+4n^2+n+\frac{1}{30}}$$ is far more accurate that the Stirling approximation when ...
23
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2answers
9k views

Series expansion of the determinant for a matrix near the identity

The problem is to find the second order term in the series expansion of the expression $\mathrm{det}( I + \epsilon A)$ as a power series in $\epsilon$ for a diagonalizable matrix $A$. Formally, we ...
14
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3answers
334 views

A curious algebraic fraction that converges to $\frac{\sqrt{2}}{2}$

I have noticed that the algebraic fraction $\frac{3a+2b}{4a+3b} $ Gives better and better approximations to $\sin 45^\circ = \frac{\sqrt{2}}{2} $ For $ a = b = 1$ we get $5/7 \approx 0.714 $ ...
13
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2answers
366 views

$f=\underset{+\infty}{\mathcal{O}}\bigr(f''\bigl)$ implies that $f=\underset{+\infty}{\mathcal{O}}\bigr(f'\bigl)$.

Let $f\in\mathcal{C}^2(\Bbb{R},\Bbb{R})$ be a positive function such that $f=\underset{+\infty}{\mathcal{O}}\bigr(f''\bigl)$ does it implies that $f=\underset{+\infty}{\mathcal{O}}\bigr(f'\bigl)$? ...
11
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0answers
395 views

A generalization of an integral related with $\zeta(2)$

It is well-known that: $$ \int_{0}^{+\infty}\frac{x}{e^{x}-1}\,dx = \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} \tag{1}$$ but what is known about $$ I_2 = \int_{0}^{+\infty}\frac{x^2}{e^x-1-x}\,dx \...
11
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1answer
2k views

Proving that: $9.9998\lt \frac{\pi^9}{e^8}\lt 10$?

I would like to use mathematical tools to prove that $$9.9998\lt \frac{\pi^9}{e^8}\lt 10$$ With an on-line calculator I got $$ \frac{\pi^9}{e^8}\approx 9.9998387978$$ But I do not know any ...
10
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2answers
1k views

Why is the polynomial best approximation to an even function itself even?

I have seen this stated and it seems intuitively obvious but I cannot prove it. I have a feeling it may be because a non-even best approximant would not satisfy the equioscillation property of the ...
8
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7answers
562 views

Why does $ \frac{2x}{2+x}$ provide a particularly tight lower bound for $\ln(1+x)$ for small positive values of $x$?

EDIT: My question was poorly worded. I wasn't asking about showing $\ln(1+x) > \frac{2x}{2+x}$ for $x>0$. What I wanted to know is why the lower bound provided by $ \frac{2x}{2+x}$ was so ...
6
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2answers
2k views

Weierstrass factorization of sine, and related questions

So the idea is that you can represent a function as a product of its zeroes, and there are some fundamental factors that often crop up. I am interested in, give this is the WF of sine : Is it ...
7
votes
5answers
462 views

What is a nice way to compute $f(x) = x / (\exp(x) - 1)$?

I want it to be stable near $f(0) = 1$. Is there a nice function that does this already, like maybe a hyperbolic trig function or something like expm1, or should I just check if $x$ is near zero and ...
6
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1answer
1k views

Multivariate Weierstrass theorem?

The Weierstrass theorem states that for any continuous function $f$ of one variable there is a sequence of polynomials that uniformly converge to $f$. To my surprise, I couldn't find any reference to ...
5
votes
1answer
238 views

prove equality with integral and series

I am stuck on one question with integral. Help me please to show that with $n=1$ the following is true $$ \int_{0}^{\infty}\left(\frac{2^n}{t^n}\left(\frac{t^n}{2^nn!}-\frac{1}{2^{n+2}}\frac{t^{n+2}}{...
5
votes
1answer
728 views

Every $K$-Lipschitz function can be uniformly approximated by $C^1$ functions with derivative bounded by $K$

The exercise states: Let $a,b\in\mathbb{R}$, $a<b$ and let $(C[a,b],\Vert\cdot\Vert)$ denote the vector space of continuous real functions on $[a,b]$ endowed with the uniform norm. Let $C^1[a,b]\...
4
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0answers
164 views

Generalizations of equi-oscillation criterion

When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...
3
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1answer
250 views

How to justify unicyclic connected graphs formula?

