PDF superior On the Krall-type discrete polynomials

Let us also point out that there are also the so-called discrete (see e.g. [8]) and q-discrete Sobolev type orthogonal polynomials associated with the classical discrete and q-classical functionals [23, 24]. In both cases the corresponding polynomials can be reduced to the Krall-type one (except for the q-case when the mass is added at zero where a more careful study is needed [23, 24]) since the differences ∆f (x) = f (x + 1) − f(x) and D q f(x) = (f (qx) − f(x))/(qx − x).

The study of some particular cases of orthogonal polynomials in Sobolev spaces has attracted the interest of several authors [1], [9], [15], [20], [21] and [25]. Particular emphasis was given to the so-called classical Sobolev polynomials of discrete type, i.e., polynomials orthogonal with respect to an inner product

is the basic hypergeometric polynomial of degree n in the variable z (throughout this paper, we will employ the standard notations of the q-special functions theory, see [9] or [10]). The q-Laguerre polynomials (2.1) satisfy two kind of orthogonality relations, an absolutely continuous one and a discrete one. The former orthogonality relation, in which we are interested in the present paper, is given by

The study of orthogonal polynomials with respect to a modication of a linear func- tional in the linear space of polynomials with real coecients via the addition of one or two delta Dirac measures has been performed by several authors. In particular, Chihara [5] has considered some properties of such polynomials in terms of the location of the mass point with respect to the support of a positive measure. More recently Marcellan and Maroni [10] analyzed a more general situation for regular ( quasi-denite ) linear functionals, i.e., such that the principal submatrices of the corresponding innite Hankel matrices associated with the moment sequences are nonsingular.

It is well known that the classical families of Jacobi, Laguerre, Hermite, and Bessel polynomials are characterized as eigenvectors of a second order linear differential operator with polynomial coefficients, Rodrigues formula, etc. In this paper we present a unified study of the classical discrete polynomials and q-polynomials of the q-Hahn tableau by using the difference calculus on linear-type lattices. We obtain in a straightforward way several characterization theorems for the classical discrete and q-polynomials of the “q-Hahn tableau”. Finally, a detailed discussion of a characterization by Marcellán et al. is presented.

In this work we will present a di<erent approach: It can be considered a pure algebraic approach and constitutes an alternative to the two previous ones, and, in some sense is the continuation of the Hahn’s work [16]. Furthermore, we will prove here that the q-classical polynomials are characterized by several relations, analogue to the ones satis1ed by the classical “continuous” (Jacobi, Bessel, Laguerre, Hermite) and “discrete” (Hahn, Meixner, Kravchuk and Charlier) orthogonal polynomials [1,13,21,22] and references therein. Besides, our point of viewis very di<erent from the previous ones based on the basic hypergeometric series and the di<erence equation, respectively. In fact we start with the distributional equation that the q-moment functionals satisfy and we will prove all the other characterizations using basically the algebraic theory developed by Maroni [23]. So, somehow, this paper is the natural continuation of the study started in [22,13] for the “continuous” and “discrete” orthogonal polynomials, respectively. Another advantage of this approach is the uni1ed and simple treatment of the q-polynomials where all the information is obtained from the coeMcients of the polynomials and of the distributional or Pearson equation (compare it with the method by the American school [20] or the Russian ones [29]).

The symmetries of quantum states play an important role in explaining the degeneracy of energy levels [1]. The connection of the energy levels of the hydrogen atom with the irreducible representation of O(4, 2) conformal symmetry was found in [2]. The q-deformed Heisenberg–Weyl symmetry was used to introduce the notion of quantum q-oscillators [3, 4]. The physical meaning of q-deformations was clarified in [5] where it was shown that classical q-oscillators and their quantum partners are standard nonlinear oscillators vibrating with a frequency depending on the amplitude. Thus, the symmetry groups and the q-deformed symmetry groups are important ingredients in the description of states in quantum optics and quantum mechanics. A general consideration of constructing the irreducible representations of Lie groups and their connection with the formalism of classical mechanics was presented in [6] in the context of symmetry applications in quantum mechanics and quantum optics.

