PDF superior On the modification of classical orthogonal polynomials: The symmetric case.

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.

but also other useful properties: they are the eigenfunction of a second order linear differential operator with polynomial coefficients, their derivatives also constitute an orthogonal family, their generating functions can be given explicitly, among others (see for instances [1,8,24,25] or the more recent work [3]). Among all these properties there are very important ones that characterize these families of polynomials.

polynomials have been obtained in [3]. However, the explicit form of these polynomials in the general case remains as an open question as well as the study of their zeros. We are trying in this paper to cover this lack. Moreover, some of the usual properties of classical orthogonal polynomials { sym- metry property, their representation as hypergeometric series and the second order linear dierential equation { are translated to the context of Sobolev-type ortogonality.

Let us point out here that the theory of orthogonal polynomials on the non-uniform lattices is based not on the Pearson equation and on the hypergeometric-type di<erence equation of the non-uniform lattices as it is shown in papers [7,26,28] and obviously it is possible to derive many properties of the q-classical polynomials from this di<erence hypergeometric equation. Our purpose is not to show howfrom the di<erence equation many properties can be obtained, but to showthat some of them characterize the q-classical polynomials, i.e., the main aim is the proof of several characterizations of these q-families as well as the explicit computations of the corresponding coeMcients in a uni1ed way. Some of these results on characterizations (e.g. the Al-Salam-Chihara or Marcell+an et al. characterization for classical polynomials) are completely new as far as we know.

Special important examples appear when is a classical linear functional and the mass points are located at the ends of the interval or orthogonal- ity. In this case we have studied the corresponding sequences of orthogonal polynomials in several papers: For the Laguerre linear functional see [2, 3], for the Bessel linear functional see [6], for the Jacobi linear functional see [7] and for the Hermite case see [1]. In particular the quasi-definiteness of e , relative asymptotics, the representation as a hypergeometric function, and the location of their zeros have been obtained.

In this paper, we consider regular Borel measures µ defined on subsets of the complex plane which are Jordan arcs, or connected finite union of Jordan arcs, and we show how the support of µ is determined by the entries of the Hessenberg matrix D associated with µ. The Hessenberg matrix is the natural generalization of the tridiagonal Jacobi matrix to the complex plane and, in the particular case of measures with support the unit circle T, the Hessenberg matrix is a Toeplitz matrix.

Here we will consider some examples. Since the classical case with one or two extra delta Dirac measures (functionals) has been studied intensively (see e.g. [4, 5]) we will focus here our attention in the q-case. For the sake of simplicity we will choose the Al-Salam & Carlitz I polynomial as the starting family. The main data of such family can be found in [25, page 113].

The term moment problem was used for the first time in T. J. Stieltjes’ clas- sic memoir [32] (published posthumously between 1894 and 1895) dedicated to the study of continued fractions. The moment problem is a question in classical analysis that has produced a rich theory in applied and pure mathematics. This problem is beautifully connected to the theory of orthogonal polynomials, spectral representation of operators, matrix factorization problems, probability, statistics, prediction of stochastic processes, polynomial optimization, inverse problems in financial mathematics and function theory, among many other areas. In the ma- trix case, M. Krein was the first to consider this problem in [21], and later on some density questions related to the matrix moment problem were addressed in [14, 15, 24, 25]. Recently, the theory of the matrix moment problem is used in [10] for the analysis of random matrix-valued measures. Since the matrix moment problem is closely related to the theory of matrix orthogonal polynomials, M. Krein was the first to consider these polynomials in [22]. Later, several researchers have made contributions to this theory until today. In the last 30 years, several known properties of orthogonal polynomials in the scalar case have been extended to the matrix case, such as algebraic aspects related to their zeros, recurrence relations, Favard type theorems, and Christoffel–Darboux formulas, among many others.

as a modication of the rst ones troughtout the addition of two mass points. All the formulas for the classical Hahn polynomials can be found in a lot of books ( see for instance the excellent monograph Orthogonal Polynomials in Discrete Variables by A.F. Nikiforov, S. K. Suslov, V. B. Uvarov [16], Chapter 2.)

The structure of the paper is as follows. In Section 2, we provide the basic properties of the classical orthogonal polynomials of discrete variable which will be needed, as well as the main data for the Meixner, Kravchuk and Charlier polynomials. In Section 3 we deduce expressions of the generalized Meixner, Kravchuk and Charlier polynomials and its rst dierence derivatives, as well as their representation as hypergeometric functions in the direction raised by Askey. In Section 4, we nd the second order dierence equation which these generalized polynomials satisfy. In Section 5, from the three term recurrence relation (TTRR) of the classical orthogonal polynomials we nd the TTRR which satisfy the perturbed ones. In Section 6, from the relation of the perturbed polynomials P An (x) as a linear combination of the classical ones, we nd the tridiagonal matrices associated with the perturbed monic orthogonal polinomial sequence (PMOPS) f P An (x) g

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.

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].

Multiwavelets have several advantages in comparison with scalar wavelets. The features such as compact support, orthogonality, symmetry, and high order vanish moments are known to be important in signal processing. A scalar wavelet can not possess all these properties at the same time but multiwavelets can.

siderably larger than ours. For the unfrustrated case, all the mean field approaches are quite inaccurate compared with much more controlled techniques like QMC. The difference in the M (Q) values of about 10%, provides, in the absence of any other quantitative evidence for the accuracy of the method as applied to this model, an indi- cation of the accuracy of the method and of all the results quoted that depend on the order parameters, including the phase boundaries. However, the mean field approach is still very useful to study gapped phases in frustrated systems. On one hand it is well known that for frus- trated systems QMC presents the famous sign problem. On the other hand, the study of quantities like energy gap requires the study of big sizes clusters and the use of exact diagonalization for small size clusters makes it very difficult to extrapolate the results.

