Expand all Collapse all | Results 1 - 20 of 20 |
1. CMB Online first
Note on the Weierstrass Theorem in Quasianalytic Local Rings Consider quasianalytic local rings of germs of smooth functions closed
under composition, implicit equation, and monomial division. We show
that if the Weierstrass Preparation Theorem holds in such a ring then
all elements of it are germs of analytic functions.
Categories:26E10, 26E05, 14P15 |
2. CMB 2012 (vol 57 pp. 178)
Quasiconvexity and Density Topology We prove that if $f:\mathbb{R}^{N}\rightarrow \overline{\mathbb{R}}$ is
quasiconvex and $U\subset \mathbb{R}^{N}$ is open in the density topology, then
$\sup_{U}f=\operatorname{ess\,sup}_{U}f,$ while
$\inf_{U}f=\operatorname{ess\,inf}_{U}f$
if and only if the equality holds when $U=\mathbb{R}^{N}.$ The first (second)
property is typical of lsc (usc) functions and, even when $U$ is an ordinary
open subset, there seems to be no record that they both hold for all
quasiconvex functions.
This property ensures that the pointwise extrema of $f$ on any nonempty
density open subset can be arbitrarily closely approximated by values of $f$
achieved on ``large'' subsets, which may be of relevance in a variety of
issues. To support this claim, we use it to characterize the common points
of continuity, or approximate continuity, of two quasiconvex functions that
coincide away from a set of measure zero.
Keywords:density topology, quasiconvex function, approximate continuity, point of continuity Categories:52A41, 26B05 |
3. CMB 2011 (vol 55 pp. 355)
Convolution Inequalities in $l_p$ Weighted Spaces Various weighted $l_p$-norm inequalities in convolutions are derived
by a simple and general principle whose $l_2$ version was obtained by
using the theory of reproducing kernels. Applications to the Riemann zeta
function and a difference equation are also considered.
Keywords:inequalities for sums, convolution Categories:26D15, 44A35 |
4. CMB 2011 (vol 54 pp. 706)
Nonconstant Continuous Functions whose Tangential Derivative Vanishes along a Smooth Curve We provide a simple example showing that the tangential derivative of a
continuous function $\phi$
can vanish everywhere along a curve while the variation of $\phi$ along
this curve is nonzero. We give additional regularity conditions on the curve
and/or the function that prevent this from happening.
Categories:26A24, 28A15 |
5. CMB 2011 (vol 54 pp. 630)
Mixed Norm Type Hardy Inequalities Higher dimensional mixed norm type
inequalities involving certain integral operators are
characterized in terms of the corresponding lower dimensional
inequalities.
Keywords:Hardy inequality, reverse Hardy inequality, mixed norm, Hardy-Steklov operator Categories:26D10, 26D15 |
6. CMB 2010 (vol 54 pp. 12)
Homotopy and the Kestelman-Borwein-Ditor Theorem
The Kestelman--Borwein--Ditor Theorem, on embedding a null sequence by
translation in (measure/category) ``large'' sets has two generalizations.
Miller replaces the translated sequence by a ``sequence homotopic
to the identity''. The authors, in a previous paper, replace points by functions:
a uniform functional null sequence replaces the null sequence, and
translation receives a functional form. We give a unified approach to
results of this kind. In particular, we show that (i) Miller's homotopy
version follows from the functional version, and (ii) the pointwise instance
of the functional version follows from Miller's homotopy version.
Keywords:measure, category, measure-category duality, differentiable homotopy Category:26A03 |
7. CMB 2010 (vol 53 pp. 327)
Multidimensional Exponential Inequalities with Weights We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality $$ \Bigl(\int_E \Phi(T_k f(x))^q u(x)\,dx\Bigr)^{1/q} \le C \Bigl (\int_E \Phi(f(x))^p v(x)\,dx\Bigr )^{1/p}, $$
where 0<$p$, $q$<$\infty$, $\Phi$ is a logarithmically convex function, and $T_k$ is an integral operator over star-shaped regions. The condition is also necessary for the exponential integral inequality. Moreover, the estimation of $C$ is given and we apply the obtained results to generalize some multidimensional Levin--Cochran-Lee type inequalities.