Discrete Fourier transform (DFT)-based systems are efficient to fight against multi-path fading channels, but they present some drawbacks, as a high peak-to-average power ratio, or the sensitivity to carrier- frequency offsets (CFO). Frequency offset interferences are mainly caused in mobile OFDM communications by mismatch or ill-stability of the local oscillators in the transmitter and the receiver, and by the movement-induced Doppler shift. As a well-known fact, the broad- ening of the Doppler spectrum caused by the speed of the terminal is a main effect that has a direct impact on the bit error probability (BEP). Different solutions to correct CFO and the rest of drawbacks are based on the use of discrete trigonometric transforms (DTTs), mainly discrete cosine transform (DCT) Type-II even, 1 as multicarrier modula-

The structure of the paper is as follows. In Section 2 we list some of the main properties of the classical Bessel polynomials which will be used later on. In Section 3 we dene the generalized polynomials and nd some of their properties. In Section 4 we obtain the representation of the generalized Bessel polynomials in terms of the hypergeometric functions. In Section 5 we obtain an asymptotic formula for these polynomials and in Section 6 we establish their quasi-orthogonality. Finally, in Sections 7 and 8 we obtain the three-term recurrence relation that such polynomials satisfy as well as the corresponding Jacobi matrices.

In 1940, H. L. Krall [19] obtained three new classes of polynomials orthogonal with respect to measures which are not absolutely continuous with respect to the Lebesgue measure. In fact, his study is related to an extension of the very well known characteri- zation of classical orthogonal polynomials by S. Bochner. This kind of measures was not considered in [28]. Moreover, in his paper H. L. Krall obtain that these three new families of orthogonal polynomials satisfy a fourth order dierential equation. The corresponding measures are given in the following table.

Surely the simplest kind of polynomials over a Banach space are the polyno- mials of finite type, those which may be written as P (x) = n i =1 γ i (x) k , where each γ i is a continuous linear form over E. Slightly larger is the class of nu- clear polynomials, made up of those which may be represented (non-uniquely) as P (x) = ∞ i =1 γ i (x) k , with ∞ i =1 γ i k < ∞. These form a non-closed vector sub- space of P ( k E) (although they do form a Banach space in the ‘nuclear’ norm, i.e. the infimum of ∞ i =1 γ i k over all possible representations of P ). The closure of the space of nuclear (or also of the finite-type polynomials) in the uniform norm is the space of approximable polynomials. Larger still are the spaces of weakly con- tinuous polynomials (those which are weakly continuous over all bounded subsets of E) and weakly sequentially continuous polynomials. We thus have the chain of vector subspaces of P( k E)

Suslov, The theory of dierene analogues of speial funtions of hypergeometri. type[r]

We have used a method [12, 16, 17] which is based only on the three-term recurrence relation satised by the involved polynomials. This method, which will be described in Section 2, is of general vality since no peculiar constraints are imposed upon the coecients of the recurrence relation. It was found in a context of tridiagonal matrices [6, 13, 14, 15] and it has been already used for the study of the distribution of zeros of q-polynomials [1, 11, 16]. Some of the results found here have been previously obtained by other means and are dispersely published, what will be mentioned in the appropiate place; they are included here for completeness, for illustrating the goodness of our procedure or because they are not accessible for the general reader [16].

In this paper, we have adapted some tools of the dynamical analysis of multivariate real discrete problems to analyze the stability of the fixed points of iterative methods with memory on quadratic polynomials. As far as we know, this kind of analysis has not been performed before on iterative methods with memory for solving nonlinear equations. From the well-known Secant method as initial inspiration, we have designed new methods with mem- ory from Steffensen’ or Traub’s schemes (in the last case we have previously transformed it in a derivative-free method), as well as a parametric family of iterative procedures of third- and fourth-order of convergence. Our statements, based on consistent discrete dynamics results and also on Feigenbaum diagrams of the family, allow us to select the most stable elements of the class and to find those that present convergence to other points different from the solution of our problem or even chaotic behavior.

