Generalized Functions Research Articles

Homogeneous generalized functions with respect to one-parametric group

We give the full description of homogeneous generalized functions along the trajectories of arbitrary one-parametric multiplicative group of linear transformations whose generator matrix has eigenvalues with positive real parts.We also study the problem of extension of such functionals from the space of test functions vanishing at the origin up to the whole space S(ℝn), and discuss the conditions of uniqueness of such extension.

Preface – Generalized Functions and Applications

Preface – Generalized Functions and Applications

Multidimensional Generalized Functions in Aeroacoustics and Fluid Mechanics-Part 1: Basic Concepts and Operations

This paper is the first part of a three part tutorial on multidimensional generalized functions (GFs) and their applications in aeroacoustics and fluid mechanics. The subject is fascinating and essential in many areas of science and, in particular, wave propagation problems. In this tutorial, we strive to present rigorously and clearly the basic concepts and the tools that are needed to use GFs in applications effectively and with ease. We give many examples to help the readers in understanding the mathematical ideas presented here. The first part of the tutorial is on the basic concepts of GFs. Here we define GFs, their properties and some common operations on them. We define the important concept of generalized differentiation and then give some interesting elementary and advanced examples on Green's functions and wave propagation problems. Here, the analytic power of GFs in applications is demonstrated with ease and elegance. Part 2 of this tutorial is on the diverse applications of generalized derivatives (GDs). Part 3 is on generalized Fourier transformations and some more advanced topics. One goal of writing this tutorial is to convince readers that, because of their powerful operational properties, GFs are essential and useful in engineering and physics, particularly in aeroacoustics and fluid mechanics.

Approximation properties of local smoothing kernels

We study some properties of smoothing kernels and their local expression as they appear in the construction of Colombeau-type generalized function algebras that are diffeomorphism-invariant.

Accurate simulations of 2-D phase shift masks with a generalized discontinuous Galerkin (GDG) method

In this paper, we apply a newly developed generalized discontinuous Galerkin (GDG) method for rigorous simulations of 2-D phase shift masks (PSM). The main advantage of the GDG method is its accurate treatment of jumps in solutions using the Dirac $\delta$ generalized functions as source terms of partial differential equations. The scattering problem of the PSM is cast with a total field/scattering field formulation while the GDG method is used to handle the inhomogeneous jump conditions between the total and scattering fields along the physical and perfectly matched layer (PML) interfaces. Numerical results demonstrate the high order accuracy of the GDG method and its capability of handling the non-periodic structures such as optical images near mask edges.

A completeness criterion for Kaniadakis, Abe and two-parameter generalized statistical theories

We recently provided a criterion of completeness valid for any generalized thermostatistics to check whether they form a bijection from ℝ + /ℝ (set of positive real numbers/all real numbers) to ℝ/ℝ + in a previous paper. In the current work, we apply this criterion to Kaniadakis, Abe and two-parameter generalized functions and obtain their respective validity ranges.

Higher order symmetric duality in nondifferentiable multi-objective programming problems involving generalized cone convex functions

A pair of Mond–Weir type multi-objective higher order symmetric dual programs over arbitrary cones is formulated. Weak, strong and converse duality theorems are established under higher order K – F -convexity assumptions. Our results generalize several known results in the literature.

Lattice path combinatorics for multiple product identities

Lattice paths are enumerated as walks on a lattice under the Delannoy criterion of vertical, horizontal and upward diagonal steps. Delannoy recursions are generalized for arbitrary weights in each of these directions, where the row elements of the Delannoy triangles correspond to lattice paths for all walks at a given length. This paper surveys known connections between generating functions for Delannoy paths, orthogonal polynomials, and multiple product identities. Generating functions for these sets of weighted lattice paths are derived for the individual triangles as well as for multiple triangle overlays. Recursive algorithms and series expansions are used to describe the basic structure of generalized Delannoy triangles. Ordinary generating functions are expanded to yield lacunary sequences in the Delannoy row summations. The lattice paths are enumerated by generalized Chebyshev polynomials of the first and second kind, as well as being evaluated by Jacobi polynomials, Legendre polynomials, and Zernike polynomials. The Jacobi triple product and Watson quintuple product identities are extended to balanced product identities. Elements of Delannoy number triangles are derived from these multiple product identities as coefficients in their power series expansions. Results for lacunary Fibonacci and Lucas number sequences are highlighted as special cases for many of the multiple product identities. For the most generalized variable selection, Jacobi polynomials and hypergeometric functions are used to derive expansion terms in the triple and quintuple products. Interpretation of the lattice path weights in exponential terms is shown to generate trigonometric theta functions.

Harmonic generalized functions in generalized function algebras

We investigate basic properties of harmonic generalized functions within the framework of J. F. Colombeau’s theory of generalized functions. In particular, we present various theorems concerning the Maximum principle, Liouville’s theorem, singularities and Poisson formula.

Solutions of nonlinear differential equations

We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in the algebra of new generalized functions. The solution of such an equation is a new generalized function. In this article we formulate necessary and sufficient conditions for when the solution of the given equation in the algebra of new generalized functions is associated with an ordinary function. Moreover, a class of all possible associated functions is described.

