Solutions Of Linear Functional Equations Research Articles

Definable Continuous Solutions of Linear Equations

In this paper, we study a generalization of a question, raised by C. Fefferman and J. Kollár, on the existence of solutions of linear functional equations. Suppose that R is a definably complete expansion of a real closed field (R;+,⋅). Let f,g1,…,gk:Rn→R be continuous functions that are definable in R. We prove that if there exist continuous functions y1,…,yk:Rn→R such that f=g1y1+⋯+gkyk, then there exist continuous functions y1,…,yk such that y1,…,yk are definable in R and f=g1y1+⋯+gkyk.

Linear Functional Equations with 𝔇H-Unstable Solutions

We establish conditions for the absolute instability of solutions of linear functional equations with self-adjoint operator coefficients.

Algebraic methods for the solution of linear functional equations

The equation $$\sum^{n}_ {i=0} a_{i}f(b_{i}x + (1 - b_{i})y) = 0$$belongs to the class of linear functional equations. The solutions form a linear space with respect to the usual pointwise operations. According to the classical results of the theory they must be generalized polynomials. New investigations have been started a few years ago. They clarified that the existence of non-trivial solutions depends on the algebraic properties of some related families of parameters. The problem is to find the necessary and sufficient conditions for the existence of non-trivial solutions in terms of these kinds of properties. One of the earliest results is due to Z. Daróczy [1]. It can be considered as the solution of the problem in case of n = 2. We are going to take more steps forward by solving the problem in case of n = 3.

Nontrivial solutions of linear functional equations: methods and examples

For a wide class of linear functional equations the solutions are generalized polynomials. The existence of non-trivial monomial terms of the solution strongly depends on the algebraic properties of some related families of parameters. As a continuation of the previous work [A. Varga, Cs. Vincze, G. Kiss, Algebraic methods for the solution of linear functional equations, Acta Math. Hungar.] we are going to present constructive algebraic methods of the solution in some special cases. Explicit examples will be also given.

Existence of nontrivial solutions of linear functional equations

In this paper we study the functional equation $$\sum_{i=1}^n a_i f(b_i x+c_i h)=0 \quad (x, h \in \mathbb{C})$$ where a i , b i , c i are fixed complex numbers and \({f \colon \mathbb{C} \to \mathbb{C}}\) is the unknown function. We show, that if there is i such that \({b_i / c_i \neq b_j /c_j}\) holds for any \({1 \leq j \leq n, j \neq i}\), the functional equation has a nonconstant solution if and only if there are field automorphisms \({\phi_1, \ldots, \phi_k}\) of \({\mathbb{C}}\) such that \({\phi_1 \cdots \phi_k}\) is a solution of the equation.

On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations

This article consists of Chapter 2 of the author's 1954 Dissertation bearing the same title and published as Courant Institute report IMM-NYU 333, available in full on the author's personal web site. Chapter 2 consists of the first complete proof of what has come to be known as the Neimark-Sacker bifurcation theorem. It includes the reduction to normal form using weighted monomials which precludes using the Center Manifold theorem which was not known to the author and was published the same year as the dissertation and called “a Reduction Principle” by V. Pliss and later named Center Manifold theorem by A. Kelley (see previous article for citations). After reduction to normal form, the resulting functional equations for the bifurcating invariant curve is solved in detail. Stand-alone decimals provide solutions of linear functional equations, a-priori estimates and interpolation inequalities between derivatives in the sup norm.

On the existence of smooth solutions of linear functional equations

On the existence of smooth solutions of linear functional equations

Integration of solutions of linear functional equations

We introduce the notion of the adjoint Ore ring and give a definition of an adjoint polynomial, operator and equation. We apply this for integrating solutions of Ore equations.*

Similar Operators and a Functional Calculus for the First-Order Linear Differential Operator

The general problem of finding solutions of linear functional equations is undoubtedly one of the main problems of applied mathematics. Among such equations the classes of linear differential equations and linear difference equations are very important and numerous theories and meth- ods for their solution have been developed. The methods based on integral transforms and the operational methods are certainly the ones used most often. In the present paper we present some general linear algebra results that may be considered as a unified approach to the transform and operational methods. We use the linear algebra foundations of the Laplace transform and the concept of similarity to generalize a description of the general solution of linear differential equations with constant coefficients in terms of certain basis for the space of exponential polynomials, and a convolution product defined in an algebraic way using the basis. The generalization gives an explicit construction of the general solution of any * Research partially supported by a fellowship from CONACYT-Mexico. ´ † Research partially supported by a grant from CONACYT-Mexico. E-mail: verde@xan- ´

An Algebraic Approach to Convolutions and Transform Methods

We show that the Hopf algebra dual of the polynomials in one variable appears often in analysis, but under different disguises that include proper rational functions, exponential polynomials, shift invariant operators, Taylor functionals, and linearly recurrent sequences. The isomorphisms from the proper rational functions to the other algebras yield an explicit and general method for the solution of linear functional equations, which can be considered as an algebraic version of the usual integral transform methods. We also explain how some of the usual convolution products in spaces of functions arise in an algebraic way.

