Classical orthogonal polynomials: A functional approach

We characterize the so-called classical orthogonal polynomials (Hermite, Laguerre, Jacobi, and Bessel) using the distributional differential equation D(u)=u. This result is naturally not new. However, other characterizations of classical orthogonal polynomials can be obtained more easily from this approach. Moreover, three new properties are obtained.     

On one set of orthogonal harmonic polynomials

A new basis of harmonic polynomials is given. Proposed polynomials are orthogonal on the unit sphere and each term of this basis consists of monomials not present in the others.

     

Classical orthogonal polynomials in two variables: a matrix approach

Classical orthogonal polynomials in two variables are defined as the orthogonal polynomials associated to a two-variable moment functional satisfying a matrix analogue of the Pearson differential equation. Furthermore, we characterize classical orthogonal polynomials in two variables as the polynomial solutions of a matrix second order partial differential equation. AMS subject classification 42C05, 33C50Partially supported by Ministerio de Ciencia y Tecnología (MCYT) of Spain and by the European Regional Development Fund (ERDF) through the grant BFM2001-3878-C02-02, Junta de Andalucía, G.I. FQM 0229 and INTAS Project 2000-272.     

Twisted exponential sums of polynomials in one variable

The twisted T-adic exponential sums associated to a polynomial in one variable are studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the C-function of the twisted T-adic exponential sums. This bound gives lower bounds for the Newton polygon of the L-function of twisted p-power order exponential sums.     

Orthonormal bases of polynomials in one complex variable

Let a sequence (P _n) of polynomials in one complex variable satisfy a recurrence relation with length growing more slowly than linearly. It is shown that (P _n) is an orthonormal basis in L ² for some measure on C, if and only if the recurrence is a 3-term relation with special coefficients. The support of lies on a straight line. This result is achieved by the analysis of a formally normal irreducible Hessenberg operator with only finitely many nonzero entries in every row. It generalizes the classical Favard's Theorem and the Representation Theorem.     

q-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials

We obtain algebraic relations (identities) for q-numbers that do not contain q ^α-factors. We derive a formula that expresses any q-number [x] in terms of the q-number [2]. We establish the relationship between the q-numbers [n] and the Fibonacci numbers, Chebyshev polynomials, and other special functions. The sums of combinations of q-numbers, in particular, the sums of their powers, are calculated. Linear and bilinear generating functions are found for “natural” q-numbers. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1055–1063, August, 1998.     

A family of discrete semi-classical orthogonal polynomials of class one

In this work, we solve the system of Laguerre–Freud equations for the recurrence coefficients \(\beta _n\), \(\gamma _{n+1} , n \ge 0\) of the \(D_{w}\)-semi-classical orthogonal polynomials sequences of class one in the case when \(\beta _{0}=-t_{0}\), \(\beta _{n+1}=t_{n}-t_{n+1}\) and \(\gamma _{n+1}=-t_{n}^{2}\) with \(t_{n}\ne 0\;n\ge 0\), where \(D_w\) is the divided difference operator. There are essentially four canonical families.     

The symmetric D ω -semi-classical orthogonal polynomials of class one

We give the system of Laguerre–Freud equations associated with the D _ω -semi-classical functionals of class one, where D _ω is the divided difference operator. This system is solved in the symmetric case. There are essentially two canonical cases. The corresponding integral representations are given.     

Rational approximation to surfaces defined by polynomials in one variable

For a Tychonoff space X, we denote by C _p (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence.

In this paper we prove that:

• If every finite power of X is Lindelöf then C _p (X) is strongly sequentially separable iff X is \({\gamma}\)-set.
• \({B_{\alpha}(X)}\) (= functions of Baire class \({\alpha}\) (\({1 < \alpha \leq \omega_1}\)) on a Tychonoff space X with the pointwise topology) is sequentially separable iff there exists a Baire isomorphism class \({\alpha}\) from a space X onto a \({\sigma}\)-set.
• \({B_{\alpha}(X)}\) is strongly sequentially separable iff \({iw(X)=\aleph_0}\) and X is a \({Z^{\alpha}}\)-cover \({\gamma}\)-set for \({0 < \alpha \leq \omega_1}\).
• There is a consistent example of a set of reals X such that C _p (X) is strongly sequentially separable but B₁(X) is not strongly sequentially separable.
• B(X) is sequentially separable but is not strongly sequentially separable for a \({\mathfrak{b}}\)-Sierpiński set X.

