Two general methods for establishing the logarithmic behavior of recursively defined sequences of real numbers are presented. One is the interlacing method, and the other one is based on calculus. Both methods are used to prove... more
Two general methods for establishing the logarithmic behavior of recursively defined sequences of real numbers are presented. One is the interlacing method, and the other one is based on calculus. Both methods are used to prove logarithmic behavior of some combinatorially relevant sequences, such as Motzkin and Schr\"oder numbers, sequences of values of some classic orthogonal polynomials, and many others. The calculus method extends also to two- (or more-) indexed sequences.
In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers have been studied, and several integer sequences related to them have been introduced. In this article other types of Sheffer polynomials are... more
In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers have been studied, and several integer sequences related to them have been introduced. In this article other types of Sheffer polynomials are considered, by introducing the adjoint Peters and adjoint Pidduk polynomials.
In this work, we introduce an algebraic operation between bounded Hessenberg matrices and we analyze some of its properties. We call this operation mm-sum and we obtain an expression for it that involves the Cholesky factorization of the... more
In this work, we introduce an algebraic operation between bounded Hessenberg matrices and we analyze some of its properties. We call this operation mm-sum and we obtain an expression for it that involves the Cholesky factorization of the corresponding Hermitian positive definite matrices associated with the Hessenberg components.This work extends a method to obtain the Hessenberg matrix of the sum of measures from the Hessenberg matrices of the individual measures, introduced recently by the authors for subnormal matrices, to matrices which are not necessarily subnormal.Moreover, we give some examples and we obtain the explicit formula for the mm-sum of a weighted shift. In particular, we construct an interesting example: a subnormal Hessenberg matrix obtained as the mm-sum of two not subnormal Hessenberg matrices.► We introduce an algebraic operation, mm-sum, between bounded Hessenberg matrices. ► The mm-sum is obtained from sums of Hermitian positive definite matrices associated. ► The mm-sum expression involves the Cholesky factors of the associated HPD matrices. ► We analyze some properties of the mm-sum as subnormality or hyponormality. ► We obtain the explicit formula for the mm-sum of a weighted shift.
This paper is concerned with the construction of a class of polynomial orthogonal with respect to the weight function w(x) = 1 − x ^2 over the interval [0, 1]. The zeros of these polynomials were employed as points of collocation for the... more
This paper is concerned with the construction of a class of polynomial orthogonal with respect to the weight function w(x) = 1 − x ^2 over the interval [0, 1]. The zeros of these polynomials were employed as points of collocation for the orthogonal collocation technique in the solution of integral equations. The method is illustrated with some numerical examples and the results obtained show that the method is effective.
A description of Orthogonal Tensor Hermite Polynomials in 3-D is presented. These polynomials, as introduced by Grad in 1949 [1], can be used to obtain a series solution to the Boltzmann Transport Equation. The properties that are... more
A description of Orthogonal Tensor Hermite Polynomials in 3-D is presented. These polynomials, as introduced by Grad in 1949 [1], can be used to obtain a series solution to the Boltzmann Transport Equation. The properties that are explored are scaling, translation and rotation. Order 6 Hermite Tensors are studied while obtaining the rotation relations. From the scaling of the independent variables of particle velocities, a criterion on temperature is obtained which implies that the equation can be applied to binary gas mixtures only if the temperature of the hotter constituent is less than four times that of the cooler one. This criterion and other properties of the tensor hermite polynomials obtained in this paper can be used to study gas dynamics in the thermosphere.
Abstract: In this paper we show how polynomial mappings of degree $\ mathfrak {K} $ from a union of disjoint intervals onto $[-1, 1] $ generate a countable number of special cases of generalizations of Chebyshev polynomials. We also... more
Abstract: In this paper we show how polynomial mappings of degree $\ mathfrak {K} $ from a union of disjoint intervals onto $[-1, 1] $ generate a countable number of special cases of generalizations of Chebyshev polynomials. We also derive a new expression for these generalized Chebyshev polynomials for any genus $ g $, from which the coefficients of $ x^ n $ can be found explicitly in terms of the branch points and the recurrence coefficients.
We consider a random walk X n in ℤ+, starting at X 0=x≥0, with transition probabilities $$\mathbb{P}(X_{n+1}=X_{n}\pm1|X_{n}=y\ge1)={1\over2}\mp{\delta\over4y+2\delta}$$ and X n+1=1 whenever X n =0. We prove $\mathbb... more
We consider a random walk X n in ℤ+, starting at X 0=x≥0, with transition probabilities $$\mathbb{P}(X_{n+1}=X_{n}\pm1|X_{n}=y\ge1)={1\over2}\mp{\delta\over4y+2\delta}$$ and X n+1=1 whenever X n =0. We prove $\mathbb {E}X_{n}\sim\mathrm{const.}\,n^{1-{\delta \over2}}$ as n ↗∞ when δ∈(1,2). The proof is based upon the Karlin-McGregor spectral representation, which is made explicit for this random walk.
