Zeros of orthogonal 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.

addition of one or more delta Dirac functions. Some examples studied by dierent authors are considered from an unique point of view. Also some properties of the Krall polynomials are studied. The three-term recurrence relation is calculated explicitly, as well as some asymptotic formulas. With special emphasis will be considered the second order dierential equations that such polynomials satisfy which allows us to obtain the central moments and the WKB approximation of the distribution of zeros. Some examples coming from quadratic transformation polynomial mappings and tridiagonal periodic matrices are also studied.

In 1940, H. L. Krall [14] 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 characterization of classical orthogonal polynomials by S. Bochner. This kind of measures was not considered in [22].

We consider the modications of the monic Hermite and Gegenbauer polynomials via the addition of one point mass at the origin. Some properties of the resulting polynomials are studied: three-term recurrence relation, dierential equation, ratio asymptotics, hypergeometric representation as well as, for large n , the behaviour of their zeros.

Our aim here is to express in terms of higher order hypergeometric Lauricella functions the corresponding asymptotic contracted measure of zeros for the sequence {P n (x)}™ =1 to be de[r]

The hypergeometric polynomials in a continous or a discrete variable, whose canonical forms are the so-called classical orthogonal polynomial systems, are objects which naturally appear in a broad range of physical and mathematical elds from quantum mechanics, the theory of vibrating strings and the theory of group representations to numerical analysis and the theory of Sturm-Liouville dierential and dierence equations. Often, they are encoun- tered in the form of a three term recurrence relation (TTRR) which connects a polynomial of a given order with the polynomial of the contiguous orders. This relation can be directly found, in particular, by use of Lanczos-type methods, tight-binding models or the appli- cation of the conventional discretisation procedures to a given dierential operator. Here the distribution of zeros and its asymptotic limit, characterized by means of its moments around the origin, are found for the continuous classical (Hermite, Laguerre, Jacobi, Bessel) polynomials and for the discrete classical (Charlier, Meixner, Kravchuk, Hahn) polynomials by means of a general procedure which (i) only requires the three-term recurrence relation and (ii) avoids the often high-brow subleties of the potential theoretic considerations used in some recent approaches. The moments are given in an explicit manner which, at times, allows us to recognize the analytical form of the corresponding distribution.

In the last decade an increasing interest on the so called q-orthogonal polynomials (or basic orthogonal polynomials) is observed ( for a review see [1], [2] and [3]). The reason is not only of purely intrinsic nature but also because of the so many applications in several areas of Math- ematics ( e.g., continued fractions, eulerian series, theta functions, elliptic functions,...; see for instance [4] and [5]) and Physics ( e.g., angular momentum [6] and [7] and its q-analog [8]-[11], q-Shrodinger equation [12] and q-harmonic oscillators [13]-[19]). Moreover, it is well known the connection between the representation theory of quantum algebras (Clebsch-Gordan coecients, 3j and 6j symbols) and the q-orthogonal polynomials, (see [20], [21] (Vol. III), [22], [23], [24] ), and the important role that these q-algebras play in physical applications (see for instance [26]-[31] and references therein).

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.

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.

After studying some examples that the literature provides us with, one may realize that, even thought it is generic to assume the perturbing matrix polynomial W ( x ) to have a nonsingular leading coefficient, many examples do have a singular matrix as its leading coefficient. This situation is a special feature of the matrix case setting since in the scalar case, having a singular leading term would mean that this coefficient is just zero (affecting, of course, to the degree of the polynomial). For this reason, when dealing with this kind of matrix polynomials talking about their degree should make no sense. The effect that this fact has on our reasoning is that since deg [ det W ( x )] ≤ Np the information encoded in the zeros (and corresponding adapted polynomials) of det W (x ) is no longer enough to make the matrices kN of the needed size. Therefore, there will be no way

We also test intermediate solutions, where left-shifting is performed but only for a few bits. Fig. 2 shows this new adder that we denote as A2 (A2H for HUB version). Comparing it with A1, the A2 design has a special leading zero detector, which detects up to two leading zeros at the output of the absolute value circuit. Furthermore, it has a barrel shifter that can perform a one-position right-shifting (in case of detecting overflow) and left-shifting up to 2-bit positions. This will increase the area and the delay of the critical path, but it will improve the error figures as we will see in section V. In this architecture the exponent has to be decremented when left- shifting is performed, and therefore underflow could happen. Although it is not depicted in Fig. 2, this situation is detected in the design and the result flushes to zero.

