# Information Theoretic Spreading Measures Of Orthogonal Functions

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### Friday April 8 - NIST

In this talk we will show two classes of measures to quantify the distribution of the special functions of Applied Mathematics all over its domain of de nition. They are de ned by means of functionals of the associated probability densities. The rst class includes the information-theoretic spreading lengths of Shannon, R enyi

### Mec´anica Cu´antica y Entrop´ıas de Polinomios Ortogonales

spreading measures of Dnl(r), which are power (variance), derivative (Fisher information) or logarithmic (Shannon entropy) functionals of the involved orthogonal polynomials. The system proﬁle can be estimated by means of the spreading measures of Πlm(Ω), which are given by the corresponding functionals of the Gegenbauer polynomials.

### Time-Frequency Foundations of Communications: Concepts and Tools

max, measures the channel s overall TF dispersion and is referred to as the channel spread. A channel is then said to be underspread if dH 1 and overspread if dH > 1. For spreading functions that do not have ﬁnite support, the channel spread can be quantiﬁed in terms of moments [32]. For multipath propagation, we have dH /1=c2 [33]. Hence,

### harmonic systems. Application to Rydberg states and high

mials, which describe the spreading quantities, leans heavily on the algebraic properties and asymptotical behavior of some weighted Lq-norms of these orthogonal functions. PACS. 89.70.Cf Entropy and other measures of information 03.65.-w Quantum Mechanics 03.65.Ge Solutions of wave equations: Bound states 1 Introduction

### A STUDY OF THE PERMUTATION SCHEMES USED IN THE MOBILE WIMAX

Figure 1. The theoretic DS for a random interleaver with N=1000. III. PERMUTATION SCHEMES USED IN WIMAX The WiMAX technology uses at its physical level a multi-carrier modulation technique, called Orthogonal Frequency Division Multiplexing (OFDM). In fact, a modified version of OFDM is defined in [3],

### arXiv:1305.4449v1 [math-ph] 20 May 2013

RelativeFisher information of discreteclassical orthogonal polynomials weight functions. are given by the corresponding information-theoretic measures of the associated

### Propuesta de Trabajo Fin de Grado en Matemáticas

-J.S. Dehesa, A. Martinez-Finkelshtein and J. Sanchez-Ruiz. Quantum information entropies and orthogonal polynomials. Journal of Computational and Applied Mathematics 133 (2001) 23-46 - J.S. Dehesa, A. Guerrero and P. Sanchez-Moreno. Information-theoretic-based spreading measures of orthogonal polynomials.

### of orthogonal polynomials arXiv:1305.3711v1 [math-ph] 16 May 2013

theory of information of the special functions. In this paper we are going to discuss the class of direct spreading measures (the standard deviation and the information-theoretic lengths of Fisher, R´enyi and Shannon types) of the Rakhmanov probability density [15] of the orthogonal

### 1 A Game-Theoretic Approach to Energy-Efﬁcient Power Control

A game-theoretic model for studying power control in multi-carrier CDMA systems is proposed. utility function considered here measures the number of reliable bits transmitted over all the

### Information-theoretic lengths of Jacobi polynomials

Jun 07, 2020 The information-theoretic lengths of the Jacobi polynomials P(α,β) n (x), which are information-theoretic measures (Renyi, Shannon and Fisher) of their associated Rakhmanov probability density, are investigated. They quantify the spreading of the polynomials along the orthogonality interval [−1,1] in a

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### Fisher information of special functions and second-order di

nation of the information-theoretic measures, such as the Fisher information and the Shannon, Renyi and Tsallis entropies, which quantify the spreading of these functions in a complementary and more appropriate manner than the familiar variance. These quantities are de ned by the corresponding spreading

### MEDIDAS ENTROPICAS Y DE COMPLEJIDAD DE FUNCIONES ESPECIALES Y

3. Information theoretic-based spreading measures of orthogonal polynomials J. S. Dehesa, A. Guerrero y P. S anchez-Moreno Complex Analysis and Operator Theory 6 (2012) 585-601 4. Upper bounds on quantum uncertainty products and complexity measures A. Guerrero, P. S anchez-Moreno y J. S. Dehesa Physical Review A 84 (2011) 042105 (8 paginas) 5.

### Cramer Rao information plane of orthogonal hypergeometric

The study of the spreading of the classical orthogonal polynomials by means of information-theoretic measures, was initiated in the 1990s with the computation of their Shannon information entropy [2,8,9]. The results found for this quantity up to 2001 are summarized in Ref. [7], where one can also see

### arXiv:math-ph/0603069v1 27 Mar 2006

lattice, and consequently ψn(t) 2 describes the spreading of the quantum wave over this space. Figure 1.3 shows the initial part of the evolution in the case of the usual Bessel functions (for which the measure µis absolutely continuous), and Figure 1.4 displays the same information in the case of a singular continuous Julia set measure.

### Information-theoretic properties of special functions*

2 Rakhmanov density of special functions 3 Information-theoretic lengths of ˆ(x) 4 Complexity measures of ˆ(x) 5 Spreading lenghts of C.O.P. (Classical Orthogonal Polynomials) 6 Complexity measures of C.O.P. 7 Conclusion and open problems Jesus S. Dehesa Information-theoretic properties of special functions 21 / 39

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4 Information-theoretic lengths of orthogonal polynomials 51 2 and 3), to the theory of special functions of math single charge spreading measures show that

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Besides modifying the network structure, an orthogonal line of works [2, 9] consider the problem of spreading positive information as the best response to limit the eventual negative inﬂuence caused by the static adversary. Other works focus on the game-theoretic version where both the players choose to

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of information that is used in the segmentation process. Previous work on interactive segmentation apply only xed motions; focus on segmenting an object from the back-ground; or do not segment during interaction, only after each action is complete. Our work addresses these issues to enable interactive segmentation in real-world scenarios. First, we

### 04- Instituto Carlos I de Física Teórica y Computacional

-Workshop Functions and Operators 2010 Lugar y fecha: Polonia, 24 Junio 2010 Nombre de la ponencia o comunicación: Spreading Lengths of Orthogonal Polynomials Ponente: Jesús Sánchez-Dehesa -11th Granada Seminar on Computational and Statistical Physics Lugar y fecha: La Herradura (Granada), 13-17 de septiembre de 2010