Non-differentiability and Hölder properties of self-affine functions

We consider the class of self-affine functions. Firstly, we characterize all nowhere differentiable self-affine continuous functions. Secondly, given a self-affine continuous function $?$, we investigate its Hölder properties. We find its best uniform Hölder exponent and when $?$ is $C^1$, we find the best uniform Hölder exponent of ${?}^{\prime }$. Thirdly, we show that the Hölder cut of $?$ takes the same value almost everywhere for the Lebesgue measure. This last result is a consequence of the Borel strong law of large numbers.     

On Hölder maps of cubes

A map of metric spaces f: X → Y satisfying the inequality $$ \left| {f(x) - f(y)} \right| \leqslant C\left| {x - y} \right|^\alpha $$ for some C and α and all x, y ∈ X is called a Hölder map with exponent α. V. I. Arnold posed the following problem: Does there exist a Höldermap from the square onto the cube with exponent 2/3? The firstmain theorem of this paper gives a general method for constructing Höldermaps of compact metric spaces. This construction yields, in particular, a dimension-raising map f: I ⁿ → I ^m with Hölder exponent arbitrarily close to m/n for m > n > 1 and a map I ¹ → I ^m with Hölder exponent 1/m. The second main theorem states the nonexistence of a regular fractal map f: I ⁿ → I ^m with Hölder exponent n/m from the n-cube onto the m-cube for m < 2n.     

Hölder exponents and box dimension for self-affine fractal functions

We consider some self-affine fractal functions previously studied by Barnsleyet al. The graphs of these functions are invariant under certain affine scalings, and we extend their definition to allow the use of nonlinear scalings. The Hölder exponent,h, for these fractal functions is calculated and we show that there is a larger Hölder exponent,h , defined at almost every point (with respect to Lebesgue measure). For a class of such functions defined using linear affinities these exponents are related to the box dimensionD _B of the graph byh2–D _Bh .Communicated by Michael F. Barnsley.     

Lipschitz and Hölder continuity results for some functions of cones

We prove the Lipschitz continuity of the maximal angle function on the set of closed convex cones in a Hilbert space. A similar result is obtained for the minimal angle function. On the other hand, we prove that the incenter of a solid cone and the circumcenter of a sharp cone behave in a locally Hölderian manner.     

On the extension of Hölder maps with values in spaces of continuous functions

We study the isometric extension problem for Hölder maps from subsets of any Banach space intoc ₀ or into a space of continuous functions. For a Banach spaceX, we prove that anyα-Hölder map, with 0<α ≤1, from a subset ofX intoc ₀ can be isometrically extended toX if and only ifX is finite dimensional. For a finite dimensional normed spaceX and for a compact metric spaceK, we prove that the set ofα’s for which allα-Hölder maps from a subset ofX intoC(K) can be extended isometrically is either (0, 1] or (0, 1) and we give examples of both occurrences. We also prove that for any metric spaceX, the above described set ofα’s does not depend onK, but only on finiteness ofK.     

On the Perron-Frobenius-ruelle operator for rational maps on the riemann sphere and for Hölder continuous functions

Let      

Hölder categories

Hölder categories are invented to provide an axiomatic foundation for the study of categories of archimedean lattice-ordered algebraic structures. The basis of such a study is Hölder’s Theorem (1908), stating that the archimedean totally ordered groups are precisely the subgroups of the additive real numbers ? with the usual addition and ordering, which remains the single most consequential result in the studies of lattice-ordered algebraic systems since Birkhoff and Fuchs to the present. This study originated with interest in W*, the category of all archimedean lattice-ordered groups with a designated strong order unit, and the ?-homomorphisms which preserve those units, and, more precisely, with interest in the epireflections on W*. In the course of this study, certain abstract notions jumped to the forefront. Two of these, in particular, seem to have been mostly overlooked; some notion of simplicity appears to be essential to any kind of categorical study of W*, as are the quasi-initial objects in a category. Once these two notions have been brought into the conversation, a Hölder category may then be defined as one which is complete, well powered, and in which (a) the initial object I is simple, and (b) there is a simple quasi-initial coseparator R. In this framework it is shown that the epireflective hull of R is the least monoreflective class. And, when I = R — that is, the initial element is simple and a coseparator — a theorem of Bezhanishvili, Morandi, and Olberding, for bounded archimedean f-algebras with identity, can be be generalized, as follows: for any Hölder category subject to the stipulation that the initial object is a simple coseparator, every uniformly nontrivial reflection — meaning that the reflection of each non-terminal object is non-terminal — is a monoreflection. Also shown here is the fact that the atoms in the class of epireflective classes are the epireflective hulls of the simple quasi-initial objects. From this observation one easily deduces a converse to the result of Bezhanishvili, Morandi, and Olberding: if in a Hölder category every epireflection is a monoreflection, then the initial object is a coseparator.     

On a Hölder covariant version of mean dimension

Let Γ be a infinite countable group which acts naturally on ${?}^p(\Gamma )$. We introduce a modification of mean dimension which is an obstruction for ${?}^p(\Gamma )$ and ${?}^q(\Gamma )$ to be Hölder conjugates. To cite this article: A. Gournay, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Hölder Error Bounds and Hölder Calmness with Applications to Convex Semi-infinite Optimization

Using techniques of variational analysis, necessary and sufficient subdifferential conditions for Hölder error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the setting of convex semi-infinite optimization and give a characterization of the Hölder calmness of the argmin mapping in terms of the level set mapping (with respect to the objective function) and a special supremum function. We also estimate the Hölder calmness modulus of the argmin mapping in the framework of linear programming.

     

Approximating periodic functions in Hölder type metrics by Fourier sums and Riesz means

Let M be a fixed space of 2π-periodic functions, L_p or C, let ω_r(f, h) be the continuity modulus of order r of the function f in the space M, and let ϕ(t) be a function such that ϕ(t) > 0 for t > 0. By S_n(f) we denote the Fourier sums and by R_n,r(f) we denote the Riesz sums (the Fejér sums for r = 1) of the function f. Set

On Jensen’s inequality and Hölder’s inequality for g-expectation

In this paper, we shall prove that for n > 1, the n-dimensional Jensen inequality holds for the g-expectation if and only if g is independent of y and linear with respect to z, in other words, the corresponding g-expectation must be linear. A Similar result also holds for the general nonlinear expectation defined in Coquet et al. (Prob. Theory Relat. Fields 123 (2002), 1–27 or Peng (Stochastic Methods in Finance Lectures, LNM 1856, 143–217, Springer-Verlag, Berlin, 2004). As an application of a special n-dimensional Jensen inequality for g-expectation, we give a sufficient condition for g under which the Hölder’s inequality and Minkowski’s inequality for the corresponding g-expectation hold true.     

Approximation by Riesz sums of periodic functions of Hölder classes

We have found asymptotic equalities for the least upper bounds of the deviations of Riesz sums on the Hölder classes W^rH, r is a nonnegative integer, (t) is an arbitrary convex modulus of continuity.Translated from Matematicheskie Zametki, Vol. 21, No. 3, pp. 341–354, March, 1977.     

On the traces of Sobolev functions on the boundary of a cusp with a Hölder singularity

For the Sobolev classes W _p ¹ on a “zero” cusp with a Hölder singularity at the vertex, we consider the question of compactness of the embedding of the traces of Sobolev functions into the Lebesgue classes on the boundary of the cusp.