Characterization of q-Dunkl-classical symmetric orthogonal q-polynomials

The Ramanujan Journal - In this paper, we show that, up to a dilatation, the $$q^2$$ -analogue of generalized Hermite and $$q^2$$ -analogue of generalized Gegenbauer polynomials are the only...     

On the q-polynomials: a distributional study

In this paper we present a unified distributional study of the classical discrete q-polynomials (in the Hahn's sense). From the distributional q-Pearson equation we will deduce many of their properties such as the three-term recurrence relations, structure relations, etc. Also several characterizations of such q-polynomials are presented.     

Integral and discrete representations of certain Brenke type orthogonal q-polynomials

We investigate the H _q -semiclassical character, duality, and the standard perturbation to characterize a well known non-symmetric MOPS {Z _n (?;b,q)} _n≥0 of Brenke type related to the Wall’s one. We prove that its corresponding regular linear form $\mathcal{Z}(b,q)$ is a perturbation of a Brenke type symmetric regular linear form $\mathcal{Y}(b,q)$ , and so $\mathcal{Z}(b,q)$ is $H_{\sqrt{q}}$ -semiclassical of class two. Consequently, moments, integral, and discrete representations of $\mathcal{Z}(b,q)$ are given.     

On the unconditional almost-everywhere convergence of general orthogonal series

The Orlicz and Tandori theorems on the unconditional almost-everywhere convergence, with respect to Lebesgue measure, of real orthogonal series defined on the interval (0; 1) are extended to general complex orthogonal series defined on a space with arbitrary measure.     

On the order of partial sums of general orthogonal series

It is shown that for convergence of every orthonormal system _n(x) given on [0, l],it is necessary and sufficient that, under the condition on tlie increasing function W(x) and for there hold almost everywhere on [0, 1].Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 451–462, October, 1969.     

On a system of extended general variational inclusions

At the present paper, a new system of extended general nonlinear variational inclusions is introduced and using the resolvent operator technique, equivalence between the aforesaid system and the fixed point problem is verified. By using this alternative equivalent formulation, the existence and uniqueness theorem for solution of the system of extended general nonlinear variational inclusions is demonstrated and two new iterative schemes for solving this system of extended general nonlinear variational inclusions are suggested and analyzed. The convergence analysis of the proposed iterative methods under some suitable conditions is studied. Some errors in a recent article by Noor et al. (J Inequal Appl 2011:10, 2011) are found and the incorrectness of the results of the cited paper is proved. Also, the results of the aforementioned paper are revised and corrected.     

On a system of general mixed variational inequalities

In this paper, we introduce and consider a new system of general mixed variational inequalities involving three different operators. Using the resolvent operator technique, we establish the equivalence between the general mixed variational inequalities and the fixed point problems. We use this equivalent formulation to suggest and analyze some new explicit iterative methods for this system of general mixed variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of mixed variational inequalities involving two operators, variational inequalities and related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.     

On a class of orthogonal series

, [0, 1], (n+1) n-. . [2]. — (. 5.4 5.6). . 6.4 2 [5]. , [4]. , , [6] [7]. [1].     

On absolute convergence of series by a general Franklin system

The paper considers general Franklin systems generated by strong regular partitions of the segment [0; 1]. For such systems we prove the following assertions: 1) the absolute convergence of a Fourier-Franklin series at a point is a local property; 2) the Fourier-Franklin series of a function of bounded variation absolute converges almost everywhere; 3) any almost everywhere finite measurable function can be represented by an almost everywhere absolutely convergent series by a general Franklin system.     

On series by general Franklin system

We prove that if the admissible sequence $\mathcal{T}$ is strongly regular or is quasi-dyadic and weakly regular, then the Paley operator by the corresponding general Franklin system is of weak type (1, 1). Examples of general Franklin system with some properties are also discussed.     

Extremal measures for a system of orthogonal polynomials

We characterize the extremal measures of an indeterminate moment problem associated with a system of orthogonal polynomials defined by a three-term recurrence relation.     

Error estimates of proper orthogonal decomposition eigenvectors and Galerkin projection for a general dynamical system arising in fluid models

We study estimates for proper orthogonal decomposition eigenvectors and eigenvalues as well as error estimates between the exact solution of a 2D Navier–Stokes model and the numerical approach when the proper orthogonal decomposition method is considered. These estimates are also extended when bifurcation diagram are calculated using the so called p-POD or SPOD methods with a new cut-off criterion to minimize noisy modes produced by the p-POD method.     

Airy's stress function in general orthogonal coordinates

The paper discusses the role of mathematics in industry from the perspective of the Swedish Institute of Applied Mathematics. The view is expressed that the new ‘generation’ of problems arising from both changes in computer technology and the increased volume of applications should be a matter of concern to mathematics educators.     

Norm asymptotics of orthogonal polynomials for general measures

Letμ be a fixed positive unit Borel measure with infinite support in the unit disk. Acarrier of μ is any Borel subsetB of the support for whichμ(B)=1, and another such measurev iscarrier-related to μ when it has the same carriers asμ. Letp _n (z, v) be the monic orthogonal polynomial of degreen forv. We describe the possible asymptotics for the sequences {(∫|p _n (z,v)|² dv)1/2n} _n≥1 which are associated to the set of measures carrier-related to μ.     

On orthogonal polynomials

Пустьw(х)∈L[-1, +1] — неотрица тельная функция така я, что $$\frac{{\log ^ + \frac{1}{{w(x)}}}}{{\sqrt {1 - x^2 } }} \in L[ - 1, + 1]$$ и пусть {(р _n (х)} — много члены, ортогональные и нормированные с весо мw(x). Мы доказываем следующие две теорем ы, являющиеся обобщен ием одного известного результа та Н. Винера. I. Для каждого δ, 0<δ<1, суще ствует числоB=B(δ, w) тако е, что если $$f_N (x) = \sum\limits_{j = 1}^N {a_j p_{v_j } (x)} $$ причем выполнено сле дующее условие лакун арности $$\begin{gathered} v_{j + 1} - v_j \geqq B(\delta ,w) (j = 1,2,...,N - 1), \hfill \\ v_1 \geqq B(\delta ,w) \hfill \\ \end{gathered} $$ , то для некоторого С(δ, w) и всехh и δ, для которых $$ - 1 \leqq h - \delta< h + \delta \leqq + 1$$ , имеет место неравенс тво $$\int\limits_{ - 1}^1 {|f_N (x)|^2 w(x)dx \leqq C(\delta ,w)} \int\limits_{h - \delta }^{h + \delta } {|f_N (x)|^2 w(x)dx} $$ каковы бы ни былиa _j ,N и h. II. Если формальный ряд $$\sum\limits_{j = 1}^\infty {b_j p_{\mu _j } (x)} $$ удовлетворяет услов ию лакунарности μ_j+1-μ_j→∞ и суммируем, например, м етодом Абеля на произвольно малом отрезке [а, Ь] ?[0,1] к ф ункцииf(x) такой, что \(f(x)\sqrt {w(x)} \in L_2 [a,b]\) , то $$\sum\limits_j {|b_j |^2< \infty } $$ Теорема I — это первый ш аг в направлении проб лемы типа Мюнтца-Саса о замкнут ости подпоследовательно сти pv_j(x)} последовател ьности {р_n(х)} на отрезке [а, Ь] в метрике С[а, Ь] (см. теорему II стать и).