User andrei mf - MathOverflow

Answer by Andrei MF for Integral of generalized Laguerre polynomials

2012-07-23T23:05:42Z

There is an expression in terms of the generalized hypergeometric function 3F2. The best references I know in this direction are:

1) J.S. Dehesa et al, "Information Theory of D-Dimensional Hydrogenic Systems: Application to Circular and Rydberg States", International Journal of Quantum Chemistry, Vol 110, 1529–1548 (2010).

2) V. F. Tarasov, "Exact numerical values of diagonal matrix elements 〈rk〉 nl , as n≤8 and −7≤k≤4, and the symmetry of Appell’s function F2(1,1)", Int J Mod Phys B (2004) 18:3177–3184.

Answer by Andrei MF for Discrete Wavelet Transform and L2 Basis

2012-06-26T08:31:23Z

Take a look also at a nice paper "Discrete Wavelet Transformations and Undergraduate Education" at http://www.ams.org/notices/201105/rtx110500656p.pdf.

Identifying a system of ODEs

2012-03-17T13:37:39Z

Studying the dynamics of the endpoints of an equilibrium measure (a minimizer of its logarithmic energy in an external field) I ran into the following system of differential equations (which I state for the case of 4 points, for simplicity): let $x_j=x_j(t)$, $j=1, \dots, 4$, be real values dependent on time $t$, all distinct at $t=0$, and satisfying the system $$ \frac{d x_j}{dt} = \frac{m_j}{q'(x_j)}=m_j \prod_{k\neq j} (x_j-x_k)^{-1}, \quad j=1, \dots, 4, $$ where $q(x)=\prod_{j=1}^4 (x-x_j)$ and $q'(x)$ is its derivative with respect to $x$. Here $m_j$ are positive numbers.

My questions (sorry if too elementary or naive) are:

1) is this kind of a system known, does it have any name attached to it?

2) I needed to prove the fact that the interior $x_j$'s collide in a finite time. Does this follow from any general fact in dynamical systems or systems of ODEs?

3) what about the more general situation, when the number of points is $n$ and the right hand sides in the system are rational functions?

Thanks in advance.

Answer by Andrei MF for Recovering a measure from its moments

2012-02-25T11:49:02Z

An addendum to Kai's answer: the moments $m_n$ of the measure $\mu$ are the coefficients of the Laurent expansion at infinity of the Cauchy transform $$\hat \mu(z)=\int \frac{d\mu(t)}{z-t}$$ of $\mu$ (just expand $(z-t)^{-1}$ there and substitute it in the integral):$$\hat \mu(z)=\sum_{n=0}^\infty \frac{m_n}{z^{n+1}}. $$ You can recover the original measure (at least its absolutely continuous part) as the difference of the boundary values of $\hat \mu$ on $(0,1)$ (Sokhotski's theorem).

From the practical point of view, you could approximate $\hat \mu$ from a finite number of moments using the (diagonal, i.e. both numerator and denominator of degree $\leq n$) Padé approximants at infinity. This will be close to optimal, and the approximation converges uniformly in the whole complex plane except the interval $[0,1]$ with a geometric rate (although the closer to the interval you are, the slower the rate of convergence is). Finally, the denominators $Q_n$ of these Padé approximants will be the orthogonal polynomials with respect to $\mu$.

Answer by Andrei MF for Upper bounds on generalized Laguerre polynomials

2012-02-04T12:13:58Z

What kind of estimates exactly do you need? It is difficult to help if you are not more specific. At any rate, I believe there are many references you could check. You might find some useful inequalities in the papers:

1) "A Sharp Inequality of Markov Type for Polynomials Associated with Laguerre Weight" by Holly Carley, Xin Li, and R N. Mohapatra, Journal of Approximation Theory 113 (2001) 221–228

2) "Some inequalities for algebraic polynomials with the Laguerre weight" by Semyon Rafalson, Journal of Approximation Theory 143 (2006) 201 – 218

3) "Inequalities for orthonormal Laguerre polynomials" by Ilia Krasikov, Journal of Approximation Theory 144 (2007) 1 – 26 (see the references therein)

Last but not least, check the NIST Digital Library of Mathematical Functions online at http://dlmf.nist.gov/, in particular, http://dlmf.nist.gov/18.14

Answer by Andrei MF for The version of Montel's theorem used in the proof of Jenkins-Strebel differential

2012-01-31T08:14:04Z

The assumption is that the modulus $M$ is bounded, so what I think Strebel is doing is applying the theorem that a uniformly bounded family of analytic functions is normal ($=$ has a uniformly convergent subsequence). The Riemann surface is endowed with a natural metrics and analytic structure. If you regard the composition of the conformal mapping $g_n$ with a local analytic chart $\varphi$, it becomes a complex-valued analytic function $ \varphi \circ g_n$, so you can apply to it standard theorems from Complex Analysis. Also the notion $|g_n-g|$ makes a perfect sense.

Answer by Andrei MF for What are other applications of difference equations in other branches of mathematics ?

2011-12-26T23:40:42Z

The three-term recurrence relation satisfied by a family of orthogonal polynomials is a crucial fact which brings together classical analysis, spectral theory and other branches of mathematics. This recurrence relation is obviously an example of a difference equation.

