38 votes
Accepted

Why the sequence of Bernstein polynomials of $\sqrt x$ is increasing?

As noted by Paata Ivanishvili, if $f$ is concave on $[0,1]$, then the Bernstein polynomials $B_n(f,p)$ are increasing in $n$. Here is a probabilistic proof: Let $I_j$ for $j \ge 1$ be independent ...
Yuval Peres's user avatar
32 votes

Examples of back of envelope calculations leading to good intuition?

Finding the primitive of logarithm Finding a primitive amounts to calculating an integral. Calculating an integral amounts to measuring the area under a curve. What is the area under the curve of $\ln$...
25 votes
Accepted

Approximation of smooth diffeomorphisms by polynomial diffeomorphisms?

The answer is 'no', because polynomial mappings with polynomial inverses preserve volumes up to a constant multiple. To see why this property holds, suppose that $p:\mathbb{R}^d\to\mathbb{R}^d$ is a ...
Robert Bryant's user avatar
24 votes

Examples of back of envelope calculations leading to good intuition?

Although requiring a bit more than undergraduate math, roughly a first course in algebraic number theory, I'd say that Pomerance's first calculations for the general number field sieve fit into this ...
20 votes
Accepted

Real polynomial bounded at inverse-integer points

Part I: $CN^{1/3}$ is enough. Start with $P(x)=\prod_{k\le N^{1/3}}(1-k^2x^2)$. Notice that it vanishes at $1/k$ with $k\le N^{1/3}$, is bounded by $1$ on $[0,N^{-1/3}]$, the degree of $P$ is $2N^{1/...
fedja's user avatar
  • 59.8k
20 votes

Why the sequence of Bernstein polynomials of $\sqrt x$ is increasing?

While nothing will beat the brilliant probabilistic proof given in Yuval Peres's answer, a more conventional argument goes as follows. Write $$a_{n,k} = \tbinom nk x^k (1-x)^{n-k} $$ and $$ p_{n,k} = \...
17 votes

Examples of back of envelope calculations leading to good intuition?

Knuth's probabilistic "proof" of the hook-length formula might qualify, though it's not an approximation as such. Here we have a partition $\lambda$ of $n$. Recall a standard Young tableau ...
16 votes

Examples of back of envelope calculations leading to good intuition?

Minkowski theorem The Poisson summation formula writes $$\sum_{n \in \mathbb Z^n} \phi(n) = \sum_{n \in \mathbb Z^n} \widehat{\phi}(n)$$ where $\hat{\phi}$ is the Fourier transform of $\phi$. Let's ...
16 votes

Why the sequence of Bernstein polynomials of $\sqrt x$ is increasing?

I would also be happy to know the converse implication, that you quoted in the last remark of your notes Let $f \in C([0,1])$. Then $p_{n+1}(f,x) \geq p_{n}(f,x)$ for all $n \in \mathbb{N}$ and all $x ...
Paata Ivanishvili's user avatar
14 votes
Accepted

Can $C^1$ mappings with derivative of low rank be approximated by smooth maps?

There is a counterexample. Example. There is $f\in C^1(\mathbb{R}^5,\mathbb{R}^5)$ with ${\rm rank}\, Df\leq 3$ that cannot be approximated in the supremum norm by mappings $g\in C^2(\mathbb{R}^5,\...
Piotr Hajlasz's user avatar
14 votes

The unreasonable effectiveness of Padé approximation

The function used in the example has an asymptotic value of $1/2$ as $x\to \infty$ A Maclaurin expansion will only either go to infinity or negative infinity as x goes to infinity. In order to match ...
David Elm's user avatar
  • 241
14 votes
Accepted

Find the maximum of $|a_{p}|$, if $a_0+a_1x+\dots+a_nx^n:[-1,1]\mapsto [-1,1]$

Let $T_{n}(x)=\sum_{\nu=0}^{n}t_{n,\nu}x^{\nu}$ denote the Chebyshev polynomial (of the first kind) of degree $n$ and let $x_{n,\nu}=\cos\nu\pi/n$ for $0\leq\nu\leq n$. The answer follows from the ...
user111's user avatar
  • 3,771
13 votes

Low-degree polynomial approximation of the piecewise-linear function $x \mapsto \max(x, 0)$ on an interval $x \in [-R,R]$

Subtracting $x/2$, and rescaling, the problem is reduced to the best uniform approximation of $|x|$ on $[-1,1]$ by polynomials, a problem already considered by Bernstein. According to Chebyshev's ...
Pietro Majer's user avatar
  • 56.6k
12 votes
Accepted

How small (in modulus) can a polynomial get?

Yes. Proved by Chebyshev. The extreme case ($\max = 2^{1-n}$) is given by $2^{n-1}\cdot f(x)$ being a Chebyshev polynomial of first kind. https://en.wikipedia.org/wiki/Chebyshev_polynomials
Max Alekseyev's user avatar
12 votes

Vector-Valued Stone-Weierstrass Theorem?

I think that you want something like this: Let $E\to X$ be a (finite rank) vector bundle over a compact, Hausdorff topological space $X$, let $\mathcal{A}\subset C(X,\mathbb{R})$ be a subalgebra that ...
Robert Bryant's user avatar
12 votes

Examples of back of envelope calculations leading to good intuition?

