218 research outputs found
Order of uniform approximation by polynomial interpolation in the complex plane and beyond
For Lagrange polynomial interpolation on open arcs in \CC, it is
well-known that the Lebesgue constant for the family of Chebyshev points
on [-1,1]\subset \RR has growth order of
. The same growth order was shown in \cite{ZZ} for the Lebesgue
constant of the family of some
properly adjusted Fej\'er points on a rectifiable smooth open arc
\gamma\subset \CC. On the other hand, in our recent work \cite{CZ2021}, it
was observed that if the smooth open arc is replaced by an -shape
arc \gamma_0 \subset \CC consisting of two line segments, numerical
experiments suggest that the Marcinkiewicz-Zygmund inequalities are no longer
valid for the family of Fej\'er points on , and that the rate of growth
for the corresponding Lebesgue constant is as fast as
for some constant .
The main objective of the present paper is 3-fold: firstly, it will be shown
that for the special case of the -shape arc consisting of two
line segments of the same length that meet at the angle of , the growth
rate of the Lebesgue constant is at least as fast as
, with ;
secondly, the corresponding (modified) Marcinkiewicz-Zygmund inequalities fail
to hold; and thirdly, a proper adjustment of the Fej\'er points on will
be described to assure the growth rate of to be exactly
.Comment: Submit to Indagationes Mathematicae, Prof. Jaap Korevaar 100-th
birthday special issue, 32 pages, no figures, keywords:Lebesgue constants;
Marcinkiewicz-Zygmund inequalities; Fejer points; Conformal Mappin
Direct Signal Separation Via Extraction of Local Frequencies with Adaptive Time-Varying Parameters
In nature, real-world phenomena that can be formulated as signals (or in
terms of time series) are often affected by a number of factors and appear as
multi-component modes. The natural approach to understand and process such
phenomena is to decompose, or even better, to separate the multi-component
signals to their basic building blocks (called sub-signals or time-series
components, or fundamental modes). Recently the synchro-squeezing transform
(SST) and its variants have been developed for nonstationary signal separation.
More recently, a direct method of the time-frequency approach, called signal
separation operation (SSO), was introduced for multi-component signal
separation. While both SST and SSO are mathematically rigorous on the
instantaneous frequency (IF) estimation, SSO avoids the second step of the
two-step SST method in signal separation, which depends heavily on the accuracy
of the estimated IFs. In the present paper, we solve the signal separation
problem by constructing an adaptive signal separation operator (ASSO) for more
effective separation of the blind-source multi-component signal, via
introducing a time-varying parameter that adapts to local IFs. A recovery
scheme is also proposed to extract the signal components one by one, and the
time-varying parameter is updated for each component. The proposed method is
suitable for engineering implementation, being capable of separating
complicated signals into their sub-signals and reconstructing the signal trend
directly. Numerical experiments on synthetic and real-world signals are
presented to demonstrate our improvement over the previous attempts
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