Class Of Special Functions Research Articles

Orthonormal bases with nonlinear phases

For adaptive representation of nonlinear signals, the bank ${\cal M}$ of real square integrable functions that have nonlinear phases and nonnegative instantaneous frequencies under the analytic signal method is investigated. A particular class of functions with explicit expressions in ${\cal M}$ is obtained using recent results on the Bedrosian identity. We then construct orthonormal bases for the Hilbert space of real square integrable functions with the basis functions from ${\cal M}$ .

Unified formulas for arbitrary order symbolic derivatives and anti-derivatives of the power-inverse hyperbolic class 1

We continue on tackling and giving a complete solution to the problem of finding the nth derivative and the nth anti-derivative, where n can be an integer, a fraction, a real, or a symbol, of elementary and special classes of functions. In general, the solutions are given through unified formulas in terms of the Fox H-function which in many cases can be simplified to less general functions. In this work, we consider two subclasses of the power-inverse hyperbolic class. Namely, the power-inverse hyperbolic sine class { f ( x ) : f ( x ) = Σ l j =1 Pj ( x α j )arcsinh(β j x γ j ), α j ∈ C, β j ∈ C\{0},γ j ∈ R\{0}, (1) and the power-inverse hyperbolic cosine class { f ( x ) : f ( x ) = Σ l j =1 Pj ( x α j )arccosh(β j x γ j ), α j ∈ C, β j ∈ C\{0},γ j ∈ R\{0}, (2) where pj's are polynomials of certain degrees. One of the key points in this work is that the approach does not depend on integration techniques The arbitrary order of differentiation is found according to the Riemann-Liouville definition, whereas the generalized Cauchy n-fold integral is adopted for arbitrary order of integration. The motivation of this work comes from the area of symbolic computation. The idea is that: Given a function f in a variable x , can CAS find a formula for the n th derivative, the n th anti-derivative, or both of f ? This enhances the power of integration and differentiation of CAS. In Maple, the formulas correspond to invoking the commands diff( f ( x ) for the n th derivative and int( f ( x ), x$n ) for the n th anti-derivative. A software exhibition will be given using Maple. Example: A unified formula for arcsinh(√ x ) in terms of the Meijer G-function (arcsinh(√ x )) (n) = x (1/2-- n over2√π G 1,2 over 1,2 (1/2,1/2over0, n --1/2│ x ) , │ x │ < 1. (3). The above G-function reduces to the original function if n = 0. It gives derivatives of any order if n > 0 and anti-derivatives of any order if n < 0.

Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials

We prove that the Ciarlet–Nečas non-interpenetration of matter condition can be extended to the case of deformations of hyperelastic brittle materials belonging to the class of special functions of bounded variation (SBV), and can be taken into account for some variational models in fracture mechanics. In order to formulate such a condition, we define the deformed configuration under an SBV map by means of the approximately differentiable representative, and we prove some connected stability results under weak convergence. We provide an application to the case of brittle Ogden materials.

A Graphical Model for Evolutionary Optimization

We present a statistical model of empirical optimization that admits the creation of algorithms with explicit and intuitively defined desiderata. Because No Free Lunch theorems dictate that no optimization algorithm can be considered more efficient than any other when considering all possible functions, the desired function class plays a prominent role in the model. In particular, this provides a direct way to answer the traditionally difficult question of what algorithm is best matched to a particular class of functions. Among the benefits of the model are the ability to specify the function class in a straightforward manner, a natural way to specify noisy or dynamic functions, and a new source of insight into No Free Lunch theorems for optimization.

Autoconvolution equations and special functions

Autoconvolution equations of the third kind are derived for some classes of special functions as generalized Mittag–Leffler functions E α, β, error functions, and generalized Volterra functions ν, μ. These equations generalize the well-known equation by Bernstein and Doetsch for the Mittag–Leffler function E α and an equation by Janno and the author for the Volterra function ν. In addition, some autoconvolution equations of the first kind for error functions and two integro-differential equations with autoconvolution integral for the derivative of Volterra functions are given.