     

Short-cut distillation columns based on orthogonal polynomials of a discrete variable

Increasing competition in the process industries enforces theapplication of mathematical simulation techniques both in thedesign phase and in the operating phase of a plant. A basicapparatus for separation processes is the distillation column.Its rigorous (tray by tray) mathematical modelling results ina system of simultaneous nonlinear equations (algebraic in thesteady-state case, differential-algebraic in the dynamic case).For high (and realistic) numbers of trays and components, thesesystems may become rather large (thousands of equations). Inaddition, realistic plant models often include several distillationcolumns. As a consequence, the numerical solution of these modelsmay become difficult and time-consuming. This has led to attemptsto model the distillation columns less rigorously with the aimof achieving a considerable reduction in the number of equations.The name shortcut distillation columns is common for modelsof this type. The present paper uses a discrete weighted residualmethod for the development of short-cut models. It suggestsa Galerkin method based on orthogonal polynomials in a discretevariable: the tray number. It is a remarkable advantage of thistechnique that even very coarse models satisfy all global balancesexactly.     

Companion orthogonal polynomials

We give some properties relating the recurrence relations of orthogonal polynomials associated with any two symmetric distributions dφ₁(x) and d₂(x) such that dφ₂(x) = (1 + kx²)d₁(x). As applications of properties, recurrence relations for many interesting systems of orthogonal polynomials are obtained.     

Kravchuk orthogonal polynomials

A survey of the principal works of Academician M. P. Kravchuk and his students in the area of orthogonal polynomials of a discrete variable is presented. The value of these studies for the further development of the theory, for drawing generalization, and for the construction of different applications of this class of special functions is noted.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 7, pp. 880–888, July, 1992.     

On orthogonal polynomials

Пустьw(х)∈L[-1, +1] — неотрица тельная функция така я, что $$\frac{{\log ^ + \frac{1}{{w(x)}}}}{{\sqrt {1 - x^2 } }} \in L[ - 1, + 1]$$ и пусть {(р _n (х)} — много члены, ортогональные и нормированные с весо мw(x). Мы доказываем следующие две теорем ы, являющиеся обобщен ием одного известного результа та Н. Винера. I. Для каждого δ, 0<δ<1, суще ствует числоB=B(δ, w) тако е, что если $$f_N (x) = \sum\limits_{j = 1}^N {a_j p_{v_j } (x)} $$ причем выполнено сле дующее условие лакун арности $$\begin{gathered} v_{j + 1} - v_j \geqq B(\delta ,w) (j = 1,2,...,N - 1), \hfill \\ v_1 \geqq B(\delta ,w) \hfill \\ \end{gathered} $$ , то для некоторого С(δ, w) и всехh и δ, для которых $$ - 1 \leqq h - \delta< h + \delta \leqq + 1$$ , имеет место неравенс тво $$\int\limits_{ - 1}^1 {|f_N (x)|^2 w(x)dx \leqq C(\delta ,w)} \int\limits_{h - \delta }^{h + \delta } {|f_N (x)|^2 w(x)dx} $$ каковы бы ни былиa _j ,N и h. II. Если формальный ряд $$\sum\limits_{j = 1}^\infty {b_j p_{\mu _j } (x)} $$ удовлетворяет услов ию лакунарности μ_j+1-μ_j→∞ и суммируем, например, м етодом Абеля на произвольно малом отрезке [а, Ь] ?[0,1] к ф ункцииf(x) такой, что \(f(x)\sqrt {w(x)} \in L_2 [a,b]\) , то $$\sum\limits_j {|b_j |^2< \infty } $$ Теорема I — это первый ш аг в направлении проб лемы типа Мюнтца-Саса о замкнут ости подпоследовательно сти pv_j(x)} последовател ьности {р_n(х)} на отрезке [а, Ь] в метрике С[а, Ь] (см. теорему II стать и).     

Constrained energy problems with applications to orthogonal polynomials of a discrete variable

Given a positive measure Σ with gs > 1, we write Με ℳ^Σ if Μ is a probability measure and Σ—Μ is a positive measure. Under some general assumptions on the constraining measure Σ and a weight functionw, we prove existence and uniqueness of a measure λ^Σ _w that minimizes the weighted logarithmic energy over the class ℳ^Σ. We also obtain a characterization theorem, a saturation result and a balayage representation for the measure λ^Σ _w As applications of our results, we determine the (normalized) limiting zero distribution for ray sequences of a class of orthogonal polynomials of a discrete variable. Explicit results are given for the class of Krawtchouk polynomials. The research done by this author is in partial fulfillment of the Ph.D. requirements at the University of South Florida. The research done by this author was supported, in part, by U.S. National Science Foundation under grant DMS-9501130.