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.

Firstly, we will consider the case when we add a point mass at x = 0. This case corresponds to the Laguerre, Charlier, Meixner and Kravchuk polynomials. Later on, we will consider the Jacobi and Hahn polynomials which involve two point masses at the ends of the interval of orthogonality. The reason of such a choice of the point in which we will add our posi- tive mass will be clear from formulas (39) and (41) from below, because in such formulas appears the value of the kernel polynomials K n (x;y) and they have a very simple analyti-

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

Head-related transfer function (HRTF) defines the spectral shaping of the sound signal on its way from a source location in the free field to the external ear. HRTFs are generally measured at discrete positions on a spherical grid equidistant from centre of the human head (or mannequin) whose measurements are being taken [1]. These measurements provide a set of HRTFs at discrete values of azimuth and elevation. The HRTFs provide a frequency dependent function of interaural intensity difference (IID) and a constant interaural time delay (ITD). The HRTF data obtained from these measurements are used for designing low-order digital filters to be made available for use in real-time audio spatialisation applications. A number of different approaches regarding the HRTF filter design process have been proposed [2][3][4].

In this work we show that, in the case of regular measures µ whose support is a Jordan arc or a connected union of Jordan arcs in the complex plane C, the limits of the values at the diagonals of the Hessenberg matrix D of µ, supposing those limits exist, determine the terms of the coefficients of the series expansion of the Riemann map φ(z) (see [20]) which applies conformally the exterior of the unit disk in the exterior of the support of the measure. As a consequence, the support of µ can be determined just knowing the limits of the values at the diagonals of its Hessenberg matrix D.

In Chapter III, we propose space-time-frequency codes for two transmit antennas over frequency selective fading channels. First, the channel model and design criteria for which a STF code guarantees multipath diversity and high coding gain over a MIMO-OFDM system are discussed. Furthermore, in the first part of the Chapter, we propose a STF code called Extended Super-Orthogonal Space-Time-Frequency Trellis Code (Ex-SOSTFTC). Here, we consider the concept of rotated constellations and we avoid parallel transitions in the trellis structure. Then, we extend the Super-Orthogonal Space-Time Trellis Codes (SOSTTCs) originally designed for the frequency-flat fading case, to the frequency-selective fading channel. Afterwards, decoding of extended-STF trellis codes is shown. In the second part of the chapter, we propose a coding scheme called Quasi-Orthogonal Space-Time-Frequency Trellis Codes (QOSTFTCs), where we systematically combine a Quasi-Orthogonal Space-Time Block Code (QOSTBC) with a trellis code, operating over a frequency selective fading channel. In addition, in order to provide the maximum coding gain for the QOSTFTCs, we will describe a systematic method to do set partitioning. Then, we derive a decoding process for the QOSTFTCs, and then the diversity of QOSTFTCs is discussed. Finally, performance simulation results of the proposed Ex-SOSTFTC and QOSTFTCs will be presented and examined. We show that our designs are able to exploit the full diversity gains available in the MIMO-OFDM channel, and can achieve high coding gain, good performance, and low decoding complexity with a low number of states in the trellis. Both proposals get full symbol rate (one symbol per frequency tone per time slot). We show with analysis and numerical simulations that our designs of STF codes outperform the best existing space-time-frequency trellis codes in the literature.

By offspring selection, the best children are chosen and become the parents of the next generation. Typically, parent selection in ES is performed randomly with no regard to fitness; survival in ESs simply saves the µ best individuals, which is only based on the relative ordering of their fitness values. Basically, there are two selection strategies for ESs:

4.- Formulas for Change of basis of chromatic polynomials. 4.1 Change from null basis to complete basis and inversely. 4.2 Change from null basis to tree basis and inversely.. ü ANALE[r]

After the measurement of the machining forces, the acquired data was processed to convert from the original signal to a sin- gle force value, in order to facilitate the comparison between different machining conditions. Therefore, the machining for- ce signal was filtered using MatLab software, and a portion of it was further selected to obtain an average value. The se- lection criteria considered avoiding instability zones, utilizing only the data from the central millimeter, from the total 5 mm in radial depth of cut (i.e. data from 2 mm after the start of the cut and 2 mm before the end of the cut was discarded). Once the points were selected and averaged, a final average force va- lue was obtained among the four (4) experimental replications. The final average values from the dynamometer force compo- nents are reported in Figures 5 and 6. The cutting force compo- nent, F c , corresponds to the z dynamometer direction.