I have read this formula in a book for $c(k,k)$, the number of unicyclic connected graphs on $k$ vertices A057500, i.e. $n=m=k$. How can one prove it? It seems that it is iterating on the length of ...
6
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2answers
2k views

Natural cubic spline interpolation error estimate

I am looking for an error estimation for natural (one with $s''(a) = s''(b) = 0$ boundary conditions) cubic spline interpolation on an evenly spaced grid. The best result I've found was $O(h^2)$ ...
5
votes
1answer
579 views

Taylor polynomial approximation

How do you determine if adding more terms to the Taylor polynomial will improve its approximation of $f(p)$ or in other words, how do you determine if a Taylor series converges for a particular value ...
5
votes
3answers
213 views

How to approximate the integral $\int_{-b}^{\infty}\log(t+b)e^{-t}e^{-e^{-t}}dt$

Suppose we have the following integral \begin{equation} \int_{-b}^{\infty}\log(t+b)e^{-t}e^{-e^{-t}}dt, \end{equation} where $b$ is a positive constant. It seems very difficult to derive the exact ...
3
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1answer
3k views

An error formula for linearization

Question: Can anyone shed some light on this formula? I can't find any information on it. It has three corollaries that I also need to understand:
0
votes
1answer
360 views

Approximating piecewise linear function

I'm trying to derive an analytic approximation to the following piecewise linear function: $$ f(x) = \left\{ \begin{eqnarray} \frac{x}{x_s} & & \text{if} & x < x_s \\ \frac{1-x}{1-...
2
votes
2answers
322 views

Stirling's Approximation Proof

I was searching for the reason why Stirling's Approximation holds true. I found the website Stirling's Approximation which apparently shows why this is the case. Does this part of the equation make ...
1
vote
0answers
71 views

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 ...
1
vote
1answer
319 views

Application of Stone-Weierstrass to approximate $f\in C(X\times Y,\mathbb{R})$ where $X$ and $Y$ are compact Hausdorff spaces?

On the wikipedia page for Stone-Weierstrass, the application section (http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem#Applications_2) says If $X$ and $Y$ are two compact Hausdorff ...
17
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2answers
8k views

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 ...
14
votes
2answers
1k views

Monotonic version of Weierstrass approximation theorem

Let $f\in\mathcal{C}^1([0,1])$ be an increasing function over $[0,1]$. Prove or disprove the existence of a sequence of real polynomials $\{p_n(x)\}_{n\in\mathbb{N}}$ with the properties: $p_n(x)$ ...
33
votes
1answer
861 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 ...
22
votes
2answers
569 views

Distance from $x^n$ to lesser polynomials

I am interested in the $L_1$ distance of $x^n$ to the $\mathbb R$-span of $\{1,x,\ldots,x^{n-1}\}$ over some interval. We can WLOG consider the interval $[0,1]$ (say) because scaling and shifting only ...
17
votes
4answers
2k views

Approximating continuous functions with polynomials

Given $\epsilon \gt 0$ and $f \in C^{0}[0,1]$, Weierstrass says that I can find at least one (how many? probably a lot?) polynomial $P$ which approximates f uniformly: $$\sup_{x \in [0,1]} |f(x) - P(x)...
10
votes
0answers
388 views

About the Gibbs phenomenon for the Legendre polynomials as an orthogonal base of $L^2(-1,1)$

In a recent question, I proved that the Fourier-Legendre expansion of the function $f(x)=\text{sign}(x)$ over $(-1,1)$ is given by: $$2\sum_{m\geq 0}\frac{4m+3}{4m+4}\cdot\frac{(-1)^m}{4^m}\binom{2m}{...
8
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2answers
3k views

How to best approximate higher-degree polynomial in space of lower-degree polynomials?

My question is: Find the best 1-degree approximating polynomial of $f(x)=2x^3+x^2+2x-1$ on $[-1,1]$ in the uniform norm(NOT in the least square sense please)? Orginially, as the title of the post ...
3
votes
2answers
376 views

How does one derive Radial Basis Function (RBF) Networks as the smoothest interpolation of points?