Solution of linear functional equations in space of functions, bounded in Rn

Solution of linear functional equations in space of functions, bounded in Rn

Convergence of Padé approximants using the solution of linear functional equations

We prove that a projection of the solutions to a linear functional equation of the Fredholm type with a compact kernel, projected into the Cini−Fubini subspaces, converge strongly to the solution in the whole space. Here either the whole sequence converges for all nonsingular points of the functional equation with at most one exceptional point, or by selecting at most two infinite subsequences we can obtain convergence for all nonsingular points. We then prove that the diagonal Padé approximants to the inner product of the solution with another element converge. For certain kernels of trace class, the numerator and denominator separately converge. As applications of these results, we prove the pointwise convergence of the Padé approximants to a wide class of meromorphic functions. We also prove the convergence, for decent potentials, of the Padé approximants to the scattering amplitudes for nonrelativistic quantum mechanical scattering problems. The numerators and denominators of the Padé approximants to the partial wave scattering amplitudes for single signed potentials converge separately to entire

Ergodic Theory and Averaging Iterations

Suppose X is a Banach space and T a continuous linear operator on X. The significance of the asymptotic convergence of T for the approximate solution of the equation (I - T)x = f by means of the Picard iterations was clearly shown in Browder's and Petryshyn's paper [1], The results of [1] have stimulated further investigation of the Picard, and more generally, averaging iterations for the solution of linear and nonlinear functional equations [2; 3; 4; 8; 9]. Kwon and Redheffer [8] analyzed the Picard iteration under the mildest possible condition on T, namely that T be continuous and linear on a normed (not necessarily complete) space X. The results of [8] (still waiting to be extended for the averaging iterations) seem to give the most complete story of the Picard iterations for the linear case. Only when T is subject to some further restrictions, such as asymptotic 4-boundedness and asymptotic A -regularity, one can agree with Dotson [4] that the iterative solution of linear functional equations is a special case of mean ergodic theory for affine operators. This thesis is rather convincingly demonstrated by results of De Figueiredo and Karlovitz [2], and Dotson [3], and most of all by Dotson's recent paper [4], in which the results of [1; 2; 3] are elegantly subsumed under the afrine mean ergodic theorem of Eberlein-Dotson.

Stability of Bounded Solutions of Linear Functional Equations

Stability of Bounded Solutions of Linear Functional Equations

Mean ergodic theorems and iterative solution of linear functional equations

Mean ergodic theorems and iterative solution of linear functional equations

Stability of bounded solutions of linear functional equations

The weak sequential compactness of reflexive Banach spaces is used to explain the fact that certain ill-posed, linear problems become well-posed if the solutions are required to satisfy a prescribed bound. Applications are made to the computability of solutions of ill-posed problems associated with elliptic and parabolic partial differential equations.

General properties of the error of the approximate solution of linear functional equations

IT is shown that the functional, independent of the method of solution, introduced in [1], describes the error of the solution for arbitrary approximate algorithms of the solution. (In the formulation of Theorem 4 of [1] there is an error: for a s ́ > 0 read a⩾ 0).

On the Solution of Linear Functional Equations by Averaging Iteration

On the Solution of Linear Functional Equations by Averaging Iteration

On the solution of linear functional equations by averaging iteration.

Curtis Outlaw and C. W. Groetsclh [4] have recently shown that if T is an asymptotically convergent continuous linear self-mapping of a Banach space E, and if f is in the range of I-T, and 0 E is a continuous linear operator, and fEE. For O 1, and for n>1, anj=(j_ )Xn'-(1 -X)j-l for 1 n. It is easily seen that AX is a lower-triangular, nonnegative, inifinite matrix with each row-sum equal to one and each column-limit equal to zero. For n > 1 we have the real polynomial a(t) defined by

An application of ergodic theory to the solution of linear functional equations in Banach spaces

An application of ergodic theory to the solution of linear functional equations in Banach spaces