User vagabond - MathOverflow

What is this space called ? And what is its character group called ?

2012-12-01T16:03:17Z

Consider the set of integers $\mathbb Z$. We define a translation invariant measure $M$ on $\mathbb Z$ in the following fashion, let $A \subseteq \mathbb Z $, then $M(A)=lim_{n\rightarrow \infty} \frac{|A \cap [-n,n]|}{2n+1}$. Now equipped with this measure consider the space of $l^2(\mathbb Z)$. What is this space called ? and what is dual group (the group of set of characters) known as ?

Connections of results in Harmonic analysis in the theory of Transcendental Numbers

2012-02-06T10:10:17Z

An entire function $f$ is said to be of exponential type if there exist constants $c$ and $k$ such that $|f(z)|\leq c e^{k |z|}$.

A famous result of Polya says if $f$ is an entire function of exponential type strictly less than $\log 2$ and take integer values at $\mathbb{Z_+}$ (the set of non-negative integers) then $f$ must be a polynomial.

A generalization of this result was used by Gelfond to prove his famous result $\alpha$ and $\beta$ are algebraic numbers with $\alpha \neq 0,1$ and if $\beta$ is not a rational number, then any value of $\alpha ^\beta = \exp(\beta \log \alpha)$ is a transcendental number.

It is well known that the theory of entire functions of exponential type is closely related to that of Harmonic analysis.

If some one can guide me to other results/method of proofs in transcendence theory where techniques or results related to harmonic analysis are used I would be very grateful.

A Question:-

Maybe the result of Polya can be seen as a manifestation of the Uncertainty Principle? In the sense that a function and its fourier transform both cannot be small.

Since, the function of exponential type is the Fourier transform of a compactly supported distribution, thus support is small there.

Now, a transcendental entire function of exponential type takes all values (with the possible exception of one) infinitely often (thus the zero set of such a function is usually infinite i.e., big and hence the support is small).

What Polya's result says is with the extra arithmetic condition on the function, that is it takes integer value at non negative integers, the function no longer can not be transcendental type (infinitely many zeros, so support is small) and hence a polynomial (only finitely many zeros i.e., support is big). (Of course this is heuristic, can this be made precise ? )

Solving PDE via Cellular Automata

2012-02-05T20:33:16Z

Is there a theory for solving PDE by using Cellular Automata ? Something which is on the line of, passing to the limit (scale) i.e., if you increase the number of grid points the solution to the cellular automata will converge to the PDE ? If so, how successful is this approach ? What are the limitations of this approach. Also, given a PDE how does one go about finding the rules for the corresponding cellular automata and vice versa ?

Answer by Vagabond for Constructing an L2-space with a given orthonormal basis

2011-11-12T13:16:20Z

Such T and measure do exist and in certain cases the measure is the restriction of usual Lebeasgue measure in $\mathbb{R}^n$. This kinds of sets are called spectral sets and spectral measures respectively, i.e., sets/ measures for which an complete orthogonal system of exponentials exist. The study of such sets began with the work of Fuglede (Fuglede, B. "Commuting Self-Adjoint Partial Differential Operators and a Group Theoretic Problem." J. Func. Anal. 16, 101-121, 1974.) who conjectured that a set is spectral if and only if it tiles and proved it in the special case when the spectrum or the tiling set is a lattice. The conjecture however has been disproved in dimension bigger than 3 following the work of Tao Tao, T. "Fuglede's Conjecture Is False in 5 and Higher Dimensions." ( http://arxiv.org/abs/math.CO/0306134.) but remain open in dimension 1,2. Further under additional hypothesis on the set T many positive results are known. The study of spectral measure began with the work of Jorgensen and Pedersen and Strichartz.

How to find the almost period of an exponential polynomial

2010-09-10T13:00:18Z

Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the exponents are $\delta$-separated.

Define $E_n$ to be the collection of all exponential polynomial of order $n$. i.e.,

$$ E_n:= \{ u : u(t) = \sum_{k=1}^n c_k e^{i \lambda_k t}, c_k \in \mathbb C, \lambda_k \in \mathbb R, and |\lambda_i - \lambda_j| > \delta\ when\ i \neq j \}. $$

Of course $u$ is an almost periodic function i.e., given $\epsilon >0$ there exists $T_\epsilon$ such that every interval of length $T_\epsilon$ contains an almost period of $u$. i.e., $\forall x \in \mathbb R,$ $\exists \tau \in (x,x+T_\epsilon)$ such that

$$ \sup_{t\in \mathbb R} |u(t)-u(t+\tau)| < \epsilon$$

Is it possible to find a bound on the $T_\epsilon$ for such an exponential polynomial, I added the separation condition hoping that it would lead to an affirmative answer.