The influence of the English language does not only affect the borrowing process, but also appears in the orthographical patterns of Spanish derivatives and neoclassical compounds. We will analyze the list of prefixed items and neoclassical compounds found in our corpus (see table 3). In fact, the use of some prefixes and combining forms in Spanish words followed by a hyphen imitates the English orthographical structure. The prefixes and combining forms used in the examples in table 3 do exist in Spanish, according to RAE, but the orthographical pattern does not follow the rules in the Spanish language. According to RAE (2010: 535), “no se consideran ortográficamente adecuadas las grafías en las que el prefijo aparece unido con guion a la palabra base (*anti-mafia, *anti-cancerígeno) o separado por ella por un espacio en blanco (anti mafia, anti cancerígeno)”. Similarly, RAE explains that the combining forms used in these formations “Si va antepuesto, se denomina elemento compositivo prefijo: biodiversidad, ecosistema; si va pospuesto, se denomina elemento compositivo sufijo: antropófago, neuralgia” (DPD) and the examples provided in the explanation clearly show that the hyphen is not used in the Spanish word-formation process, for example, biodegradable is recorded in RAE, but without the hyphen. Similarly, Fundeú 22 (2017)

The English use of the term ‘wealth’ can be dated back to the thirteenth and fourteenth centuries, with parallel formations in the Middle Dutch weelde, welde, Middle Low German welede, and Old High German welida. From then right through to the seventeenth century in English it conveyed a wide sense of human well-being, not limited to material possessions: ‘They sayd howe the noble men of the realme of Fraunce, knyghtes and squyers shamed the realme, and that it shulde be a great welth to dystroy them all’ (1523); ‘Christe . . . liued . . . and . . . suffred . . . for our sakes, and for our welthe’ (1537); ‘She . . . pro- cured both suche as was for the welthe of his soule, and prepared holsome meates for his body’ (1541); ‘In all tyme of our tribulacion, in all tyme of our wealth’ (1548–9); ‘The inuentyon of feates, helpynge annye thynge to the adu- antage and wealthe of lyffe’ (1551). The notion of wealth as material property is however also in use from the earliest times: ‘For here es welth inogh to win, To make vs riche for euermore’ (1352); ‘all men shoulde haue and enioye equall portions of welthes and commodities’ (1551; from Thomas More’s Utopia); ‘To [sic] late you shall repent the act When all my realme, and all your wealthes are sackt’ (1574); ‘Iulia. What think’st thou of the rich Mercatio? Lucetta. Well of his wealth; but of himselfe, so, so’ (1591; William Shake- speare); ‘Wealth, howsoever got, in England makes Lords of Mechanicks, Gen- tlemen of Rakes’ (1703; all from OED: ‘wealth’). The OED provides just two prior instances of that phrase given perpetual currency by Smith’s title: ‘The winds were hush’d, the waves in ranks were cast, As awfully as when God’s people past: Those, yet uncertain on whose sails to blow, These, where the wealth of Nations ought to flow’ (Dryden 1667: prefatory ‘Verses to . . . the Dutchess’). ‘To be poor, in the epick language, is only not to command the wealth of nations’ (Johnson 1752: 238). 35

To do this, we take a certain viewpoint, which is suitable not only for this proof, but also for similar questions: The R-matrix can be viewed as a twist that takes the coproduct into the coopposite coproduct. However, while twisting leaves the antipode unchanged, the coopposite coproduct naturally comes endowed with the inverse antipode. The so-called Drinfel’d element now appears as the element that connects these two choices for the antipode of the coopposite quasi-Hopf algebra. Viewing the Drinfel’d element in this way enables us not only to give a relatively easy proof of our claim, but also allows us to give a new derivation of the funda- mental properties of the Drinfel’d element in a comparatively short and conceptual way.

[8] R. Koekoek and R. F. Swarttouw The Askey-scheme of hypergeometric orthog- onal polynomials and its q-analogue. Reports of the Faculty of Technical Math- ematics and Informatics No. 94-05. Delft University of Technology. Delft 1994. [9] T.H. Koornwinder: Compact quantum groups and q-special functions. In Rep-

the permitted lines are fairly weak and we had to use a three- point running average to smooth their pro fi les in Figure 3. Even then the FWZIs of these lines are dif fi cult to ascertain with satisfactory accuracy, but it is clear that they are narrower than the forbidden lines and have an FWZI of ∼ 4000 – 5000 km s −1 . These values are satisfactorily consistent with generally observed pro fi le velocities in novae. In the classi fi cation scheme by Williams ( 1992 ) , the Fe II novae have FWZI < 5000 km s − 1 whereas the He / N novae generally have FWZI > 5000 km s −1 . Based on this criterion alone, however, it is dif fi cult to say whether VVV-WIT-06, if indeed a CN, is of the Fe II type or He / N type. But we would favor the former class because of the extended climb to maximum seen in the light curve and also since the line pro fi les are not signi fi cantly fl at-topped, as expected for He / N novae.

Section 4 summarizes low-dimensional complete results for certain NIEP variants. General sufficient conditions for the realizability of spectra are given in section 5. It has long been known that n-fold spectra that meet simple nec- essary conditions, but are not realizable, may be made realizable by append- ing of 0 eigenvalues. Information about this phenomenon is given in section 6. We turn to what is known about the graph-NIEP’s in section 7. The new idea of Perron similarities - studying the diagonalizable NIEP’s via the diagonal- izing similarities - is discussed in section 8, and the role of Jordan structure in the NIEP and R-NIEP in section 9.