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Search: All articles in the CJM digital archive with keyword measures

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1. CJM Online first

Samart, Detchat
Mahler Measures as Linear Combinations of $L$-values of Multiple Modular Forms
We study the Mahler measures of certain families of Laurent polynomials in two and three variables. Each of the known Mahler measure formulas for these families involves $L$-values of at most one newform and/or at most one quadratic character. In this paper, we show, either rigorously or numerically, that the Mahler measures of some polynomials are related to $L$-values of multiple newforms and quadratic characters simultaneously. The results suggest that the number of modular $L$-values appearing in the formulas significantly depends on the shape of the algebraic value of the parameter chosen for each polynomial. As a consequence, we also obtain new formulas relating special values of hypergeometric series evaluated at algebraic numbers to special values of $L$-functions.

Keywords:Mahler measures, Eisenstein-Kronecker series, $L$-functions, hypergeometric series
Categories:11F67, 33C20

2. CJM 2011 (vol 64 pp. 1075)

Raja, Chandiraraj Robinson Edward
A Stochastic Difference Equation with Stationary Noise on Groups
We consider the stochastic difference equation $$\eta _k = \xi _k \phi (\eta _{k-1}), \quad k \in \mathbb Z $$ on a locally compact group $G$ where $\phi$ is an automorphism of $G$, $\xi _k$ are given $G$-valued random variables and $\eta _k$ are unknown $G$-valued random variables. This equation was considered by Tsirelson and Yor on one-dimensional torus. We consider the case when $\xi _k$ have a common law $\mu$ and prove that if $G$ is a distal group and $\phi$ is a distal automorphism of $G$ and if the equation has a solution, then extremal solutions of the equation are in one-one correspondence with points on the coset space $K\backslash G$ for some compact subgroup $K$ of $G$ such that $\mu$ is supported on $Kz= z\phi (K)$ for any $z$ in the support of $\mu$. We also provide a necessary and sufficient condition for the existence of solutions to the equation.

Keywords:dissipating, distal automorphisms, probability measures, pointwise distal groups, shifted convolution powers
Categories:60B15, 60G20

3. CJM 2010 (vol 62 pp. 827)

Ouyang, Caiheng; Xu, Quanhua
BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces
This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.

Keywords:BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spaces
Categories:46E40, 42B25, 46B20

4. CJM 2009 (vol 61 pp. 124)

Dijkstra, Jan J.; Mill, Jan van
Characterizing Complete Erd\H os Space
The space now known as {\em complete Erd\H os space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the closed subspace of the Hilbert space $\ell^2$ consisting of all vectors such that every coordinate is in the convergent sequence $\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of $\cerdos$. As an application we determine the class of factors of $\cerdos$. In another application we determine precisely which of the spaces that can be constructed in the Banach spaces $\ell^p$ according to the `Erd\H os method' are homeomorphic to $\cerdos$. A novel application states that if $I$ is a Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$, $\Z\times2^\omega$, or $\cerdos$. This last result answers a question that was asked by Stevo Todor{\v{c}}evi{\'c}.

Keywords:Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on $\omega$, $\R$-trees, line-free groups in Banach spaces
Categories:28C10, 46B20, 54F65

5. CJM 2006 (vol 58 pp. 401)

Kolountzakis, Mihail N.; Révész, Szilárd Gy.
On Pointwise Estimates of Positive Definite Functions With Given Support
The following problem has been suggested by Paul Tur\' an. Let $\Omega$ be a symmetric convex body in the Euclidean space $\mathbb R^d$ or in the torus $\TT^d$. Then, what is the largest possible value of the integral of positive definite functions that are supported in $\Omega$ and normalized with the value $1$ at the origin? From this, Arestov, Berdysheva and Berens arrived at the analogous pointwise extremal problem for intervals in $\RR$. That is, under the same conditions and normalizations, the supremum of possible function values at $z$ is to be found for any given point $z\in\Omega$. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to $\RR^d$ and to non-convex domains as well. Here we present another approach to the problem, giving the solution in $\RR^d$ and for several cases in~$\TT^d$. Actually, we elaborate on the fact that the problem is essentially one-dimensional and investigate non-convex open domains as well. We show that the extremal problems are equivalent to some more familiar ones concerning trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relationship between the problem for $\RR^d$ and that for $\TT^d$ is given, showing that the former case is just the limiting case of the latter. Thus the hierarchy of difficulty is established, so that extremal problems for trigonometric polynomials gain renewed recognition.

Keywords:Fourier transform, positive definite functions and measures, Turán's extremal problem, convex symmetric domains, positive trigonometric polynomials, dual extremal problems
Categories:42B10, 26D15, 42A82, 42A05

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