Optimal polynomial approximation method for a class of functional equations

In this paper, we consider a class of functional equations and prove three theorems which give the approximation properties and error estimates of the optimal polynomial approximation of slution of those functional equations.     

Nonlinear functional equations satisfied by orthogonal polynomials

Let c be a linear functional defined by its moments c(xⁱ)=c_i for i=0,1,…. We proved that the nonlinear functional equations P(t)=c(P(x)P(αx+t)) and P(t)=c(P(x)P(xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. This equation relates the polynomials of the family with those of the scaled and shifted family. Other types of nonlinear functional equations whose solutions are formal orthogonal polynomials are also presented. Applications to Legendre and Chebyshev polynomials are given. Then, orthogonality with respect to a definite inner product is studied. When c is an integral functional with respect to a weight function, the preceding functional equations are nonlinear integral equations, and these results lead to new characterizations of orthogonal polynomials on the real line, on the unit circle, and, more generally, on an algebraic curve.     

Generating differential equations satisfied by products of functions

Based on the coefficients of two homogeneous linear differential equations, a method is proposed to construct a third homogeneous linear differential equations which is satisfied by all products of the form uv, where u and v satisfy, respectively, the first and the second given differential equation. The method was used recently in the computation of rapidly oscillatory integrals with kernels which are products of Bessel functions and their variants.     

On continuous descent functions for polynomial equations

In his recent book, Henrici (1974) gave an axiomatic treatment of the method of descent applied to the solution of polynomial equations, dealing in particular with the non-existence of continuous descent functions defined on the whole complex plane. This note presents an alternative account of this question, in which a somewhat stronger theorem is proved. At the same time, a certain problematical step, to which Henrici himself drew attention, is avoided.

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Zusammenfassung Henrici (1974) gibt in einem kürzlich erschienenen Buch eine axiomatische Behandlung der Absteigungsmethode zur Lösung von Polynomialgleichungen. Dort wird insbesondere die Nichtexistenz von stetigen Absteigungsfunktionen, die auf der ganzen komplexen Ebene definiert sind, behandelt. In dieser Arbeit wird das gleiche Problem von einem anderen Standpunkt aus betrachtet, und es wird ein etwas stärkerer Satz bewiesen. Dabei wird eine kleine Schwierigkeit vermieden, auf die Henrici selber aufmerksam gemacht hat.

     

Non-linear algebraic differential equations satisfied by a certain family of elliptic functions

Kurokawa and Wakayama (Ramanujan J. 10:23–41, 2005) studied a family of elliptic functions defined by certain q-series. This family, in particular, contains the Weierstrass ?-function. In this paper, we prove that elliptic functions in this family satisfy certain non-linear algebraic differential equations whose coefficients are essentially given by rational functions of the first few Eisenstein series of the modular group.     

On differential equations satisfied by modular forms

We use the theory of modular functions to give a new proof of a result of P. F. Stiller, which asserts that, if t is a non-constant meromorphic modular function of weight 0 and F is a meromorphic modular form of weight k with respect to a discrete subgroup of SL ₂ () commensurable with SL ₂ (), then F, as a function of t, satisfies a (k+1)-st order linear differential equation with algebraic functions of t as coefficients. Furthermore, we show that the Schwarzian differential equation for the modular function t can be extracted from any given (k+1)-st order linear differential equation of this type. One advantage of our approach is that every coefficient in the differential equations can be relatively easily determined. Mathematics Subject Classification (2000):11F03, 11F11.     

On a class of abstract neutral functional differential equations

By using the theory of semigroups of growth α, we discuss the existence of mild solutions for a class of abstract neutral functional differential equations. A concrete application to partial neutral functional differential equations is considered.     

On Hyers-Ulam stability for a class of functional equations

Summary In this paper we prove some stability theorems for functional equations of the formg[F(x, y)]=H[g(x), g(y), x, y]. As special cases we obtain well known results for Cauchy and Jensen equations and for functional equations in a single variable. Work supported by M.U.R.S.T. Research funds (60%).     

On polynomial equations

The set of all triples of positive integers (α, β, γ) for which there exist polynomials f, g, h (with or without common factors) which satisfy the equationf ^α + ^β =h ^γ , is determined.     

On a new class of nonlinear wave equations

In this paper we prove the existence and uniqueness of regular solutions for the Cauchy problem for the evolution equation $\text{u″ + A}{}^2\text{u + (α + M(¦A}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{u¦}{}^2\text{) Au = 0}$, suggested by the study of beams and plates. We represent by A a linear operator of a Hilbert space H with norm ∥, α is a real number, and M(λ) > 0 a real function, for λ ? 0.     

On a class of dual integral equations involving trigonometrical functions

In this paper we consider certain dual integral equations involving trigonometrical functions whose closed form solutions are obtained. Solutions are obtained by using the properties of Mehler-Fock transforms and the inversion theorem of the generalized Mehler-Fock transforms. The solutions of these dual integral equations have applications in engineering.