Narrow Results

Let U be either the classical or quantized universal enveloping algebra of the Lie algebra extended over the field of fractions of the Cartan subalgebra. We suggest a PBW basis in U over the extended Cartan subalgebra diagonalizing the contravariant Shapovalov form on generic Verma module. The matrix coefficients of the form are calculated and the inverse form is explicitly constructed.

Let the self-similar measure ${\mu }_{M,D}$ be generated by an expanding real matrix $M={\rho }^{-1}I\in M_2(ℝ)$ and a digit set $D=\{{(0,0)}^t,{(1,0)}^t,{(0,1)}^t,{(-1,-1)}^t\}$ in space ${ℝ}^2$. In this paper, we only consider $\rho >0$ and the case $\rho <0$ is similar. We show that there exists an infinite orthogonal set of exponential functions in $L^2({\mu }_{M,D})$ if and only if $\rho ={(q/(2p))}^{\frac{1}{r}}$ for some $p,q,r\in \mathbb{N}$ with $gcd(2p,q)=1$. Furthermore, for the cases that $L^2({\mu }_{M,D})$ does not admit any infinite orthogonal set of exponential functions, the exact cardinality of orthogonal exponential functions in $L^2({\mu }_{M,D})$ is given.

In this paper, we discuss the existence of o-basis of the symmetry classes of polynomials associated to the irreducible characters of the semidihedral groups.

The offset linear canonical transform (OLCT) has proven to be a novel and effective method in signal processing and optics. Many important properties and results of the OLCT have been well studied and published. In this work, the sampling theorem of the OLCT bandlimited signals based on reproducing kernel Hilbert space has been proposed. First, we show that the bandlimited signals in the OLCT domain form a reproducing kernel Hilbert space. Then, an orthogonal basis for the OLCT bandlimited signals has been obtained based on the reproducing kernel Hilbert space. By using the orthogonal basis, the uniform sampling theory for bandlimited signals associated with the OLCT has been obtained. Furthermore, the nonuniform sampling of the OLCT bandlimited signals also has been attained. Finally, the simulations are provided to prove the usefulness and correctness of the derived results.

We show that data-adaptive orthogonal wavelet bases can be obtained from a generalization of principal component analysis. These bases are not self-similar in general. They are optimal for data-compression and denoising.

A novel method for identification of non-linear systems is proposed to solve the problem of the estimating the unknown parameters of non-linear systems. For Haar wavelet is a set of complete, orthogonal basis and is easy to computation, so it has significant advantage in system identification and also has been applied to many other fields. Therefore, we as well utilize Haar wavelet to analyses all time functions and estimate the unknown parameters. Finally, simulation results with or without noise respectively verify the effectiveness of our method.