Answer by Andrei MF for Approximation:- Algorithmic considerations

2011-12-06T04:19:02Z

First of all, if you need a polynomial approximation, I'd rather rescale $f$ to put it on $(-1,1)$ and interpolate it at the Chebyshev nodes http://en.wikipedia.org/wiki/Chebyshev_nodes. Furthermore, instead of using the monomial basis, as for Taylor polynomials, it is better to expand the function in a series of Chebyshev polynomials, and truncate it at a certain value. Next, multipoint Padé approximants (rational interpolants) work well at the Chebyshev nodes too. Finally, if you can allow for some algebraic functions as interpolants, you can try to approximate first a function of the form $h= f g$, where $g$ is a known function such that $h$ has no singularities at the end points.

Btw, you might want to check out the Chebfun project http://www2.maths.ox.ac.uk/chebfun/ where several of the ideas mentioned here are nicely implemented.

Answer by Andrei MF for Uniform continuity and boundedness

2011-12-04T09:34:52Z

The theorem you mention is kind of strange. You don't need to assume uniform continuity, it is enough to suppose that your function $f$ is continuous: every continuous function on a compact subset of $\mathbb R$ is automatically uniformly continuous. Then, what you are trying to prove is that continuity on a compact $\Rightarrow$ boundedness (so called, extreme value theorem, see http://en.wikipedia.org/wiki/Extreme_value_theorem where a the standard proof is outlined).

Answer by Andrei MF for gaussian quadrature

2011-03-17T23:40:16Z

Look at it this way: what you are computing is a PROJECTION. In this sense, you are not surprised that in order to compute the value of the function $f(x_1, x_2)=x_1$ you need only 1 parameter. Something similar happens here. Instead of taking as parameters the coefficients $a_j$ of the expansion of the given polynomial in the basis of monomials, take the $a_j$ to be the expansion of this polynomial in terms of the orthogonal polynomials with respect to the integration weight. Then the quadrature formula is given by the linear combination of the coefficients $a_n, \dots, a_{2n-1}$ only.

Answer by Andrei MF for Distribution of eigenvalue spacings

2011-02-05T23:43:10Z

I suggest you to take a look at the paper "How to Generate Random Matrices from the Classical Compact Groups" by F. Mezzadri, Notices AMS 54 (5), 592-604 (2007). Can be downloaded freely from here: http://www.google.es/url?sa=t&source=web&cd=1&ved=0CBcQFjAA&url=http%3A%2F%2Fwww.ams.org%2Fnotices%2F200705%2Ffea-mezzadri-web.pdf&rct=j&q=How%20to%20Generate%20Random%20Matrices%20from%20the%20Classical%20Compact%20Groups&ei=W-BNTb6nFZaShAfS2bWTDw&usg=AFQjCNESHv00uIyATuY1NETwTfH5IO1fhQ&cad=rja

Answer by Andrei MF for Why fourier transform tell us energy of any frequency of f(t)

2010-08-29T11:30:17Z

In signal processing, the energy of a continuous-time signal is defined as the square of its $L^2$ norm. Hence, the spectral energy of this signal is the square of the $L^2$ norm of this signal in the spectral domain, i.e. of its Fourer transform. By Parceval's theorem, these two energies are equal. But as a consequence, $|x(t)|^2$ is the energy density of the signal at the moment $t$, and $|F(x)(\tau)|^2$ is the spectral energy density at the frequency $\tau$.

This is in complete analogue with the discrete case: in your notation, $a_n^2 + b_n^2$ is equal (up to a multiplicative constant) to the square of the absolute value of $\int f(t) e^{-i \theta t} dt$ for $\theta=n$ (for the real-valued $f$).

Answer by Andrei MF for eBook readers for mathematics

2010-07-10T13:42:49Z

I use IPad as an ebook reader with iAnnotate, which allowes to annotate the PDF files, and I am ver pleased.

Comment by Andrei MF

2012-05-13T19:17:44Z

Thanks Robert, this is a nice argument.

Comment by Andrei MF

2012-04-03T19:01:12Z

Thanks, very interesting. Could you clarify which Ruelle's book you have in mind? I saw in Amazon "Statistical Mechanics: Rigorous Results", and "Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics".

Comment by Andrei MF

2012-03-18T00:16:33Z

I found a couple of papers by H. Aref on the arxiv, chao-dyn/9907038 might have something I could use, although I have to look more carefully. As you say, not clear that it answers my questions, but it is interesting anyway. Thanks.

Comment by Andrei MF

2012-02-05T12:30:34Z

@Noam: don't leave Abramowitz-Stegun behind either... :-)

Comment by Andrei MF

2011-12-08T10:49:14Z

Sorry, I probably don't get your question right, but Chebyshev nodes are explicit, see e.g. en.wikipedia.org/wiki/Chebyshev_nodes So, don't see why you need the root finding.

Comment by Andrei MF

2011-12-07T00:41:46Z

Actually, Chebyshev-Pade approximation could do better in your context (these are rational functions $r$ whose Chebyshev series should match that of $f$ as far as possible). I think that Ricardo Pachon has implemented many of these algorithms in Matlab, you might want to check his webpage, maths.ox.ac.uk/node/10864 Take a look especially at his recent publications listed there.

Comment by Andrei MF

2011-12-04T10:52:32Z

You wrote $[a,b]$, this is closed and bounded $\Rightarrow$ compact.