Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn alternative wavelet reconstruction formula: a case study of approximate wavelets
158
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The application of the continuous wavelet transform to study of a wide class of physical processes with oscillatory dynamics is restricted by large central frequencies due to the admissibility condition. We propose an alternative reconstruction formula for the continuous wavelet transform, which is applicable even if the admissibility condition is violated. The case of the transform with the standard Morlet wavelet, which is an important example of such analyzing functions, is discussed.
rate research
Read More
A simple proof of Ramanujans formula for the Fourier transform of the square of the modulus of the Gamma function restricted to a vertical line in the right half-plane is given. The result is extended to vertical lines in the left half-plane by solving an inhomogeneous ODE. We then use it to calculate the jump across the imaginary axis.
We compare frameworks of nonstationary nonperiodic wavelets and periodic wavelets. We construct one system from another using periodization. There are infinitely many nonstationary systems corresponding to the same periodic wavelet. Under mild conditions on periodic scaling functions, among these nonstationary wavelet systems, we find a system such that its time-frequency localization is adjusted with an angular-frequency localization of an initial periodic wavelet system. Namely, we get the following equality $ lim_{jto infty} UC_B(psi^P_j) = lim_{jto infty}UC_H(psi^N_j), $ where $UC_B$ and $UC_H$ are the Breitenberger and the Heisenberg uncertainty constants, $psi^P_j in L_2(mathbb{T})$ and $psi^N_jin L_2(mathbb{R})$ are periodic and nonstationary wavelet functions respectively.
In the paper we obtain sufficient conditions for a trigonometric polynomial to be a refinement mask corresponding to a tight wavelet frame. The condition is formulated in terms of the roots of a mask. In particular, it is proved that any trigonometric polynomial can serve as a mask if its associated algebraic polynomial has only negative roots (at least one of them, of course, equals $-1$).
In this note, we look at some of the less explored aspects of the gamma function. We provide a new proof of Eulers reflection formula and discuss its significance in the theory of special functions. We also discuss a result of Landau concerning the determination of values of the gamma function using functional identities. We show that his result is sharp and extend it to complex arguments. In 1848, Oskar Schlomilch gave an interesting additive analogue of the duplication formula. We prove a generalized version of this formula using the theory of hypergeometric functions.
Using the wavelet theory introduced by the author and J. Benedetto, we present examples of wavelets on p-adic fields and other locally compact abelian groups with compact open subgroups. We observe that in this setting, the Haar and Shannon wavelets (which are at opposite extremes over the real numbers) coincide and are localized both in time and in frequency. We also study the behavior of the translation operators required in the theory.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