Keywords:multidimensional inequalities, geometric mean operators, exponential inequalities, star-shaped regions Categories:26D15, 26D10 |
8. CMB 2009 (vol 52 pp. 295)
On Functions Whose Graph is a Hamel Basis, II We say that a function $h \from \real \to \real$ is a Hamel function
($h \in \ham$) if $h$, considered as a subset of $\real^2$, is a Hamel
basis for $\real^2$. We show that $\A(\ham)\geq\omega$, \emph{i.e.,} for
every finite $F \subseteq \real^\real$ there exists $f\in\real^\real$
such that $f+F \subseteq \ham$. From the previous work of the author
it then follows that $\A(\ham)=\omega$.
Keywords:Hamel basis, additive, Hamel functions Categories:26A21, 54C40, 15A03, 54C30 |
9. CMB 2006 (vol 49 pp. 438)
Unimodular Roots of\\ Special Littlewood Polynomials We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1}$ a Littlewood
polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ self-reciprocal
if $\alpha(z) = z^{n-1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if
$n = 2m+1$ and $a_{m+j} = (-1)^j a_{m-j}$ for all $j$. It has been observed
that Littlewood polynomials with particularly high minimum modulus on
the unit
circle in $\bC$ tend to be skewsymmetric. In this paper, we prove that a
skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle,
as well as providing a new proof of the known result that a self-reciprocal
Littlewood polynomial must have a zero on the unit circle.
Categories:26C10, 30C15, 42A05 |
10. CMB 2006 (vol 49 pp. 256)
A Bernstein--Walsh Type Inequality and Applications A Bernstein--Walsh type inequality for $C^{\infty }$ functions of several
variables is derived, which then is applied to obtain analogs and
generalizations of the following classical theorems: (1) Bochnak--Siciak
theorem: a $C^{\infty }$\ function on $\mathbb{R}^{n}$ that is real
analytic on every line is real analytic; (2) Zorn--Lelong theorem: if a
double power series $F(x,y)$\ converges on a set of lines of positive
capacity then $F(x,y)$\ is convergent; (3) Abhyankar--Moh--Sathaye theorem:
the transfinite diameter of the convergence set of a divergent series is
zero.
Keywords:Bernstein-Walsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power series Categories:32A05, 26E05 |
11. CMB 2006 (vol 49 pp. 82)
Embeddings and Duality Theorem for Weak Classical Lorentz Spaces We characterize the weight functions
$u,v,w$ on $(0,\infty)$ such that
$$
\left(\int_0^\infty f^{*}(t)^
qw(t)\,dt\right)^{1/q}
\leq
C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t),
$$
where
$$
f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{-1}
\int_{0}^{t}f^*(s)u(s)\,ds.
$$
As an application we present a~new simple characterization of
the associate space to the space $\Gamma^ \infty(v)$, determined by the
norm
$$
\|f\|_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t),
$$
where
$$
f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds.
$$
Keywords:Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequality Categories:26D10, 46E20 |
12. CMB 2005 (vol 48 pp. 333)
Monotonicity Properties of the Hurwitz Zeta Function Let
$$
\zeta(s,x)=\sum_{n=0}^{\infty}\frac{1}{(n+x)^s} \quad{(s>1,\, x>0)}
$$
be the Hurwitz zeta function and let
$$
Q(x)=Q(x;\alpha,\beta;a,b)=\frac{(\zeta(\alpha,x))^a}{(\zeta(\beta,x))^b},
$$
where $\alpha, \beta>1$
and $a,b>0$ are real numbers. We prove:
(i) The function $Q$ is decreasing on $(0,\infty)$ if{}f $\alpha a-\beta b\geq \max(a-b,0)$.