Impulse controls in models of hybrid systems

We consider a mathematical model of a hybrid system in which the continuous dynamics induced at each instant of time by a certain system of differential equations from a given finite family is alternated with discrete operations sending commands either for an instantaneous switching from one system to another, or for an instantaneous reset from online coordinates to other coordinates, or for both actions together. We present a description of a hybrid system via systems with impulses. We suggest a new scheme, which permits one to reduce the problem of synthesis of controls for a hybrid system to an impulse control problem. The aim of the present paper is to find a control containing both ordinary components (that are not generalized functions) and higher-order generalized functions.

Delta distributions supported on quadratic O(p,q)-invariant surfaces

An important subset of distributions encountered in physics is the set of multiplet delta distributions δcy(k), with support a quadratic O(p,q)-invariant surface cy≜{x∊Rn:P(x)=y}, n=p+q. The evaluation of these distributions for a general test function is not always sufficiently detailed in the classical literature, especially for those distributions that are defined as a regularization and/or when one needs their causal or anticausal version (for p=1 or q=1). This work intends to improve this situation by deriving explicit expressions for ⟨δcy(k),φ⟩, ∀p,q∊Z+, ∀k∊N, ∀y∊R, and ∀φ∊D(Rn), in a form suitable for practical applications. In addition, we also apply to these distributions a new approach to regularization. Distributions that need to be regularized are (equivalently) defined as extensions of a partial distribution. This extension process reveals in a natural way that regularized multiplet delta distributions are in general, uncountably multivalued. In the work of Gel’fand and Shilov [Generalized Functions (Academic, New York, 1964), Vol. 1] four types of multiplet delta distributions with support the null space c0 were introduced: δ1(k)(P), δ2(k)(P) and δ(k)(P+), δ(k)(P−). Our regularization method explains why this particular nonuniqueness was observed and further discloses the full extent of this nonuniqueness.

A global approach to generalized function algebras

We consider distributions on a closed compact manifold as maps on smoothing operators to smooth functions. For any subgroup of the diffeomorphism group, we produce a full-type algebra of generalized functions and an equivariant embedding of distributions into this algebra. As examples, we consider groups of Riemannian isometries and the symplectic group.

Nonautonomous differential equations in the algebra of new generalized functions

We study a nonautonomous equation with generalized coefficients in an algebra of generalized functions. The solutions of the equation can be rather different depending on the interpretation of the equation. We show that all these solutions can be obtained from the solution of this equation in the algebra of generalized functions.

Vanishing Theorems in Colombeau Algebras of Generalized Functions

AbstractUsing a canonical linear embedding of the algebra of Colombeau generalized functions in the space of -valued ℂ-linear maps on the space of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class . These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.

A concise introduction to Colombeau generalized functions and their applications in classical electrodynamics

The objective of this introduction to Colombeau algebras of generalized functions (in which distributions can be freely multiplied) is to explain in elementary terms the essential concepts necessary for their application to basic nonlinear problems in classical physics. Examples are given in hydrodynamics and electrodynamics. The problem of the self-energy of a point electric charge is worked out in detail: the Coulomb potential and field are defined as Colombeau generalized functions, and integrals of nonlinear expressions corresponding to products of distributions (such as the square of the Coulomb field and the square of the delta function) are calculated. Finally, the methods introduced in Gsponer (2007 Eur. J. Phys. 28 267, 2007 Eur. J. Phys. 28 1021 and 2007 Eur. J. Phys. 28 1241), to deal with point-like singularities in classical electrodynamics are confirmed.

Algebraic and Geometric Theory of the Topological Ring of Colombeau Generalized Functions

AbstractWe continue the investigation of the algebraic and topological structure of the algebra of Colombeau generalized functions with the aim of building up the algebraic basis for the theory of these functions. This was started in a previous work of Aragona and Juriaans, where the algebraic and topological structure of the Colombeau generalized numbers were studied. Here, among other important things, we determine completely the minimal primes of and introduce several invariants of the ideals of (Ω). The main tools we use are the algebraic results obtained by Aragona and Juriaans and the theory of differential calculus on generalized manifolds developed by Aragona and co-workers. The main achievement of the differential calculus is that all classical objects, such as distributions, become C∞-functions. Our purpose is to build an independent and intrinsic theory for Colombeau generalized functions and place them in a wider context.

Pointwise characterizations in generalized function algebras

We define the algebra of Colombeau generalized functions on the space of generalized points of {\mathbb R}^d which naturally contains the tempered generalized functions. The subalgebra of \mathscr{S}-regular generalized functions of this algebra is characterized by a pointwise property of the generalized functions and their Fourier transforms. We also characterize the equality in the sense of generalized tempered distributions for certain elements of this algebra (namely those with so-called slow scale support) by means of a pointwise property of their Fourier transforms. Further, we show that (contrary to what has been claimed in the literature) for an open subset \Omega of {\mathbb R}^d, the algebra of pointwise regular generalized functions \dot{\mathcal G}^\infty(\Omega) equals {\mathcal G}^\infty(\Omega) and give several characterizations of pointwise {\mathcal G}^\infty-regular generalized functions.

Generalized functions obtained by the regularization of nonintegrable functions

Generalized functions obtained by the regularization of nonintegrable functions

Foundations of the Calculus of Variations in Generalized Function Algebras

We propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we introduce the first and second variation of a variational problem. We then derive necessary (Euler-Lagrange equations) and sufficient conditions for extremals. The concept of association is used to obtain connections to a distributional description of singular variational problems. We study variational symmetries and derive an appropriate version of Nother’s theorem. Finally, a number of applications to geometry, mechanics, elastostatics and elastodynamics are presented.