Equations fonctionnelles et estimations de normes de matrices

We solve independently the equations 1/θ(x)θ(y)=ψ(x)−ψ(y)+φ(x−y)/θ(x−y) and 1/θ(x)θ(y)=σ(x)−σ(y)/θ(x−y)+τ(x)τ(y), τ(0)=0. In both cases we find θ²=aθ⁴+bθ²+c. We deduce estimates for the spectral radius of a matrix of type(1/θ(x _r −x _s )) (the accent meaning that the coefficients of the main diagonal are zero) and we study the case where thex _r are equidistant.

Dédié to à Monsieur le Professeur Otto Haupt à l'occasion de son cententiare avec les meilleurs voeux     

Relations de dépendance entre les équations fonctionnelles de Cauchy

Résumé Afin d'examiner les relations entre les différentes équations de Cauchy, nous résolvons, sans aucune hypothèse de régularité, l'équation fonctionnellea f(xy) + b f(x)f(y) + c f(x + y) + d (f(x) + f(y)) = 0, pour des fonctionsf, définies sur un anneau unifère divisible par deux et prenant leurs valeurs dans un corps, Les coefficientsa, b, c, etd appartiennent au centre de ce corps. Entre autres applications, nous en déduisons qu'une seule équation, à savoirf(xy) + f(x + y) = f(x)f(y) + f(x) + f(y), caractérise les endomorphismes des corps dont la caractéristique est différente de 2. En introduisant la notion d'équations fonctionnelles étrangères et d'équations fonctionnelles fortement étrangères, nous concluons à l'indépendance, au sens de cette notion, des équations classiques de Cauchy.

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Summary In order to study the inter-relations between the four Cauchy functional equations, we solve the functional equationa f(xy) + b f(x) f(y) + c f(x + y) + d(f(x) + f(y)) = 0. The functionf is defined over a ring which is divisible by 2 and which possesses a unit, while the values off are in a(skew)-field. The constantsa, b, c andd belong to this field and commute with all elements of thes-field. No regularity assumption is made onf. Among other applications, we show that the single equationf(xy) + f(x + y) = f(x)f(y) + f(x) + f(y), is enough to characterize field endormophisms in fields of characteristic different from 2. We introduce the notion of alien functional equations and that of strongly alien functional equations, to conclude that for such notions, Cauchy equations are indeed largely independent.

Dédié avec nos meilleurs voeux à Monsieur le Professeur Otto Haupt à l'occasion de son centenaire     

Algèbre de Clifford symplectique. Revêtements du groupe symplectique. Indices de Maslov et spineurs symplectiques

Sans résumé

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This article is the developed text of a talk given at the Symposium on Differential Geometry in Debrecen, Hungary, on August 28–September 3, 1975.     

Equations on real intervals

A (finite or infinite) set ∑ of equations, in operation symbols F_t (t ∈T) and variables x_i, is said to be compatible with iff there exist continuous operations F_t^A on such that the algebra satisfies the equations ∑ (with the variables x_i understood as universally quantified). It is proved that there is no algorithm to decide -compatibility for all finite ∑. If the definition is restricted to C¹ idempotent operations F_t^A , then there does exist an algorithm for compatibility. Received August 9, 2005; accepted in final form February 14, 2006.     

On two mean value properties and functional equations associated with them

Summary Motivated by different mean value properties, the functional equationsf(x) – f(y)/x–y=[(x, y)], (i)xf(y) – yf(x)/x–y=[(x, y)] (ii) (x y) are completely solved when, are arithmetic, geometric or harmonic means andx, y elements of proper real intervals. In view of a duality between (i) and (ii), three of the results are consequences of other three.The equation (ii) is also solved when is a (strictly monotonic) quasiarithmetic mean while the real interval contains 0 and when is the arithmetic mean while the domain is a field of characteristic different from 2 and 3. (A result similar to the latter has been proved previously for (i).)     

Ona-Wright convexity and the converse of Minkowski's inequality

Summary Leta (0, 1/2] be fixed. A functionf satisfying the inequalityf(ax + (1 – a)y) + f((1 – a)x + ay) f(x) + f(y), called herea-Wright convexity, appears in connection with the converse of Minkowski's inequality. We prove that every lower semicontinuousa-Wright convex function is Jensen convex and we pose an open problem. Moreover, using the fact that 1/2-Wright convexity coincides with Jensen convexity, we prove a converse of Minkowski's inequality without any regularity conditions.     

Boundary Behavior of Functions in the de Branges–Rovnyak Spaces

This paper deals with the boundary behavior of functions in the de Branges–Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges–Rovnyak spaces. This criterion generalizes a result of Ahern–Clark. Then we prove that the continuity of all functions in a de Branges–Rovnyak space on an open arc I of the boundary is enough to ensure the analyticity of these functions on I. We use this property in a question related to Bernstein’s inequality. Received: May 10, 2007. Revised: August 8, 2007. Accepted: August 8, 2007.     

Generalized sine equations,III

Summary We solve the equationf(x + y)f(x – y) = P(f(x), f(y)) under various conditions on the unknown functionsf, P.     

Further results on supernumary polylogarithmic ladders

Summary With the help of the PARI computer program a number of matters left unresolved from previous work have now been settled. It will be recalled that a ladder is a rational sum of polylogarithms, with predetermined coefficients, of powers of a given algebraic base. The simplest bases considered are the roots in (0, 1) ofu ^p +u ^q = 1 for various integersp andq.. They possess a number of generic results, together with some additional equations, termed supernumary for certain specific values ofp andq. In particular, ladders of the base (see [1]) have been extended to the sixth order, and involve a new index, 60, found by the PARI program. The base from (p, q) = (11, 7) has an additional index 20, and this combines with earlier results to produce a valid ladder. The apparent barren feature of certain equations is now explained in terms of a need to work with a sufficient number of results. It is confirmed that the equation with (p, q) = (5, 3) indeed does not possess any supernumary results.A complete investigation of the smallest Salem number of degree four is given: it possesses results to the 8th order. An introduction is given to similar studies for the smallest known Salem number, which has now been shown to extend to the 16th order.Some ladder results for combined bases are found, with one such formula deducible from a three-variable dilogarithmic functional equation. Formulas of a new type are developed in which summation over conjugate roots enables ladders to be extended fromn = 2 to 3.