(ii) $Q$ is increasing on $(0,\infty)$ if{}f $\alpha a-\beta b\leq
\min(a-b,0)$.
An application of part (i) reveals that for all $x>0$ the function $s\mapsto [(s-1)\zeta(s,x)]^{1/(s-1)}$ is decreasing on $(1,\infty)$. This settles
a conjecture of Bastien and Rogalski.
Categories:11M35, 26D15 |
13. CMB 2005 (vol 48 pp. 133)
Estimates of Henstock-Kurzweil Poisson Integrals If $f$ is a real-valued function on $[-\pi,\pi]$ that
is Henstock-Kurzweil integrable, let $u_r(\theta)$ be its Poisson
integral. It is shown that $\|u_r\|_p=o(1/(1-r))$ as $r\to 1$
and this estimate is sharp for $1\leq p\leq\infty$.
If $\mu$ is a finite Borel measure and $u_r(\theta)$ is its Poisson
integral then for each $1\leq p\leq \infty$ the estimate
$\|u_r\|_p=O((1-r)^{1/p-1})$ as $r\to 1$ is sharp.
The Alexiewicz
norm estimates $\|u_r\|\leq\|f\|$ ($0\leq r<1$) and $\|u_r-f\|\to 0$
($r\to 1$) hold. These estimates lead to two uniqueness theorems for
the Dirichlet problem
in the unit disc with Henstock-Kurzweil integrable boundary data.
There are similar growth estimates when $u$ is in the harmonic Hardy
space associated with the Alexiewicz
norm and when $f$ is of bounded variation.
Categories:26A39, 31A20 |
14. CMB 2004 (vol 47 pp. 540)
Compactness of Hardy-Type Operators over Star-Shaped Regions in $\mathbb{R}^N$ We study a compactness property of the operators between weighted
Lebesgue spaces that average a function over certain domains involving
a star-shaped region. The cases covered are (i) when the average is
taken over a difference of two dilations of a star-shaped region in
$\RR^N$, and (ii) when the average is taken over all dilations of
star-shaped regions in $\RR^N$. These cases include, respectively,
the average over annuli and the average over balls centered at origin.
Keywords:Hardy operator, Hardy-Steklov operator, compactness, boundedness, star-shaped regions Categories:46E35, 26D10 |
15. CMB 2003 (vol 46 pp. 323)
Characterizing Two-Dimensional Maps Whose Jacobians Have Constant Eigenvalues Recent papers have shown that $C^1$ maps $F\colon \mathbb{R}^2
\rightarrow \mathbb{R}^2$
whose Jacobians have constant eigenvalues can be completely
characterized if either the eigenvalues are equal or $F$ is a
polynomial. Specifically, $F=(u,v)$ must take the form
\begin{gather*}
u = ax + by + \beta \phi(\alpha x + \beta y) + e \\
v = cx + dy - \alpha \phi(\alpha x + \beta y) + f
\end{gather*}
for some constants $a$, $b$, $c$, $d$, $e$, $f$, $\alpha$, $\beta$ and
a $C^1$ function $\phi$ in one variable. If, in addition, the function
$\phi$ is not affine, then
\begin{equation}
\alpha\beta (d-a) + b\alpha^2 - c\beta^2 = 0.
\end{equation}
This paper shows how these theorems cannot be extended by constructing
a real-analytic map whose Jacobian eigenvalues are $\pm 1/2$ and does
not fit the previous form. This example is also used to construct
non-obvious solutions to nonlinear PDEs, including the Monge--Amp\`ere
equation.