Luigi Gatteschi’s work on asymptotics of special functions and their zeros

A good portion of Gatteschi’s research publications—about 65%—is devoted to asymptotics of special functions and their zeros. Most prominently among the special functions studied figure classical orthogonal polynomials, notably Jacobi polynomials and their special cases, Laguerre polynomials, and Hermite polynomials by implication. Other important classes of special functions dealt with are Bessel functions of the first and second kind, Airy functions, and confluent hypergeometric functions, both in Tricomi’s and Whittaker’s form. This work is reviewed here, and organized along methodological lines.

Operator Valued Functions with Pick Operators Having Negative Subspaces of Bounded Dimensions

Functions whose values are bounded linear Hilbert space operators (each operator may be defined on its own subspace of the ambient Hilbert space), the domain of definition is contained in the open unit disc, and having the following property κ, are studied. (κ): All Pick operators associated with the function have the dimensions of their spectral subspace corresponding to the negative part of the spectrum bounded above by a fixed nonnegative integer κ, and the bound κ is attained. No a priori hypotheses concerning regularity of the functions are assumed. A particular class of functions, called standard functions, is introduced, and the corresponding nonnegative integer κ is identified for standard functions. It is proved that every function with property (κ) can be extended to a standard function with property (κ), for the same κ. This result is interpreted as a result on interpolation. As an application, maximal (with respect to the extension relation) functions with the property κ, for a fixed κ, are studied in terms of standard functions.

Solution of a class of Volterra integral equations with singular and weakly singular kernels

A class of Volterra integral equations are solved in this paper, where the involved kernel may be weakly singular, singular, and hypersingular. For various cases, existence and uniqueness of such Volterra integral equations are established. Furthermore, by reducing such equations to ordinary differential equations, analytic solutions have been determined explicitly. When restricting the solutions to certain special function classes, the number of suitable solutions is given.

Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials

Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. The best-known example of this type is the Hermite functions, i.e., the Hermite polynomials multiplied by exp(-x 2 /2), which are eigenfunctions of the Fourier transform. In this paper, we introduce two new examples of finite systems of this type and obtain their orthogonality relations. We also estimate a complicated integral and propose a conjecture for a further example of finite orthogonal sequences.

Transcendence measures and algebraic growth of entire functions

In this paper we obtain estimates for certain transcendence measures of an entire function f. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial P(z,w) in ℂ2 along the graph of f. These inequalities provide, in turn, estimates for the number of zeros of the function P(z,f(z)) in the disk of radius r, in terms of the degree of P and of r.Our estimates hold for arbitrary entire functions f of finite order, and for a subsequence {n j } of degrees of polynomials. But for special classes of functions, including the Riemann ζ-function, they hold for all degrees and are asymptotically best possible. From this theory we derive lower estimates for a certain algebraic measure of a set of values f(E), in terms of the size of the set E.

Rational approximation of vertical segments

In many applications, observations are prone to imprecise measurements. When constructing a model based on such data, an approximation rather than an interpolation approach is needed. Very often a least squares approximation is used. Here we follow a different approach. A natural way for dealing with uncertainty in the data is by means of an uncertainty interval. We assume that the uncertainty in the independent variables is negligible and that for each observation an uncertainty interval can be given which contains the (unknown) exact value. To approximate such data we look for functions which intersect all uncertainty intervals. In the past this problem has been studied for polynomials, or more generally for functions which are linear in the unknown coefficients. Here we study the problem for a particular class of functions which are nonlinear in the unknown coefficients, namely rational functions. We show how to reduce the problem to a quadratic programming problem with a strictly convex objective function, yielding a unique rational function which intersects all uncertainty intervals and satisfies some additional properties. Compared to rational least squares approximation which reduces to a nonlinear optimization problem where the objective function may have many local minima, this makes the new approach attractive.

Second-order superintegrable quantum systems

A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n − 1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, then the system is second-order superintegrable, the most tractable case and the one we study here. Such systems have remarkable properties: multi-integrability and separability, a quadratic algebra of symmetries whose representation theory yields spectral information about the Schrodinger operator, and deep connections with expansion formulas relating classes of special functions. For n = 2 and for conformally flat spaces when n = 3, we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here, we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension.