I have this suspicion that the almost period may have to do something with the number theoretical properties of the set of exponents. After all if all the exponents are in a lattice then the function is periodic.

and may I even dare hope to find a bound which will work for the entire class? Some nice subclass maybe ?

If that is too much to ask then what is a good question to ask ?

Has some one studied related questions or variations of it ? I would be glad to know.

Answer by Vagabond for Most memorable titles

2010-11-01T19:00:50Z

Fractured Fractals And Broken Dreams. Self-similar Geometry through Metric and Measure. Guy David, Stephen Semmes.

http://www.oup.com.au/titles/academic/maths/9780198501664

This is the most unusual title of a book which I have ever come across. I discovered this while randomly browsing through books in the library and got hooked. I was an undergraduate then and it had a strange attraction to me, even though I could not figure out anything that was written in it then.

I was not the only one!

Our university used to put out a list of courses (in the good old days) which were going to be offered and students would choose from it. Some of us managed to add the name of this book against the fractal geometry course as a course material. A record number of students enlisted.

Within a week a record number of them wanted to opt out. So there were inquiries: it turned out most of the students cited that they found the title of the book mentioned in the course material attractive which prompted them to enlist.

(We had found in the previous year that the instructor did not care about teaching, insist on taking class at 8 in the morning and would religiously take attendance for 10 minutes, by the end of the class half the class would be snoring. The assignments were to be submitted on A4 paper, we were supposed to write on one side with appropriate margin.

It was a case of a pun / warning which had gone horribly wrong. )

How many minors I need to check to conclude all minors will vanish ?

2010-10-28T12:06:22Z

Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.

I remember hearing that one really does not need to check all possible minors in order to conclude that all of them would vanish.

If such a result is true, how many minors will do the job and which ones ?

I am wondering if it is even possible to calculate the value of all minors based on the value of a nicely chosen "generating subset" ?

Edit:- The question which I had asked does not have an affirmative answer as explained
by Steven Sam. But matrix minors do satisfy some relationships see the answer by Sheikraisinrollbank below. If someone can modify the question to a more appropriate one (in light of Steven Sam and Sheikraisinrollbank answers ) please feel free to do so.

I have often come across a situation (more so at present than ever before) where in order to answer a problem in my subject area I am led to questions which are totally different areas about which I have absolutely no familiarity. Most often these are quite basic and I would suppose well known to any one who works in those areas. It is natural that a person who is not familiar with a given field will end up asking for "a result of the following kind" rather than a precise question. For a person who is knowledgeable I understand the question may be irritating or look ill posed but mind you the hapless fellow is not a graduate student in the given field and please do not judge him accordingly. I think its desirable that if someone knows how to reformulate the question to something so that it becomes well posed or meaningful it should be done. Why not edit the question to something so that it becomes a valid well posed question, to something which is obviously much more interesting than which was originally posed ?

structure of singular matrices whose entries have modulus one

2010-10-08T11:41:06Z

Let $A$ be a $n \times n$ matrix all of whose entries has modulus 1.

Suppose the matrix $A$ is singular.

We will assume without loss of generality that all the entries in the first row and the first column of the matrix are 1.

Observe when $n=2$ the matrix $A$ can be then singular if and only if $a_{2,2}=1$ as well.

A slightly less trivial observation is that the same thing happens when $n=3$, that is the matrix $A$ is singular if and only if two of the rows or columns are identical.

\begin{equation} \left|\begin{array}{ccc} 1 & 1 & 1 \ 1 & \alpha_{2,2} & \alpha_{2,3} \ 1 & \alpha_{3,2} & \alpha_{3,3} \ \end{array}\right| = 0 \end{equation}

So the matrix $A$ is singular iff $(\alpha_{2,2}-1)(\alpha_{3,3}-1)=(\alpha_{2,3}-1)(\alpha_{3,2}-1)$.

Let us assume without loss of generality that $\alpha_{2,2} \neq 1$ and $\alpha_{3,2} \neq 1$.

Consider the circle $C_1(t)= (\alpha_{2,2}-1) (e^{2 \pi i t}-1) $ and $C_2(t)=(\alpha_{2,3}-1) (e^{2 \pi i t}-1), t\in [0,1]$.

Since, the two circles either are identical and in that case $\alpha_{i,2}=\alpha_{i,3}$ that is the second and third columns are identical, or else as two distinct circles can intersect in at most two points we get similarly two of the rows or columns are identical.

Now, probably it is too much to expect the same result for all $n$.

But my requirement is only for $n=4$, is it true that a similar result holds for $n=4$ ?