I think Flory's argument for the exponent for the mean-square displacement for the self-avoiding walk (SAW) qualifies as a back-of-the-envelope calculation which is surprisingly good. Let $\omega(n)$ ...
12 votes
Accepted

Bounding the discrete $l^p$ norm by the continuous $L^p$ norm for trigonometric polynomials

Yes, this goes back to the work of Plancherel, M.; Pólya, George, Fonctieres entières et intégrales de Fourier multiples, Comment. Math. Helv. 9, 224-248 (1937). ZBL0016.36004. (see for instance ...
Terry Tao's user avatar
  • 109k
11 votes
Accepted

approximating the $|x|$ function

Denote the minimal approximation error to the function $f(x)=|x|$ in the uniform norm on $[-1,1]$ by $$ E_{mn}(f,[-1,1])=\inf_{r\in\mathcal{R}_{mn}}\|f-r\|_{\infty,[-1,1]}, $$ where $\mathcal{R}_{mn}$...
user111's user avatar
  • 3,771
11 votes
Accepted

Are the polynomials in $\{1/t\}$ dense in $L^2(0,1)$?

They are not. The function $g(t) = \begin{cases}\frac{-1}{(1-t)^2}& \frac{1}{3} < t < \frac{1}{2}\cr 1& \frac{1}{2} < t < 1\end{cases}$ is orthogonal to all of them. That is ...
Nik Weaver's user avatar
  • 42.1k
11 votes

Examples of back of envelope calculations leading to good intuition?

There's a back of an envelope calculation by Beckenstein in thinking how the area of a black hole can be interpreted as a measure of entropy, the underlying assumption being that the laws of ...
11 votes
Accepted

Stone-Weierstrass theorem: coefficients of approximating sequence bounded?

The answer is negative. Suppose we approximate a continuous function on $[-1,1]$ with ordinary polynomialss $P_n$. If the coefficients are bounded, say $|a_{n,k}|\leq C$, then $$|P_n(z)|\leq C(1-|z|)^{...
Alexandre Eremenko's user avatar
10 votes
Accepted

Low-degree polynomial approximation of the piecewise-linear function $x \mapsto \max(x, 0)$ on an interval $x \in [-R,R]$

This is done by using Remes's algorithm. It is implemented in Mathematica's command MiniMaxApproximation[]. Below is an image of the corresponding Mathematica notebook, giving a polynomial ...
Iosif Pinelis's user avatar
10 votes
Accepted

Optimal polynomial approximation of rational function $\frac{1}{1-x}$

This problem has an exact solution, written in the book N. I. Akhiezer, Theory of approximation. Dover Publications, Inc., New York, 1992, Chap II section 37. The error is $$\frac{(1-\sqrt{1-\rho^2})^...
Alexandre Eremenko's user avatar
10 votes
Accepted

Approximation of pseudogeometric progression

For natural $n$ and $x\in(0,1)$, one has $$L_n(x)\le f_n(x)\le U_n(x), \tag{10}\label{10}$$ where $$U_n(x):=1+I_n(x), \tag{20}\label{20}$$ $$I_n(x):=\int_0^n x^{\sqrt t}\,dt =2\frac{1+x^{\sqrt{n}} \...
Iosif Pinelis's user avatar
10 votes
Accepted

Approximation for complex variables

The theory of approximation in the complex plane is almost as rich as the theory on the real interval. Some of the good books are: D. Gaier, Lectures on complex approximation. Translated from the ...
Alexandre Eremenko's user avatar
9 votes
Accepted

What are some of the surprising results of finite sample statistical estimation?

First of all I have to express my opinion that the gap between large sample behavior and the finite sample behavior should be considered "unsurprising". Basically speaking, the theory of asymptotics ...
Henry.L's user avatar
  • 7,961
9 votes

Approximating power series coefficients --- Why does a clearly illegitimate method (sometimes) work so well?

The paper 'A New Method for Computing Asymptotics of Diagonal Coefficients of Multivariate Generating Functions' by A. Raichev and M. Wilson has the precise machinery that can solve this problem. Get ...
skbmoore's user avatar
  • 884
9 votes

Uniformly approximating a function and its derivative using polynomials

The span of $\{x^{2k}\colon k=0,1,2,\dots\}$ is dense in $C([0,1])$. But all their derivatives vanish at $0$.
Ilya Bogdanov's user avatar
9 votes

Upper bound an integral with exponential function

The integral in question can be rewritten as $$ \begin{aligned} I&:=\frac1{\sqrt n}\,\int_{-a\sqrt n}^{(1-a)\sqrt n} e^{-u^2}\Big(1-\exp\Big\{-\frac{u^4/n}{1-u^2/n}\Big\}\Big)\,du \\ &\le\...
Iosif Pinelis's user avatar
8 votes
Accepted

Subspaces of $L^2(0,1)$ dense on every truncation $L^2(c,1)$

It is not a priori clear to me whether there is any tame and dense subspace of $L^2(0,1)$. Indeed, no such space exists. To see it, choose any sequence of functions $f_k\in L^2([0,1])$ such that $f_k|...
fedja's user avatar
  • 59.8k

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