A Fast Algorithm for the Calculation of the Roots of Special Functions

We describe a procedure for the determination of the roots of functions satisfying second-order ordinary differential equations, including the classical special functions. The scheme is based on a combination of the Prufer transform with the classical Taylor series method for the solution of ordinary differential equations and requires $O(1)$ operations for the determination of each root. When the functions in question are classical orthogonal polynomials (Legendre, Hermite, etc.), the techniques presented here also provide tools for the evaluation of the weights for the associated Gaussian quadratures. The performance of the scheme for several classical special functions (prolate spheroidal wave functions, Bessel functions, and Legendre, Hermite, and Laguerre polynomials) is illustrated with numerical examples.

Minimax inequality for a special class of functionals and its application for a two point boundary value problem

Minimax inequality for a special class of functionals and its application for a two point boundary value problem

Special functions: Approximations and bounds

The Steffensen inequality and bounds for the Cebysev functional are utilised to obtain bounds for some classical special functions. The technique relies on determining bounds on integrals of products of functions. The above techniques are used to obtain novel and useful bounds for the Bessel function of the first kind, the Beta function, and the Zeta function.

Wave catastrophes: Types of focusing in diffraction and propagation of electromagnetic waves

Theoretical and applied results obtained in the theory of diffraction and propagation of electromagnetic waves with the use of methods of wave catastrophes are reviewed. The basic concepts of the theory are considered: classification of wave catastrophes, methods for constructing uniform asymptotics applied to describe fields in such regions, and analysis of the field structure. A general description of classes of special functions involved in uniform asymptotic decompositions of wave fields, properties of these functions, and methods of their calculation are presented. Applied problems that are solved efficiently by means of methods of wave catastrophes are discussed.

Generalized Voigt functions and their derivatives

This paper deals with a special class of functions called generalized Voigt functions H ( n ) ( x , a ) and G ( n ) ( x , a ) and their partial derivatives, which are useful in the theory of polarized spectral line formation in stochastic media. For n = 0 they reduce to the usual Voigt and Faraday–Voigt functions H ( x , a ) and G ( x , a ) . A detailed study is made of these new functions. Simple recurrence relations are established and employed for the calculation of the functions themselves and of their partial derivatives. Asymptotic expansions are given for large values of x and a. They are used to examine the range of applicability of the recurrence relations and to construct a numerical algorithm for the calculation of the generalized Voigt functions and of their derivatives valid in a large ( x , a ) domain. It is also shown that the partial derivatives of the usual H ( x , a ) and G ( x , a ) can be expressed in terms of H ( n ) ( x , a ) and G ( n ) ( x , a ) .

Isospectral potentials: Special functions partners and factorization operators

In supersymmetric quantum mechanics (SUSY QM), the isospectral potentials are those involved in a Schrödinger equation, which remains invariant under the standard Darboux transform. In this work, we propose a reciprocal and a generalized Darboux transform to find the isospectral potentials having eigenfunctions related to special functions (SF) partners. To that, the differential equation for SF is converted into a Schrödinger-like equation allowing SF solutions for the corresponding potential-like involved. This in turn is related with the equivalent of the Witten superpotential, occurring in SUSY QM. The equivalent of the generalized Witten superpotential consequently leads to the generalized SF as well as to their factorization operators. As a useful application of the proposal, we explicitly consider the SF partners, factorization operators and isospectral potentials for some classical SF: Hermite, Laguerre and Legendre. However, the method is general and can be applied to obtain new isospectral potentials, which could be useful in quantum chemical applications.

On confluences of general hypergeometric systems

We extend the well-known results about the process of confluence for the Gauss hypergeometric differential equation to the case of general hypergeometric systems. We see that the process of confluence comes from the geometry of the set of regular elements of the Lie algebra of complex general linear group. As a consequence, we give a geometric and group-theoretic view on the process of confluence for classical special functions.

Grothendieck’s dessins d’enfants, their deformations, and algebraic solutions of the sixth Painlevé and Gauss hypergeometric equations

Dedicated to Ludwig Dmitrievich Faddeev on the occasion of his 70th birthday Abstract. Grothendieck's dessins d'enfants are applied to the theory of the sixth Painleve and Gauss hypergeometric functions, two classical special functions of iso- monodromy type. It is shown that higher-order transformations and the Schwarz table for the Gauss hypergeometric function are closely related to some particular Bely˘ i functions. Moreover, deformations of the dessins d'enfants are introduced, and it is shown that one-dimensional deformations are a useful tool for construction of algebraic sixth Painleve functions.