Edit: I forgot to mention that I am interested in the case when the matrix is singular > > and none of its sub matrices are singular. (thanks @ Gerry Myerson for pointing it out)

Thankyou,

Answer by Vagabond for two sequences whose difference converges to zero

2010-10-27T20:06:53Z

To define convergence one needs a "metric" or a concept of "distance", and there can be many different notion of "distances". For example one can consider $\lim_{n -> \infty} \frac {1}{n} \sum_{k=1}^n |(A_k - B_k)|^p$. Or alternatively $\lim_{n -> \infty} \frac {1}{n} \sum_{k=n}^{2n} |(A_k - B_k)|^p$.

Though your notion of distance is much stronger than the above, to be precise its $\limsup_{n->\infty}|A_n-B_n|$. So if the "distance" between two sequences is zero one can define an equivalence relation in a natural way and then you do actually get a proper metric. As everyone has mentioned this is how we go about constructing the real number system using Cauchy sequences.

So, I would suggest the sequence ~~{A eventually converge to B or}~~, as per Willie Wong's suggestion A is asymptotically equivalent to B.

Introductory text book for Linear Recurrence Sequences

2010-10-25T14:19:45Z

What is a good introductory text for linear recurrence sequences?

What all are the necessary prerequisite for it? (My background is in Euclidean Fourier Analysis.) After browsing through several books, my perception is one is supposed to know a fare bit of algebraic number theory , algebraic geometry, Diophantine Equations etc. (I am not sure if the subjects I mentioned are the only or even the appropriate areas, so please correct me if I am wrong) Is there a book which builds/gives the necessary material as it progresses.

I will appreciate any suggestion which you may think is going to be helpful.

Thank you

Answer by Vagabond for Group which "resembles" the free product of a cyclic group of order two and a cyclic group of order three, but isn't.

2010-10-12T18:27:59Z

I am hazarding a guess, I believe this should do the job

A = { {1, x, 0}, {0, -1, 0}, {0, y, 1} }

It really does not matter what x and y are, they can be chosen arbitrarily and can even be two formal symbols

B = { {0, 0, -$i$}, {$i$, 0, 0}, {0, 1, 0} }

Then A.A= B.B.B = Id

the order of $A.B$ would be infinite when $x$ and $y$ are suitably chosen, for example one can choose $x$ and $y$ so that the coefficient of the matrices $(A.B)^n$ unbounded ?

Algebraic integers on the unit circle

2010-10-01T09:41:35Z

Consider a set of algebraic integers which lie on the unit circle, they will generate a multiplicative subgroup of $\mathbb S^1$. Do these objects have a name?

I would guess they contain useful arithmetic/number theoretic information, for example if the generating set is the set of roots of an irreducible polynomial, what kind of information would they contain?

Has the group structure of the elements of a number field which lie on the unit circle $\mathbb S^1$ and the subgroup of it which is made up of algebraic integers been studied.

Would greatly appreciate if you could suggest a reference.

Regards Vagabond

It would be real nice if you could answer keeping in mind that I do not know much Algebra/ commutative algebra/ algebraic geometry. But I hope that does not stop you from answering.

A question regarding polynomials whose roots satisfy certain algebraic relation

2010-09-27T07:34:18Z

Suppose I know the following information about a function :

1) Its a polynomial (not an explicit equation, neither the roots nor the degree is known)

2) I have managed to find an algebraic relation between some of the roots (mind you I do not know the roots explicitly, just the form of the algebraic relation is known to me).

Now given this information can one say something about the polynomial itself ?

Now what do I seek for? Well, information on something like the divisors of the degree of the polynomial, or say something about the Galois group of the polynomial may be .... so you can say am asking an inverse question.

I understand that under these very general condition the problem may not even be well posed. I actually have more information about the polynomial in the particular case I encountered it ... the polynomial is a 0-1 polynomial ...some of the roots lie in the unit circle... etc. etc.

But certainly there would be instances of similar problems (with more information available about the polynomial/ the nature and number of algebraic relations that are available etc.) which has been dealt with ?

So, I wanted to ask the question in a more general setting. Any variant of this I would say is quite interesting. So you can assume different kind of condition on the roots, coefficient algebraic relation,

I will greatly appreciate if some one can point out where I should be looking. Reference to literature where such a problem has been dealt with would be great.

Regards

Vagabond

Answer by Vagabond for Is there a simple criterion to determine if two parallelograms intersect?

2010-09-16T09:42:14Z

Observe Two parallelogram $A$ and $B$ can intersect if and only if one edge of $A$ and one edge of $B$ intersects (crosses over) unless one parallelogram is sitting inside the other.

I am assuming the second case does not happen.

So the problem reduces to answering when does two line segment of finite edges intersect.

Here is how to check ...

