14 research outputs found
Cauchy's functional equation and extensions: Goldie's equation and inequality, the Go{\l}\k{a}b-Schinzel equation and Beurling's equation
The Cauchy functional equation is not only the most important single
functional equation, it is also central to regular variation. Classical
Karamata regular variation involves a functional equation and inequality due to
Goldie; we study this, and its counterpart in Beurling regular variation,
together with the related Go{\l}\k{a}b-Schinzel equation.Comment: Companion paper to: Additivity, subadditivity and linearity:
automatic continuity and quantifier weakenin
Multivariate Popa groups and the Goldie Equation
We give a necessary and sufficent condition which characterizes which
continuous solutions of a multivariate Goldie functional equation are Popa
homomorphisms and so deduce that all continuous solutions are homomorphisms
between multivariate Popa groups. We use this result to characterize as Popa
homomorphisms smooth solutions of a related more general equation, also of
Levi-Civit\`a type
Cauchy’s functional equation and extensions: Goldie’s equation and inequality, the Gołąb–Schinzel equation and Beurling’s equation
The Cauchy functional equation is not only the most important single functional equation, it is also central to regular variation. Classical Karamata regular variation involves a functional equation and inequality due to Goldie; we study this, and its counterpart in Beurling regular variation, together with the related Gołąb–Schinzel equation
Multivariate general regular variation: Popa groups on vector spaces
We extend to multi-dimensional (or infinite-dimensional) settings the general
regular variation of `General regular variation, Popa groups and quantifier
weakening', J. Math. Anal. Appl., to appear (arXiv1901.05996). The theory
focuses on extension of the treatment there of Popa groups. The applications
focus on multivariate extreme-value theory
Beurling regular variation, Bloom dichotomy, and the Gołąb–Schinzel functional equation
The class of 'self-neglecting' functions at the heart of Beurling slow variation is expanded by permitting a positive asymptotic limit function λ(t), in place of the usual limit 1, necessarily satisfying the following 'self-neglect' condition:(Formula presented.)known as the Goła{ogonek}b-Schinzel functional equation, a relative of the Cauchy equation (which is itself also central to Karamata regular variation). This equation, due independently to Aczél and Goła{ogonek}b, occurring in the study of one-parameter subgroups, is here accessory to the λ -Uniform Convergence Theorem (λ-UCT) for the recent, flow-motivated, 'Beurling regular variation'. Positive solutions, when continuous, are known to be λ(t) = 1 + at (below a new, 'flow', proof is given); a = 0 recovers the usual limit 1 for self-neglecting functions. The λ-UCT allows the inclusion of Karamata multiplicative regular variation in the Beurling theory of regular variation, with λ (t) = 1 + t being the relevant case here, and generalizes Bloom's theorem concerning self-neglecting functions
On a system of simultaneous iterative functional equations
A system of two simultaneous functional equations in a single
variable, related to a generalized Gołąb-Schinzel functional equation, is considered
Additivity, subadditivity and linearity: automatic continuity and quantifier weakening
We study the interplay between additivity (as in the Cauchy functional
equation), subadditivity and linearity. We obtain automatic continuity results
in which additive or subadditive functions, under minimal regularity
conditions, are continuous and so linear. We apply our results in the context
of quantifier weakening in the theory of regular variation completing our
programme of reducing the number of hard proofs there to zero.Comment: Companion paper to: Cauchy's functional equation and extensions:
Goldie's equation and inequality, the Go{\l}\k{a}b-Schinzel equation and
Beurling's equation Updated to refer to other developments and their
publication detail
Report of Meeting: The Twenty-first Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities Brenna (Poland), February 2–5, 2022
The Twenty-first Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities was held in Hotel Kotarz in Brenna, Poland, from February 2 to February 5, 2022. The meeting was organized by the Institute of Mathematics of the University of Silesia. 11 participants came from the University of Debrecen (Hungary), 7 from
the University of Silesia in Katowice (Poland), 2 from the Pedagogical University
of Krakow (Poland), 1 from Budapest University of Technology and Economics (Hungary), 1 from the University of Rzeszów (Poland) and 1 with a dual affiliation University of Silesia (Poland) and Chernivtsi National University (Ukraine)