[18] F. A. Gr¨ unbaum, L. Haine, and E. Horozov, On the Krall-Hermite and Krall-Bessel polynomials, Internat. Math. Res. Not. 19 (1997), 953-966. [19] L. Haine, The Bochner-Krall problem: some new perspectives. In Special Functions 2000: Current Perspective and Future Directions, J. Bustoz et al. (Eds.) NATO ASI Series, Dordrecht, Kluwer (2002),141-178. [20] L. Haine and P. Iliev, Askey-Wilson type functions, with bound states,

The analysis of properties of polynomials orthogonal with respect to a perturbation of a measure via the addition of mass points was introduced by P.Nevai [35]. There the asymp- totic properties of the new polynomials have been considered. In particular, he proved the dependence of such properties in terms of the location of the mass points with respect to the support of the measure. Particular emphasis was given to measures supported in [ 1;1] and satisfying some extra conditions in terms of the parameters of the three-term recurrence relation that the corresponding sequence of orthogonal polynomials satises.

computations are very hard and cumbersome so the use of the symbolic pack- age Mathematica is again a very useful tool (see [3, 5, 7]). Finally, let us point out that from the SODE the moments of the zero distribution easily follows using the approach described in [10].

Markett [37] for symmetric orthogonal polynomials, has designed a method which be- gins with the three-term recurrence relation of the involved orthogonal polynomial system to set up a partial dierence equation for the (orthogonal) polynomial, in case of connec- tion problems, or for the product of two (orthogonal) polynomials, in case of linearization problems, to be expanded; then, this equation has to be solved in terms of the initial data. Ronveaux et al. [47, 48] for classical and semiclassical orthogonal polynomials and Lewanowicz [30, 34] for classical orthogonal polynomials have proposed alternative, sim- pler techniques of the same type although they require the knowledge of not only the recurrence relation but also the dierential-dierence relation or/and the second-order dierence equation, respectively, satised by the polynomials of the orthogonal set of the expansion problem in consideration. See also [5, 11, 23, 31, 45, 46] for further description and applications of this method in the discrete case, and [19, 32] in the continuous case as well as [3, 33] for the q -discrete orthogonality. Koepf and Schmersau [25] has proposed a computer-algebra-based method which, starting from the second order dierence hyper- geometric equation, produces by symbolic means and in a recurrent way the expansion coecients of the classical discrete orthogonal hypergeometric polynomials (CDOHP) in terms of the falling factorial polynomials (already obtained analytically by Lesky [28]; see also [41], [42]) as well as the expansion coecients of its corresponding inverse problem. The combination of these two simple expansion problems allows these authors to solve the connection problems within each specic CDOHP set.

The classical orthogonal polynomials are very interesting mathematical objects that have attracted the attention not only of mathematicians since their appearance at the end of the XVIII century connected with some physical problems. They are used in several branches of mathematical and physical sciences and they have a lot of useful properties: they satisfy a three-term recurrence relation (TTRR), they are the solution of a second order linear differential (or difference) equation, their derivatives (or finite differences) also constitute an orthogonal family, their generating functions can be given explicitly, among others (for a recent review see e.g. [1]). Among such properties, a fundamental role is played by the so-called characterization theorems, i.e., such properties that completely define and characterize the classical polynomials. Obviously not every property characterize the classical polynomials and as an example we can use the TTRR. It is well-known that, under certain conditions—by the so-called Favard Theorem (for a review see [7])—, the TTRR characterizes the orthogonal polynomials (OP) but there exist families of OP that satisfy a TTRR but not a linear differential equation with polynomial coefficients, or a Rodrigues-type formula, etc. In this paper we will complete the works [3,10] proving a new characterization for the classical discrete [3,6] and the q-classical [4,10] polynomials. For the continuous case see [8,9].

Actions due to impact are listed in the “Technical Building Code” (CTE, 2006), in paragraph 4.3.2. of its document “Loading for Buildings”. The indications given in this document are consistent with those included in Part 1.7 of Eurocode 1, described earlier in this document. However, in this case there is a significant difference between these two codes: while the part 1.7 of Eurocode 1 limits the maximum vehicle speed to determine the type of road involved and the range of values for the equivalent static load, Spanish rule limits the maximum mass of the vehicle for which a parking was designed. The reader should notice that during the impact there is an important energy exchange between the vehicle and the structure. The initial energy is the kinetic energy of the vehicle before the impact, which depends quadratically on the speed. Therefore it is more logical to limit the speed of the impact instead that the vehicle mass.