Take edge $A_{12}$ joing $a_1$ to $a_2$ and $B_{34}$ joining $b_3$ to $b_4$

so in order to check if $A_{12}$ intersects $B_{34}$

take the line defined by $A_{12}$, $b_3$ and $b_4$ should lie on two different sides of this and further ${a_1}, {a_2}$ should lie on opposite side of $B_{34}$.

So one check this by taking some inner products .... we need to check if appropriate sign change happens ...(intermediate value theorem) if it does then lines cross.

Answer by Vagabond for Can an algebraic number on the unit circle have a conjugate with absolute value different from 1?

2010-09-14T13:51:15Z

One interesting set of example is Salem numbers. These are real algebraic integer all of whose conjugates has absolute value less than or equal to one and at least one of the conjugates lies on the unit circle.

So you have a scenario of, one which flew over the cuckoo's nest, at least one in the edge and the rest are all in the nest (caged ?) .

http://en.wikipedia.org/wiki/Salem_number

Also if all its conjugates lies on the unit circle then it has to be a root of unity, its known as Kronecker Theorem.

http://mathoverflow.net/questions/33169/good-effective-version-of-kroneckers-theorem

Queries about the Skolem-Mahler-Lech theorem (integer zeros of exponential polynomials)

2010-09-12T03:46:15Z

The Skolem-Mahler-Lech Theorem says that the integer zeros of an exponential polynomial are the union of complete arithmetic progressions and a finite number of exceptional zeros. http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-46954 (thanks to Qiaochu Yuan for pointing out the link).

*What I am really interested in is the description of the exceptional set. About how and where they are located ? Is there some kind of an estimate on how many of them are there ? conditions under which the exceptional set is empty / rather small / far from the origin etc. etc. in short any description about the geometry of the zero set. *

Here is an observation

Apparently all the known proofs use p-adic methods. Incidentally a lemma due to Turan comes very close to proving Skolem Mahler Lech theorem. It says if one knows the values of an exponential polynomial along an n length AP then one can find an estimate of it at the n+1 th point. Here is the exact statement

$\underline {Turan's Lemma}$ Let $ z_1,\dots,z_n$ be complex numbers, $|z_j|\geq 1, j=1,\dots,n.$ Let $ b_1,\dots, b_n \in \mathbb C $ and $$S_j:= \Sigma_{k=1}^n b_k z_k^j$$ Then $$|S_0| \leq {\frac{4 e (m+n-1)}{n}}^{n-1} \max_{j=m+1}^{m+n} |S_j|.$$ As a simple consequence of this result when the value of an exponential polynomial (with constant coefficient) is known for $n$ consecutive term of an arithmetic progression, then one can get an estimate of the value of the polynomial along that arithmetic progression. i.e., Let $p(t)=\Sigma_{k=1}^n c_k e^{i \lambda_k t}$ and assume that the value of the polynomial $p(t)$ is known for $t_j=t_0+j \delta$ for $ j= m+1,...,m+n$. Then substitute $b_k=c_k e^{i \lambda_j t_0}$ and $z_k= e^{i \lambda_k \delta}$ and apply Turan's lemma.

Applied to the the specific case under consideration ( i.e. an exponential polynomial of order n) it says if the exponential polynomial vanishes at an arithmetic progression of length n then it vanishes on the complete arithmetic progression.

So Szemeredi's Theorem now tells us that the structure of the zero set has to be union of complete AP plus a set of density zero set (actually more the exceptional set has no AP of length n).

Though it does not prove Skolem Mahler Lech theorem it's tantalizingly close and provides the extra information about the geometry of the zero sets .... that the exceptional set do not have an n length AP and using estimates from Ramsey theory one can find a bound on how many zeros can be there in an interval ?

Can one use all of these to give an analytic proof of Skolem Mahler Lech theorem maybe ? That would be real nice.

Having said all that what is a good reference to what is known about the nature of the exceptional set. Can somebody suggest a good reference ? a good review article maybe.

My problem is though the statement about Skolem Mahler Lech Theorem seems like a problem in analysis/ complex analysis the relevant literature is in other fields about which I have very little familiarity. My present ambition is just to understand the results (not the proofs) that is what can one infer about the nature of the zero set (may be with some additional condition imposed on the set of exponents).

Relation between entire function of exponential type and exponential polynomials

2010-09-11T19:41:20Z

Is it true in general that the theory of entire function of exponential type and and that of exponential polynomials (with purely imaginary exponents) are analogous ?

Can one derive results about entire function of exponential type by using results about exponential polynomials ?

For example I am wondering if it is possible to derive sampling theorems about band-limited functions by studying properties of exponential polynomials ?

What about the distribution of zeros ?

A question about self-affine tiles

2010-09-09T05:58:14Z

A self-affine tile is a compact set $T$ in $\mathbb R^n$ of positive Lebesgue measure for which there is an $n\times n$ expanding matrix $A$ (i.e. all its eigenvalues have modulus greater than 1) such that the affinely inflated copy $A(T)$ of $T$ can be perfectly tiled with essentially disjoint translates of $T$.

