Extreme Markov operators and the orbits of Ryff

A problem in convexity leading to the analysis of two functional equations

“…A case in point is the equation It is understood that (i) means Tf > 0 if / > 0, while in (ii), 1 denotes the constant function. This class has been studied extensively [5,8,9]. It is clearly convex and is also compact in a sufficiently weak topology [4].…”

“…Then any function / orthogonal to the class {$n} on [0,6) must satisfy (9) / exp(-x)f(x)dx=-expf-xj f(x)dx, We may recover the functions fu from a solution of (14) and therefore a solution /. Moreover, an 7/2-solution of (14) produces L2-solutions of (13) and vice versa.…”

A Problem in Convexity Leading to the Analysis of Two Functional Equations

“…It is understood that (i) means Tf > 0 if / > 0, while in (ii), 1 denotes the constant function. This class has been studied extensively [5,8,9]. It is clearly convex and is also compact in a sufficiently weak topology [4].…”

“…We shall work with specific choices of these numbers, chosen so that exp(n/o) = exp(n/6), n = ±1, ±2,_In other words, a-1 = 6_1 + N for a natural number N. The simplest case occurs when N -1 giving a = §(3 -v5) and 6 = ^(v^5 -1)-Assume that a particular value of N has been chosen. Then any function / orthogonal to the class {$n} on [0,6) must satisfy (9) / exp( where A = a/6 as before and 0 < x < 6. The symbol /" is used for both the functions fv and their periodic extensions over intervals of length a.1 Care must be taken not to lose track of the variables, to wit: We may recover the functions fu from a solution of (14) and therefore a solution /.…”

The functional equation 𝑎𝑓(𝑎𝑥)+𝑏𝑓(𝑏𝑥+𝑎)=𝑏𝑓(𝑏𝑥)+𝑎𝑓(𝑎𝑥+𝑏) extensions and almost periodic solutions

“…That is, does there exist an extreme element of V such that whenever g is an extreme point of £l(f), then Tg is not extreme? This should hold true for all nonconstant ƒ G L l {I) (see [5]). …”