WAVELET PACKETS ASSOCIATED WITH NONUNIFORM MULTIRESOLUTION ANALYSIS ON POSITIVE HALF LINE

Gabardo and Nashed have studied nonuniform multiresolution analysis based on the theory of spectral pairs in a series of papers, see Refs. 4 and 5. Farkov,3 has extended the notion of multiresolution analysis on locally compact Abelian groups and constructed the compactly supported orthogonal p-wavelets on L2(ℝ+). We have considered the nonuniform multiresolution analysis on positive half-line. The associated subspace V0 of L2(ℝ+) has an orthonormal basis, a collection of translates of the scaling function φ of the form {φ(x ⊖ λ)}λ∈Λ+ where Λ+ = {0, r/N} + ℤ+, N > 1 (an integer) and r is an odd integer with 1 ≤ r ≤ 2N - 1 such that r and N are relatively prime and ℤ+ is the set of non-negative integers. We find the necessary and sufficient condition for the existence of associated wavelets and derive the analogue of Cohen's condition for the nonuniform multiresolution analysis on the positive half-line.

Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal.158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ , and the translation factor is $2\textrm{N}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$ . Next we introduce $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$ -NUMRA.

In this article, we introduce the notion of biorthgonoal nonuniform multiresolution analysis on the spectrum , where N ≥ 1 is an integer and r is an odd integer with 1 ≤ r ≤ 2N − 1 such that r and N are relatively prime. We first establish the necessary and sufficient conditions for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families. Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases.

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

The main purpose of this paper is to provide a characterization of scaling functions for non-uniform multiresolution analysis (NUMRA, in short). Some necessary and sufficient conditions for scaling functions of wavelet NUMRA in the frequency domain are also obtained.

We construct a class of generalized nonlinear coherent states by means of a newly obtained class of 2D complex orthogonal polynomials. The associated Bargmann-type transform is discussed. A polynomials realization of the basis of the quantum states Hilbert space is also obtained. Here, the entire structure owes its existence to a certain measure on the positive real half line, of finite total mass, together with all its moments. We illustrate this method with the measure $$r^\beta e^{-r}dr$$, where $$\beta $$ is a non-negative constant, which leads to a new generalization of the true-polyanalytic Bargmann transform.

Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators

Recently, the author determined the higher-order differential operator having the Jacobi-Sobolev polynomials as its eigenfunctions for certain eigenvalues. These polynomials form an orthogonal system with respect to an inner product equipped with the Jacobi measure on the interval [ − 1 , 1 ] with parameters α ∈ N 0 , β > − 1 and two point masses N, S>0 at the right end point of the interval involving functions and their first derivatives. The first purpose of the present paper is to reveal how the Jacobi-Sobolev equation reduces to the differential equation satisfied by the Laguerre-Sobolev polynomials on the positive half line via a confluent limiting process as β → ∞ . Secondly, we explicitly establish the differential equation for the symmetric Gegenbauer-Sobolev polynomials by employing a quadratic transformation of the argument. Each of the three differential operators involved is of order 4 α + 10 and symmetric with respect to the corresponding Sobolev inner product.

In this paper, we establish sufficient conditions for the existence of positive solutions for a two-point boundary value problem of a third order differential equation at resonance on the positive half line. Our results are based on the Leggett-Williams norm-type theorem of coincidence. As an application, an example is given to demonstrate our results.

The Krein condition has been used as a qualitative result to show the M-indeterminacy of some kind of densities.In this work we use results from the theory of the Hilbert transform to construct families of densities having all the same finite moment sequence as a density f with finite logarithmic integral.Actually, our approach explicitly gives Stieltjes classes with center at f and perturbations involving the Hilbert transform of ln f .We consider densities supported on the whole real line or the positive half line.

In this chapter, we discuss the basic properties of Laplace functionals of random measures, which provide an important tool in the study of measure-valued processes. In particular, we give some characterizations of the convergence of random measures in terms of their Laplace functionals. Based on these results, a general representation for the distributions of infinitely divisible random measures is established. We also give some characterizations of continuous functions on the positive half line with Levy–Khintchine type representations.

The double gamma function and related Pick functions

Given a smooth real function f on the positive half line consider the induced map \(\) on the set of positive Hilbert space operators. Let f(k be the kth derivative of the real function f and >E5>Dkf the kth Frechet derivative of the operator map f. We identify large classes of functions for which \(\), for k= 1,2,... . This reduction of a noncommutative problem to a commutative one makes it easy to obtain perturbation bounds for several operator maps. Our techniques serve to illustrate the use of a formalism for “quantum analysis” that is like the one recently developed by M. Suzuki.

In the 21st century researchers redefine the role of universities in the life and development of society. Activities in this area have become so important that they are considered as a special mission along with the traditional educational and research missions. At the same time, no consensus has been formed yet on which areas of university activities should be in the centre of the third mission. Experts from Western universities put emphasis on the commercialization of the products of educational and especially research missions in the interests of the society. Our study examines the vision of the third mission at St. Petersburg State University, as well as at universities in the Republic of Korea. The material for the study make the speeches of the participants of the 9th Forum of Rectors of the leading Universities of Russia and the Republic of Korea as well as a wide range of information on this topic on the St. Petersburg University portal. The analysis showed the similarity of the approaches of St. Petersburg State University and Korean universities to interaction with society and industry. St. Petersburg State University meets the society’s expectations about education and development. Representatives of employers are involved in the development of specialized programs and evaluation of learning outcomes. Internships in research institutes and enterprises provide students with employment and future career growth. University pays a special attention to creating comfortable working conditions at the university, in which students and teachers are part of the local society. In addition, access to the university’s cultural and museum resources is open to all citizens.

The study of uranium-bearing natural and synthetic compounds has been one of the primary research focuses at the Department of Crystallography, St. Petersburg State University, for the past 25 years. During this period, the department's researchers have discovered three new uranium minerals, determined and refined the structures of 15 known mineral species, and synthesized and characterized over 370 new uranium compounds. These efforts have resulted in the publication of more than 200 scientific works, including three monographs and over 190 articles in leading international journals in the fields of mineralogy, crystallography, and chemistry. This review highlights the most significant and intriguing results, in the authors' view, from crystal-chemical studies of uranium compounds conducted by the Department of Crystallography at St. Petersburg State University over the past quarter-century.