Thus we have $$ A(T) = \cup_{i=1}^m (T+d_i); \mathcal D= d_1,d_2,\dots,d_m $$

where $|det(A)| =|\mathcal D|= m$

Results of Kenyon (Projecting the one-dimensional Sierpinski gasket Projecting the one-dimensional Sierpinski gasket. Israel J. Math. 97 (1997), 221--238.) and Lagarias and Wang (Self-affine tiles in $R^n$. Adv. Math. 121 (1996), no. 1, 21--49) tells that such sets always can be used to give a translational tiling of $R^n$ and has boundary of measure zero and has nonempty interiors.

Thus in one dimension we can think of them as a union of intervals (possibly infinitely many ).

My question is :-

Is there a characterization of self-affine tiles in $\mathbb R$ which are union of finitely many intervals ?

Lecture notes by Thurston on tiling

2010-09-07T10:13:35Z

I am looking for a copy of the following

W. Thurston, Groups, tilings, and finite state automata, AMS Colloquium Lecture Notes.

I see that a lot of papers in the tiling literature refer to it but I doubt it was ever published. May be some notes are in circulation ?

Does anyone have access to it? I would be extremely grateful if you can send me a copy or tell me where can I find it.

Answer by Vagabond for Nice orthonormal basis for L^2(Cantor set)

2010-08-03T16:43:18Z

Though not always, for certain Cantor measures $\mu$ there exists orthonormal basis for $L^2(\mu)$ consisting of complex exponentials $\{e^{2 \pi i \lambda_n t}: \lambda \in \Lambda \}$ where $\Lambda \subset \mathbb R$. These are called spectral cantor measures.

see the following papers for details

P. E. T. Jorgensen and S. Pedersen , Dense analytic subspaces in fractal L2-spaces. J. Anal. Math. 75 (1998), pp. 185–228. http://www.springerlink.com/content/211651p66833m7j7/

R. Strichartz , Remarks on Dense analytic subspaces in fractal L2-spaces. J. Anal. Math. 75 (1998), pp. 229–231 http://www.springerlink.com/content/a53685h15g17x509/

Izabella Laba and Yang Wang On Spectral Cantor Measures Journal of Functional Analysis Volume 193, Issue 2, 20 August 2002, Pages 409-420 http://www.math.ubc.ca/~ilaba/preprints/meas1.dvi

How large (small) can be the measure of a set where a polynomial takes small values ?

2010-07-21T16:57:53Z

A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question how large ( and small) can be the measure of a set where a polynomial takes small values ?

This, and other interesting variation of this must have been studied in depth.

I would really appreciate any reference to the relevant literature.

Also, if there are some interesting variation of this problem I would like to know.

Thank you.

Example of a pair which is not weakly annihilating

2010-07-19T10:58:35Z

Let $\mathcal F$ denotes the Fourier transform $\mathcal{F} :L^2(\mathbb R)\rightarrow L^2(\mathbb R)$ and $E, \Sigma$ be two measurable sets in $\mathbb R$.

The pair $(E,\Sigma)$ is called a weakly annihilating pair, if for any $f \in L^2(\mathbb{R})$, $support (f) \subseteq E$, $support(\mathcal F f)\subseteq \Sigma$, implies $f \equiv 0$.

The pair $(E,\Sigma)$ is called a strongly annihilating pair, if there exists a constant $C$ such that for any $f \in L^2(\mathbb{R})$,

$$\|f\|_2^2 \leq C \left(\int_{\mathbb R \setminus E} |f|^2 dx + \int_{\mathbb R \setminus \Sigma} |\mathcal F f|^2 d\xi \right).$$ The notion of annihilating pair arises in the study of uncertainty property in Fourier Analysis. For example Benedicks's Theorem says if $E$, $\Sigma$ are both sets of finite measures then they form a weakly annihilating pair, whereas Theorem of Amrein and Berthier says they form a strongly annihilating pair.

I am looking for examples of

~~1) A weakly annihilating pair which is $\underline{not}$ a strongly annihilating pair.~~ ( Willie Wong has already answered this and I realise this was rather easy and I should have been able to figure it out myself, so my apologies.)

1') Sets $E$, $\Sigma$ both have infinite measure such that $(E,\Sigma)$ is a strongly annihilating pair.

2) Sets $E$, $\Sigma$ such that $E^c$ and $\Sigma^c$ have nonzero measure and $(E,\Sigma)$ is $\underline{not}$ a weakly annihilating pair.

I realise the standard reference for this topic is the book by Havin and Jöricke, which unfortunately our library does not have a copy of!! Is there any alternative reference someone can suggest ?