Keywords:Jacobian Conjecture, injectivity, Monge--AmpÃ¨re equation Categories:26B10, 14R15, 35L70 |
16. CMB 2001 (vol 44 pp. 61)
The Inequalities for Polynomials and Integration over Fractal Arcs The paper is dealing with determination of the integral $\int_{\gamma}
f \,dz$ along the fractal arc $\gamma$ on the complex plane by terms
of polynomial approximations of the function~$f$. We obtain
inequalities for polynomials and conditions of integrability for
functions from the H\"older, Besov and Slobodetskii spaces.
Categories:26B15, 28A80 |
17. CMB 1999 (vol 42 pp. 478)
A Remark On the Moser-Aubin Inequality For Axially Symmetric Functions On the Sphere Let $\scr S_r$ be the collection of all axially symmetric functions
$f$ in the Sobolev space $H^1(\Sph^2)$ such that $\int_{\Sph^2}
x_ie^{2f(\mathbf{x})} \, d\omega(\mathbf{x})$ vanishes for $i=1,2,3$.
We prove that
$$
\inf_{f\in \scr S_r} \frac12 \int_{\Sph^2} |\nabla f|^2 \, d\omega
+ 2\int_{\Sph^2} f \, d\omega- \log \int_{\Sph^2} e^{2f} \, d\omega > -\oo,
$$
and that this infimum is attained. This complements recent work of
Feldman, Froese, Ghoussoub and Gui on a conjecture of Chang and Yang
concerning the Moser-Aubin inequality.
Keywords:Moser inequality, borderline Sobolev inequalities, axially symmetric functions Categories:26D15, 58G30 |
18. CMB 1999 (vol 42 pp. 417)
Some Properties of Rational Functions with Prescribed Poles Let $P(z)$ be a polynomial of degree not exceeding $n$ and let
$W(z) = \prod^n_{j=1}(z-a_j)$ where $|a_j|> 1$, $j =1,2,\dots, n$.
If the rational function $r(z) = P(z)/W(z)$ does not vanish in $|z|
< k$, then for $k=1$ it is known that
$$
|r'(z)| \leq \frac{1}{2} |B'(z)| \Sup_{|z|=1} |r(z)|
$$
where $B(Z) = W^\ast(z)/W(z)$ and $W^\ast (z) = z^n \overline
{W(1/\bar z)}$. In the paper we consider the case when $k>1$ and
obtain a sharp result. We also show that
$$
\Sup_{|z|=1} \biggl\{ \biggl| \frac{r'(z)}{B'(z)} \biggr| +\biggr|
\frac{\bigl(r^\ast (z)\bigr)'}{B'(z)} \biggr| \biggr\} =
\Sup_{|z|=1} |r(z)|
$$
where $r^\ast (z) = B(z) \overline{r(1/\bar z)}$, and as a consequence of
this result, we present a generalization of a theorem of O'Hara and
Rodriguez for self-inversive polynomials. Finally, we establish a similar
result when supremum is replaced by infimum for a rational function which
has all its zeros in the unit circle.
Category:26D07 |
19. CMB 1998 (vol 41 pp. 497)
On the construction of HÃ¶lder and Proximal Subderivatives We construct Lipschitz functions such that for all $s>0$ they are
$s$-H\"older, and so proximally, subdifferentiable only on dyadic
rationals and nowhere else. As applications we construct Lipschitz
functions with prescribed H\"older and approximate subderivatives.
Keywords:Lipschitz functions, HÃ¶lder subdifferential, proximal subdifferential, approximate subdifferential, symmetric subdifferential, HÃ¶lder smooth, dyadic rationals Categories:49J52, 26A16, 26A24 |
20. CMB 1997 (vol 40 pp. 88)
The multidirectional mean value theorem in Banach spaces Recently, F.~H.~Clarke and Y.~Ledyaev established a
multidirectional mean value theorem applicable to lower
semi-continuous functions on Hilbert spaces, a result which
turns out to be useful in many applications. We develop a
variant of the result applicable to locally Lipschitz functions
on certain Banach spaces, namely those that admit a
${\cal C}^1$-Lipschitz continuous bump function.
Categories:26B05, 49J52 |