Expand all Collapse all | Results 1 - 6 of 6 |
1. CMB 2011 (vol 56 pp. 194)
On the Smallest and Largest Zeros of MÃ¼ntz-Legendre Polynomials MÃ¼ntz-Legendre
polynomials $L_n(\Lambda;x)$ associated with a
sequence $\Lambda=\{\lambda_k\}$ are obtained by orthogonalizing the
system $(x^{\lambda_0}, x^{\lambda_1}, x^{\lambda_2}, \dots)$ in
$L_2[0,1]$ with respect to the Legendre weight. If the $\lambda_k$'s
are distinct, it is well known that $L_n(\Lambda;x)$ has exactly $n$
zeros $l_{n,n}\lt l_{n-1,n}\lt \cdots \lt l_{2,n}\lt l_{1,n}$ on $(0,1)$.
First we prove the following global bound for the smallest zero,
$$
\exp\biggl(-4\sum_{j=0}^n \frac{1}{2\lambda_j+1}\biggr) \lt l_{n,n}.
$$
An important consequence is that if the associated MÃ¼ntz space is
non-dense in $L_2[0,1]$, then
$$
\inf_{n}x_{n,n}\geq
\exp\biggl({-4\sum_{j=0}^{\infty} \frac{1}{2\lambda_j+1}}\biggr)\gt 0,
$$
so
the elements $L_n(\Lambda;x)$ have no zeros close to 0.
Furthermore, we determine the asymptotic behavior of the largest zeros; for $k$ fixed,
$$
\lim_{n\rightarrow\infty} \vert \log l_{k,n}\vert \sum_{j=0}^n
(2\lambda_j+1)= \Bigl(\frac{j_k}{2}\Bigr)^2,
$$
where $j_k$ denotes the $k$-th zero of the Bessel function $J_0$.
Keywords:MÃ¼ntz polynomials, MÃ¼ntz-Legendre polynomials Categories:42C05, 42C99, 41A60, 30B50 |
2. CMB 2011 (vol 55 pp. 424)
Convergence Rates of Cascade Algorithms with Infinitely Supported Masks We investigate the solutions of refinement equations of the form
$$
\phi(x)=\sum_{\alpha\in\mathbb
Z^s}a(\alpha)\:\phi(Mx-\alpha),
$$ where the function $\phi$
is in $L_p(\mathbb R^s)$$(1\le p\le\infty)$, $a$ is an infinitely
supported sequence on $\mathbb Z^s$ called a refinement mask, and
$M$ is an $s\times s$ integer matrix such that
$\lim_{n\to\infty}M^{-n}=0$. Associated with the mask $a$ and $M$ is
a linear operator $Q_{a,M}$ defined on $L_p(\mathbb R^s)$ by
$Q_{a,M} \phi_0:=\sum_{\alpha\in\mathbb
Z^s}a(\alpha)\phi_0(M\cdot-\alpha)$. Main results of this paper are
related to the convergence rates of $(Q_{a,M}^n
\phi_0)_{n=1,2,\dots}$ in $L_p(\mathbb R^s)$ with mask $a$ being
infinitely supported. It is proved that under some appropriate
conditions on the initial function $\phi_0$, $Q_{a,M}^n \phi_0$
converges in $L_p(\mathbb R^s)$ with an exponential rate.
Keywords:refinement equations, infinitely supported mask, cascade algorithms, rates of convergence Categories:39B12, 41A25, 42C40 |
3. CMB 2009 (vol 52 pp. 627)
On $L^{1}$-Convergence of Fourier Series under the MVBV Condition Let $f\in L_{2\pi }$ be a real-valued even function with its Fourier series $%
\frac{a_{0}}{2}+\sum_{n=1}^{\infty }a_{n}\cos nx,$ and let
$S_{n}(f,x) ,\;n\geq 1,$ be the $n$-th partial sum of the Fourier series. It
is well known that if the nonnegative sequence $\{a_{n}\}$ is decreasing and
$\lim_{n\rightarrow \infty }a_{n}=0$, then%
\begin{equation*}
\lim_{n\rightarrow \infty }\Vert f-S_{n}(f)\Vert _{L}=0
\text{ if
and only if }\lim_{n\rightarrow \infty }a_{n}\log n=0.
\end{equation*}%
We weaken the monotone condition in this classical result to the so-called
mean value bounded variation (MVBV) condition. The generalization of the
above classical result in real-valued function space is presented as a
special case of the main result in this paper, which gives the $L^{1}$%
-convergence of a function $f\in L_{2\pi }$ in complex space. We also give
results on $L^{1}$-approximation of a function $f\in L_{2\pi }$ under the
MVBV condition.
Keywords:complex trigonometric series, $L^{1}$ convergence, monotonicity, mean value bounded variation Categories:42A25, 41A50 |
4. CMB 2008 (vol 51 pp. 561)
Expansion of the Riemann $\Xi$ Function in Meixner--Pollaczek Polynomials In this article we study in detail the expansion of the Riemann
$\Xi$ function in Meixner--Pollaczek polynomials. We obtain explicit
formulas, recurrence relation and asymptotic expansion for the
coefficients and investigate the zeros of the partial sums.
Categories:41A10, 11M26, 33C45 |
5. CMB 2008 (vol 51 pp. 236)
6. CMB 2007 (vol 50 pp. 434)
MKZ Type Operators Providing a Better Estimation on $[1/2,1)$ In the present paper, we introduce a modification of the Meyer-K\"{o}nig and
Zeller (MKZ) operators which preserve the test functions $f_{0}(x)=1$ and
$f_{2}(x)=x^{2}$, and we show that this modification provides a better estimation
than the classical MKZ operators on the interval $[\frac{1}{2},1)$ with
respect to the modulus of continuity and the Lipschitz class functionals.
Furthermore, we present the $r-$th order generalization of our operators and
study their approximation properties.
Keywords:Meyer-KÃ¶nig and Zeller operators, Korovkin type approximation theorem, modulus of continuity, Lipschitz class functionals Categories:41A25, 41A36 |