Thankyou.

Answer by Vagabond for Set of real numbers with positive measure containing no midpoints

2010-07-16T10:43:30Z

No, such a set cannot exist and one can prove this using Lebesgue Density Theorem and a simple pegionhole argument. Infact all points $x$ which are density points of $E$ will be a midpoint for some $y,z \in E$ i.e., $x=\frac{y+z}{2}$.

Let $F \subseteq E$ be the set of density points of E, and $x \in F$.

Then there exists a $\epsilon > 0$ such that $m( B_{\epsilon}(x)\cap F) > \epsilon$. Now if $x$ is not a midpoint of $E$ then $\forall d \in (0,\epsilon)$, atleast one of $x-d$ or $x+d$ does not belong to $F$.

But then $m( B_{\epsilon}(x)\cap F)= \int_0^{\epsilon} |F\cap \lbrace x-t,x+t\rbrace| dt < \epsilon$, a contradiction !!

A set $A$ of real number is called Universal if every measurable set of positive measure necessarily contains an affine image of $A$. A simple variation of the above argument will give that all finite set $A$ are infact Universal. However, no example of an infinite Universal set is knwon and its a conjecture of Erdos that no infinite universal sets exists.

This paper has a nice discussion and references to this problem

M. Kolountzakis: Infinite Patterns That Can Be Avoided by Measure, Bull. London Math. Soc. 29 (1997), 4, 415-424. http://fourier.math.uoc.gr/~mk/ps/universal.pdf

As Gerry and Benoît Kloeckner has mentioned the problem becomes interesting when one considers Hausdroff measure instead of Lebesgue measure.

Recently I. Laba and M. Pramanik proved existence of 3 term arithmetic progression even in closed sets which has Hausdroff dimension close to 1, `under the condition that E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions'

I. Laba and M. Pramanik: "Arithmetic progressions in sets of fractional dimension",, Geom. Funct. Anal. 19 (2009), 429-456. http://www.math.ubc.ca/~ilaba/preprints/progressions-may15.pdf

Approximation by exponential polynomials

2010-06-27T09:35:24Z

Let $u(t) = \Sigma_{k=1}^n c_k e^{\lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb C) $ be an exponential polynomial of $\underline{order}$ $n$.

Define $E_n$ to be the collection of all exponential polynomial of order $n$, i.e.,

$$ E_n:= { u : u(t) = \sum_{k=1}^n c_k e^{\lambda_k t}, c_k \in \mathbb C, \lambda_k \in \mathbb C }. $$ Notice, that two elememts of $E_n$ may have different (disjoint) set of exponents. Only requirement for $u$ to be in $E_n$ is that it has order at most $n$.

Let $\mathbf{P}_n$ be the collection of all polynomials of $\underline{degree}$ at most $n$.

Consider a function $f = \sum_{j=1}^{M} p_{m_j}(t) e^{\lambda_j t}, p_{m_j} \in \mathbf{P}_{m_j}, \sum_{j=1}^{M} (m_j+1) \leq n$.

My question is, can one find a sequence $u_m \in E_n$ such that, $$\sup_{x\in[0,1]}| f(x)- u_m(x) |\rightarrow 0. $$

If possible, then how should one go about constructing such a sequence ?

Extension of harmonic function at infinity

2010-06-22T17:36:05Z

Can a harmonic function defined on the upper half-plain (or any domain which is unbounded) be extended to the point at infinity. If so, under what condition. What happens to the mean value property then ? Do we still get a integral representation of some sort. Please suggest a reference.

Thank you.

some questions about properties of harmonic measure

2010-06-21T05:06:56Z

The original post

The following argument appears in a paper of Nazarov (Lemma 1.2) "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" http://www.math.msu.edu/~fedja/Published/paper.ps where he proves a (weak type) Bernstein Inequality using certain properties of `harmonic measure'.

I am not familiar with harmonic measure (I have checked wiki and got hold of a mammoth book by koosis (logarithmic integral) which has a section about harmonic measure). What I am looking for is the statements of theorems and principles about harmonic measures which are being used in the following argument.

Let $h(\xi)$ be the harmonic measure of the set $\mathbb R$ \ $[-y,y]$ with respect to the upper half-plane and a point $\xi \in \mathbb C_+$.

Let $z_1, z_2, \dots, z_{n_1}$ be such that $ Im (z_j) \leq 0$, and define $\sum_1(z):=\sum_{j=1}^{n} \frac{1}{z-z_j}$.

Define $u(z) := h(-\sum_1(z))$.