I was reading/watching CalTech's ML course and it said that one could derive the RBF Gaussian kernel from the solution to smoothest interpolation that minimizes squared loss. i.e. one can derive the ...
4
votes
2answers
378 views

maybe this sum have approximation $\sum_{k=0}^{n}\binom{n}{k}^3\approx\frac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty$

prove or disprove this $$\sum_{k=0}^{n}\binom{n}{k}^3\approx\dfrac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty?$$ this problem is from when Find this limit $$\lim_{n\to\infty}\dfrac{\displaystyle\sum_{k=...
3
votes
1answer
160 views

The magnitude of the difference between the integral and the Riemann sums for continuous functions

How do I prove that $$\lim_{n\to\infty}n\left[\int_{0}^{1}f(x){d}x - \frac{1}{n}\sum_ {i=0}^{n}f\left(\frac{i}{n}\right)\right] = -\frac{1}{2}$$ for any $ f(x)$ in $C^0([0,1])$ such that $f(0)=0$ and $...
2
votes
2answers
889 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 ...
7
votes
1answer
597 views

What is “Approximation Theory”?

What exactly is "Approximation Theory"? If I read the wikipedia-article I doesn't get much clearer. Why are "pure" mathematicians interested in it? I see a lot of people that do harmonic analysis also ...
3
votes
2answers
2k views

Proof of non-strictly convexity of $l_1$ and $l_{\infty}$

Given the Banach space $l_p = \left\{f\in R^n : \|f\| \leq \infty\right\}$ for $1\leq p \leq \infty$, we define the following norms $\|f\|_p = \left(\displaystyle\sum_{j=0}^{\infty} |f_j|^p\,w_j\...
3
votes
1answer
39 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 ...
2
votes
1answer
732 views

differentiability of $f(x,y)=xy\sin\left({1\over x^2+y^2}\right)$

Let $f(x,y)=xy\sin\left({1\over x^2+y^2}\right)$ if $(x,y)\neq (0,0)$ and $0$ if $(x,y)=(0,0)$. Determine the points in which $f$ is differentiable I know that $f(x,y)$ is differentiable at $(x_0,y_0)...
1
vote
3answers
2k views

Least square fit using Legendre polynomials

I would like to apply Legendre polynomials to least square approximation. Therefore I would like the function: $$L_n (x)=\sum_{k=0}^n a_k P_k (x)$$ to fit $f(x)$ defined over $[-1,1]$ in a least ...
1
vote
2answers
3k views

Interpolation, Extrapolation and Approximations rigorously

A foreign book mentioned that "when the Lagrange's interpolation formula fails (for example with large sample due to Runge's phenomenon), you should use approximation methods such as Least-squares-...
4
votes
0answers
159 views

Weighted polynomial approximation on the half-line

Let's $w : \mathbb R_+ \to \mathbb R_+^*$ a continuous function (I will only be interested in the case $w : t \mapsto e^{-t}$). We use $w$ to define the Banach space $C^0_w(\mathbb R_+)$ of continuous ...
4
votes
1answer
2k views

Smooth approximation of characteristic function of a bounded open set

Let $U$ be an open bounded set of $\mathbb{R}^n$. Is it possible to approximate $\chi_U$ as almost everywhere limit of increasing sequence of smooth functions?
3
votes
0answers
58 views

Instantaneous smoothing effect of sphere-valued maps

The instantaneous smoothing effect of the heat equation is the property that the solution to $$\begin{cases} \partial_t u= \Delta u, & t>0 \\ u(0, x)=f(x), & x\in \mathbb R^d,\end{cases}$$ ...
2
votes
1answer
619 views

Is the dual of a reflexive Banach space strictly convex?

Is the dual of a reflexive Banach space strictly convex? Why? This is a question that arouse trying to understand the theory behind approximation by finite element methods.
2
votes
1answer
106 views

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 ...
1
vote
1answer
241 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 $...
1
vote
1answer
2k views

Examples of strictly convex normed spaces that are not uniformly convex

I am studying approximation theory, just to learn basically, and in different books I saw a theorem about uniqueness of best approximation. One book uses strict, the other uses uniform convexity with ...