By convention, the translation and scale invariant functions of Legendre moments are achieved by using a combination of the corresponding invariants of geometric moments. They can also be accomplished by normalizing the translated and/or... more
$\textbf{D}_{u}$-classical orthogonal polynomial sequences are defined through the $\textbf{D}_{u}$-Hahn's property: sequences that are orthogonal and their $\textbf{D}_{u}$-first derivative, where $\textbf{D}_{u}(p)=p'+u\theta_0p,$ for... more
$\textbf{D}_{u}$-classical orthogonal polynomial sequences are defined through the $\textbf{D}_{u}$-Hahn's property: sequences that are orthogonal and their $\textbf{D}_{u}$-first derivative, where $\textbf{D}_{u}(p)=p'+u\theta_0p,$ for all $p\in\mathbb{C}[X]$. We characterize them by means of a functional equation, a $\textbf{D}_{u}$-second order linear differential equation, the first and the second structure relation. A $\textbf{D}_{u}$-classical orthogonal sequence is especially a $\textbf{D}$-Laguerre-Hahn sequence of class less than or equal to two. A complete classification of the $\textbf{D}_{u}$-classical sequences is obtained. The functional equation coefficients, the structure relation coefficients and the class are whenever given.
Abstract In this paper we study polynomials that are orthogonal with respect to a weight function which is zero on a set of positive measure. These were initially introduced by Akhiezer as a generalization of the Chebyshev polynomials... more
Abstract In this paper we study polynomials that are orthogonal with respect to a weight function which is zero on a set of positive measure. These were initially introduced by Akhiezer as a generalization of the Chebyshev polynomials where the interval of orthogonality is [− 1, α]∪[β, 1]. Here, this concept is extended and the interval is the union of g+ 1 disjoint intervals,[− 1, α1]∪ g− 1 j= 1 [βj, αj+ 1]∪[βg, 1], denoted by E.
We consider the problem of reconstructing an unknown function f on a domain X from samples of f at n randomly chosen points with respect to a given measure ρ. Given a sequence of linear spaces (V_m)m>0 with dim(V_m) = m <= n, we study the... more
We consider the problem of reconstructing an unknown function f on a domain X from samples of f at n randomly chosen points with respect to a given measure ρ. Given a sequence of linear spaces (V_m)m>0 with dim(V_m) = m <= n, we study the least squares approximations from the spaces V_m. It is well known that such approximations can be inaccurate when m is too close to n, even when the samples are noiseless. Our main result provides a criterion on m that describes the needed amount of regularization to ensure that the least squares method is stable and that its accuracy, measured in L2(X;ρ), is comparable to the best approximation error of f by elements from V_m. We illustrate this criterion for various approximation schemes, such as trigonometric polynomials, with ρ being the uniform measure, and algebraic polynomials, with ρ being either the uniform or Chebyshev measure. For such examples we also prove similar stability results using deterministic samples that are equispaced with respect to these measures.
We obtain an extension of the Christoffel–Darboux formula for matrix orthogonal polynomials with a generalized Hankel symmetry, including the Adler–van Moerbeke generalized orthogonal polynomials.
Abstract: We revisit the ladder operators for orthogonal polynomials and re-interpret two supplementary conditions as compatibility conditions of two linear over-determined systems; one involves the variation of the polynomials with... more
Abstract: We revisit the ladder operators for orthogonal polynomials and re-interpret two supplementary conditions as compatibility conditions of two linear over-determined systems; one involves the variation of the polynomials with respect to the variable $ z $(spectral parameter) and the other a recurrence relation in $ n $(the lattice variable). For the Jacobi weight
Second kind Chebyshev polynomials are modified set of defined Chebyshev polynomials by a slightly different generating function. This paper presents new and efficient algorithm for achieving an analytical approximate solution to optimal... more
Second kind Chebyshev polynomials are modified set of defined Chebyshev polynomials by a slightly different generating function. This paper presents new and efficient algorithm for achieving an analytical approximate solution to optimal control problems. The proposed solution is based on state parameterization, such that the state variable is approximated by the second kind Chebyshev polynomials with unknown coefficients. At first, the equation of motion, boundary conditions and performance index are changed into some algebraic equations. This task converts the optimal control problem into an optimization problem, which can then be solved easily. The presented technique approximates the control and state variables as a function of time. After optimizing, the system is converted into a feedback mode for having the closed loop profits. The results proved the algorithm convergence. Finally by analyzing two numerical examples, the reliability and effectiveness of the proposed method by comparing two different methods is demonstrated.