The function $u(z)$ is harmonic in $C_+$, $0\leq u(z) \leq 1$, $u(it) \lim_{t\rightarrow + \infty} 0$, and $u(z) \geq 1/2$ if $|\sum_1(z)| \geq y$ (the latter fact follows from the geometric description of the harmonic measure as a ratio to $\pi$ of the angle at which a subset of $\mathbb R$ is seen from the point $\xi$).

Moreover, we have

$$\lim_{t\rightarrow +\infty} \pi t u(it) =\int_{\mathbb R} u(x) dx \ \ (Why?) $$

$$\geq \frac{1}{2} \mu { x \in \mathbb R : |\sum_1(x)|>y }.$$

On the other hand, an easy computation shows that

$\lim_{t\rightarrow \infty} \pi t u(it) =\lim_{t \rightarrow +\infty} \pi t h(\iota n/t + O (1/t^2)) = 2n/y.$

(As you can see by the end of this the author has obtained a weak type Bernstein Inequality).

Thankyou for your time and patience.

Question

The only place I am stuck now is the place which I have highlighted(see above). My guess is it is some version of mean value property of harmonic function. So, one has to extend the harmonic function at infinity then ? It should be possible, as in this case $u(it)->0$ as $t -> \infty$. Is it so ? Can someone suggest a reference ?

Any suggestion?

I would like to know if there are some lecture notes about harmonic measures available which is self contained and fits the category `every analysis student must know'. The wiki article is not very helpful there are many treatise available but I couldnot find an exposition which introduces the concept and its importance.

Afterthought

I now realise the first part of the argument is quite easy, they simply follows from properties of harmonic functions (like maximum modulous principle, composition of harmonic functions etc. ) and the geometric description mentioned within quotes about harmonic measure of the set $\mathbb R \setminus [−y,y].$

For example to check that $u(z)\geq 1/2$ iff $\sum_1(z) \geq y$ just draw a semicircle of radius $y$ centered at $0$ and observe that for any point which is outside it the angle (which is the harmonic measure !!) is more than $\frac{\pi}{2}$.

The geometric description is easy to obtain:- The harmonic measure of an interval $[a,b]$ is simply the harmonic extension of $\chi_[a,b]$ on the upper half plane, so

$\int_a^b P_y(x-t) dt = \frac{1}{\pi} \int_a^b \frac{y}{(t-x)^2+y^2} = \int_a^b \frac{1}{\pi} Im(\frac{1}{t-z})= \frac{1}{\pi} Im (log (\frac{b-z}{a-z}) )$

Using limiting argument one can find harmonic measure of $\chi_{\mathbb R} = 1$ and hence get the geometric description of harmonic measure of $\mathbb R \setminus [-y,y]$.

Salem Inequality

2010-06-15T18:04:07Z

I have come across this inequality in the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" http://www.math.msu.edu/~fedja/Published/paper.ps by Nazarov and he calls it by the name of Salem Inequality (which according to him is well known but I cant find a reference).

If I have understood it correctly the Inequality says that if $p$ is an exponential polynomial whose exponents are well separated, then the average value of square of the modulus of $p$ over a sufficiently large interval dominates the sum of the square of the modulus of its coefficients.

Let $p(t) = \Sigma_{k=1}^n c_k e^{ i \lambda_k t}$, where $ \lambda_1<\lambda_2\dots<\lambda_n \in \mathbb R$ and $\lambda_k$'s satisfies a separation condition i.e., $\lambda_{k+1}-\lambda_k \geq \Delta >0$. Let $I$ be an interval of length bigger than $4\pi / \Delta$, then $$\sum_{k=1}^{n} |c_k|^2 \leq \frac{4}{|I|} \int_I |p(t)|^2 dt. $$ How can one prove this Inequality? This surely would have a lot of appliction (and as he says must be well known !! may be by a different name ?). I would appreciate some references to such inequalities in general. Also I find curious that the length of the interval does not seem to depend on $n$ and depends only on $\Delta$.

estimates of exponential polynomials

2010-06-14T07:32:40Z

Let $ p(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t}$ be an exponential polynomial.

In the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" http://www.math.msu.edu/~fedja/Published/paper.ps Nazarov proves an estimate on the maximum value attained by the polynomial $p$ in an interval $I$, in terms of the maximum of $p$ in a subset $E \subset I$.

To be precise he obtains the following estimate:-

$$ \sup_{t \in I} |p(t)| \leq ( \frac{A \mu(I)}{\mu(E)} )^{n-1} \sup_{t\in E} |p(t)|.$$

At one point he mentions that the result holds true for more general functions of the type $$p(t) = \Sigma_{k=1}^n q_k(t) e^{i \lambda_k t}$$ where $q_k(t)$ are algebraic polynomials of degree d_k$; by an "obvious" approximation argument.

It is not clear to me, what exactly is the argument he is suggesting ?