We show that for many families of OPUC, one has ‖φn′‖2/n→1, a condition we call normal behavior. We prove that this implies |αn|→0|αn|→0 and that it holds if ∑n=0∞|αn|<∞. We also prove it is true for many sparse sequences. On the other... more
We show that for many families of OPUC, one has ‖φn′‖2/n→1, a condition we call normal behavior. We prove that this implies |αn|→0|αn|→0 and that it holds if ∑n=0∞|αn|<∞. We also prove it is true for many sparse sequences. On the other hand, it is often destroyed by the insertion of a mass point.
We derive lower bounds for the L p (µ) norms of monic extremal poly-nomials with respect to compactly supported probability measures µ. We obtain a sharp universal lower bound for all 0 < p < ∞ and all measures in the Szeg˝ o class and an... more
We derive lower bounds for the L p (µ) norms of monic extremal poly-nomials with respect to compactly supported probability measures µ. We obtain a sharp universal lower bound for all 0 < p < ∞ and all measures in the Szeg˝ o class and an improved lower bound on L 2 (µ) norm for several classes of orthogonal polynomials including Jacobi polynomials, isospectral torus of a finite gap set and orthogonal polynomials with respect to the equilibrium measure of an arbitrary non-polar compact subset of R.
Multiple orthogonality is considered in the realm of a Gauss–Borel factorization problem for a semi-infinite moment matrix. Perfect combinations of weights and a finite Borel measure are constructed in terms of M-Nikishin systems. These... more
Multiple orthogonality is considered in the realm of a Gauss–Borel factorization problem for a semi-infinite moment matrix. Perfect combinations of weights and a finite Borel measure are constructed in terms of M-Nikishin systems. These perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution, that can be obtained from the solution of a Gauss–Borel factorization problem for a semi-infinite matrix, which plays the role of a moment matrix. This leads to sequences of multiple orthogonal polynomials, their duals and second kind functions. It also gives the corresponding linear forms that are bi-orthogonal to the dual linear forms. Expressions for these objects in terms of determinants from the moment matrix are given, recursion relations are found, which imply a multi-diagonal Jacobi type matrix with snake shape, and results like the ABC theorem or the Christoffel–Darboux formula are re-derived in this context (using the factorization problem and the generalized Hankel symmetry of the moment matrix). The connection between this description of multiple orthogonality and the multi-component 2D Toda hierarchy, which can be also understood and studied through a Gauss–Borel factorization problem, is discussed. Deformations of the weights, natural for M-Nikishin systems, are considered and the correspondence with solutions to the integrable hierarchy, represented as a collection of Lax equations, is explored. Corresponding Lax and Zakharov–Shabat matrices as well as wave functions and their adjoints are determined. The construction of discrete flows is discussed in terms of Miwa transformations which involve Darboux transformations for the multiple orthogonality conditions. The bilinear equations are derived and the τ-function representation of the multiple orthogonality is given.
Digital Video Broadcasting – Terrestrial (DVB-T) has become a very popular technology for terrestrial digital television services. DVB-T is based on Orthogonal Frequency Division Multiplexing (OFDM) technique. OFDM is considered suitable... more
Digital Video Broadcasting – Terrestrial (DVB-T) has become a very popular technology for terrestrial digital television services. DVB-T is based on Orthogonal Frequency Division Multiplexing (OFDM) technique. OFDM is considered suitable for DVB-T system because of its low Inter-Symbol Interference (ISI). DVB-T has some limitations too including large dynamic signal range and sensitivity to frequency error. To overcome these limitations a good DVB-T receiver is a must. In this paper we address these issues. This paper has two-fold objectives (i) to investigate the performances of DVB-T system under different channel conditions, and (ii) to improve performance of DVB-T system by selecting suitable filters in receiver. To investigate the performance of DVB-T system we have considered some popular channel models namely AWGN, Rayleigh, and Ricean. In order to improve the system performance some classic filters like Butterworth, Chebyshev, and elliptic have been included in the receiver. The simulation results show that a careful selection of filter is a must for a DVB-T system. It is also shown that the filter selection should be based on the underlying channel conditions.
This study introduces a new set of orthogonal polynomials and moments and the set's application in signal and image processing. This polynomial is derived from two well-known orthogonal polynomials: the Tchebichef and Krawtchouk... more
We show that for many families of OPUC, one has ‖ϕ ′ n‖2/n → 1, a condition we call normal behavior. We prove that this implies |αn | → 0 and that it holds if ∑∞ n=0 |αn | < ∞. We also prove it is true for many sparse sequences. On the... more
We show that for many families of OPUC, one has ‖ϕ ′ n‖2/n → 1, a condition we call normal behavior. We prove that this implies |αn | → 0 and that it holds if ∑∞ n=0 |αn | < ∞. We also prove it is true for many sparse sequences. On the other hand, it