One of the method he uses to obtain an estimate as mentioned above is by using Turan's Lemma. Although in that case one gets exponent $2 n^2$ instead of $n-1$.

Let $p(t)=\Sigma_{k=1}^n c_k e^{i \lambda_k t}$ and assume that the value of the polynomial $p(t)$ is known for $t_j=t_0+j \delta$ for $ j= m+1,...,m+n$. Then substitute $b_k=c_k e^{i \lambda_j t_0}$ and $z_k= e^{i \lambda_k \delta}$ and apply Turan's lemma.

The result now follows by Lebesgue's density theorem and some averaging argument.

We might want to get a similar result like Turan's Lemma for the more general type of exponential polynomial $p(t) = \Sigma_{k=1}^n q_k(t) e^{i \lambda_k t}$ where $q_k(t)$ are algebraic polynomials of degree d_k$.

But I doubt this is what he is suggesting here, as later in order to get the sharper result (the one with exponent $n-1$) he uses some weak type estimates it seems. (I have not read this part of the proof yet).

So, what exactly is the obvious approximation argument he is trying to suggest here?

bound for binomial coefficients

2010-06-12T07:17:14Z

How can one show $\displaystyle\frac{(m+n-1)!}{m ! (n-1)!}\leq \left[\frac{e (m+n-1)}{n}\right]^{n-1}$ ?

Comment by Vagabond

2012-12-02T08:54:58Z

All I wanted to ask was has anyone studied harmonic analysis on Z when it has density measure or whatever would be appropriate. Now it seems like a Catch 22 situation !! You can only ask if you know the answer and not otherwise ? Strange.

Comment by Vagabond

2012-12-01T17:32:03Z

Todd Trimble, Many thanks.

Comment by Vagabond

2012-12-01T16:36:02Z

Todd Trimble so, instead of countable additivity, if we just have finite additivity ? can this work out some way so as one is able to do some reasonable analysis in this setup ?

Comment by Vagabond

2012-12-01T16:32:30Z

true, but is there a way to bypass it, say if we only consider subsets having infinite number of elements ? or do not require the countable additivity for $M$ ?

Comment by Vagabond

2012-02-08T12:24:16Z

I have changed the question as per S. Carnahan's suggestion.

Comment by Vagabond

2012-02-08T10:20:49Z

Thankyou, Prof. Carnahan for your suggestion. I will try to modify my question as per your suggestion.

Comment by Vagabond

2012-02-05T22:02:36Z

So, if the grid size goes to zero does the solution ``converges'' to the actual solution ?

Comment by Vagabond

2012-02-05T21:59:51Z

Actually, what I am looking for is discrete models of a problem which converges in the limit to the continuous problem.

Comment by Vagabond

2012-02-05T06:05:09Z

see equation 10 page 6 of the above mentioned paper.

Comment by Vagabond

2012-02-05T06:02:31Z

I believe he is referring to the recent result of Alex Iosevich, Mihail N. Kolountzakis ``Periodicity of the spectrum in dimension one'' arxiv.org/abs/1108.5689 where they prove such a result. In the context of that paper `spectral gap' would mean the existence of a > 0 such that supp (\widehat{\delta_\Lambda}) \cap (0,a) = \phi .

Comment by Vagabond

2011-12-06T08:39:14Z

Try applying pigeonhole principle you should be able to get what you are looking for. Its also called Dirichlet's Box Principle.

Comment by Vagabond

2010-11-02T10:28:44Z

You can see it in books.google.co.in/…

Comment by Vagabond

2010-11-02T09:18:18Z

Also see. Rectangles as sums of squares. Walters, Mark Discrete Math. 309 (2009), no. 9, 2913–2921.ams.org/leavingmsn?url=http://dx.doi.org/10.1016/… Where precisely this question is discussed and is ascribed to Laczkovich (i.e., the minimum number of squares needed to tile an integer sided rectangle ( the squares are not assumed to be integer sided though)). Of course you are asking for an efficient algorithm which is a different question ... Kenyon paper does discuss some algorithm(greedy) though.

Comment by Vagabond

2010-11-02T09:03:10Z

Tiling a rectangle with the fewest squares. Kenyon, Richard J. Combin. Theory Ser. A 76 (1996), no. 2, 272–291. ams.org/leavingmsn?url=http://dx.doi.org/10.1006/… "We show that a square-tiling of a p×q rectangle, where p and q are relatively prime integers, has at least log 2p squares. If q>p we construct a square-tiling with less than q/p+Clogp squares of integer size, for some universal constant C.''

Comment by Vagabond

2010-11-02T07:31:00Z

(cntd) which are polytopes with symmetries acting transitively on their facets (fair by symmetry). The authors characterize a second type of fair dice (fair by continuity). The question of further fair dice is still open.