Approximate convexity of Takagi type functions

Given a bounded function Φ:R→R, we define the Takagi type function TΦ:R→R by

     

An inequality for sums of binary digits, with application to Takagi functions

Let ?(x)=2inf{|x−n|:n∈Z}, and define for α>0 the function

     

Extensions of convex and semiconvex functions and intervally thin sets

We call A⊂RNintervally thin if for all x,y∈RN and ε>0 there exist x^′∈B(x,ε), y^′∈B(y,ε) such that [x^′,y^′]∩A=∅. Closed intervally thin sets behave like sets with measure zero (for example such a set cannot “disconnect” an open connected set). Let us also mention that if the (N−1)-dimensional Hausdorff measure of A is zero, then A is intervally thin. A function f is preconvex if it is convex on every convex subset of its domain. The consequence of our main theorem is the following: Let U be an open subset ofRNand let A be a closed intervally thin subset of U. Then every preconvex functioncan be uniquely extended (with preservation of preconvexity) onto U. In fact we show that a more general version of this result holds for semiconvex functions.     

Approximately convex functions and approximately monotonic operators

We present characterizations of some generalized convexity properties of functions with the help of a general subdifferential. We stress the case of lower semicontinuous functions. We also study the important case of marginal functions and we provide representation results.     

On the establishment of the Takagi Lectures

The Takagi Lectures are the first series of lectures in mathematics to be crowned with a Japanese mathematician’s name. The author provides some historical background of the Takagi Lectures.     

Distribution of the maxima of random Takagi functions

This paper concerns the maximum value and the set of maximum points of a random version of Takagi’s continuous, nowhere differentiable function. Let F(x):=∑ _n=1^∞ ε _n ϕ(2 ⁿ⁻¹ x), x ∈ R, where ɛ ₁, ɛ ₂, ... are independent, identically distributed random variables taking values in {−1, 1}, and ϕ is the “tent map” defined by ϕ(x) = 2 dist (x, Z). Let p:= P (ɛ ₁ = 1), M:= max {F(x): x ∈ R}, and := {x ∈ [0, 1): F(x) = M}. An explicit expression for M is given in terms of the sequence {ɛ _n }, and it is shown that the probability distribution μ of M is purely atomic if p < , and is singular continuous if p ≧ . In the latter case, the Hausdorff dimension and the multifractal spectrum of μ are determined. It is shown further that the set is finite almost surely if p < , and is topologically equivalent to a Cantor set almost surely if p ≧ . The distribution of the cardinality of is determined in the first case, and the almost-sure Hausdorff dimension of is shown to be (2p − 1)/2p in the second case. The distribution of the leftmost point of is also given. Finally, some of the results are extended to the more general functions Σa ^{n − 1} ɛ _n ϕ(2 ^{n − 1} x), where 0 < a < 1.      

Representation theorems for t-Wright convexity

In the present paper we prove some representation theorems for t-Wright convex functions, as a consequence of a support theorem, which was proved by the author in earlier paper.     

Sum of graphs of continuous functions and boundedness of additive operators

Assume that and are uniformly continuous functions, where D₁,D₂⊂X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f(x)=x^∗(x)+a and g(x)=x^∗(x)+b with some x^∗∈X^∗ and a,b∈R or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X×R treated as a normed space with a norm .     

The approximate and the Clarke subdifferentials can be different everywhere

We prove that, for a Lipschitz function on , n2, the approximate and the Clarke subdifferentials can differ everywhere. This completely answers a question by A.D. Ioffe, which was partially answered by G. Katriel.     

On some relative convexities

We study two types of relative convexities of convex functions f and g. We say that f is convex relative to g   in the sense of Palmer (2002, 2003), if f=h(g)$f=h(g)$, where h   is strictly increasing and convex, and denote it by f_?(1)g$f{?}_{(1)}g$. Similarly, if f is convex relative to g   in the sense studied in Rajba (2011), that is if the function f−g$f-g$ is convex then we denote it by f_?(2)g$f{?}_{(2)}g$. The relative convexity relation _?(2)${?}_{(2)}$ of a function f   with respect to the function g(x)=cx²$g(x)=c{x}^2$ means the strong convexity of f. We analyze the relationships between these two types of relative convexities. We characterize them in terms of right derivatives of functions f and g, as well as in terms of distributional derivatives, without any additional assumptions of twice differentiability. We also obtain some probabilistic characterizations. We give a generalization of strong convexity of functions and obtain some Jensen-type inequalities.     

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations is performed. First, complete infinite-order approximate symmetry classification of the equation is obtained by means of the method originated by Fushchich and Shtelen. An optimal system of one-dimensional subalgebras is derived and used to construct general formulas of approximate symmetry reductions and similarity solutions. Second, we study approximate homotopy symmetry of the equation and construct connections between the two symmetry methods for the first-order and higher-order cases, respectively. The series solutions derived by the two methods are compared.     

New integral representations of nth order convex functions

In this paper we give an integral representation of an n-convex function f in general case without additional assumptions on function f. We prove that any n-convex function can be represented as a sum of two (n+1)-times monotone functions and a polynomial of degree at most n. We obtain a decomposition of n-Wright-convex functions which generalizes and complements results of Maksa and Páles (2009) [13]. We define and study relative n-convexity of n-convex functions. We introduce a measure of n-convexity of f. We give a characterization of relative n-convexity in terms of this measure, as well as in terms of nth order distributional derivatives and Radon-Nikodym derivatives. We define, study and give a characterization of strong n-convexity of an n-convex function f in terms of its derivative f(n+1)(x) (which exists a.e.) without additional assumptions on differentiability of f. We prove that for any two n-convex functions f and g, such that f is n-convex with respect to g, the function g is the support for the function f in the sense introduced by W?sowicz (2007) [29], up to polynomial of degree at most n.     

Support-type properties of convex functions of higher order and Hadamard-type inequalities

It is well known that every convex function (where I⊂R is an interval) admits an affine support at every interior point of I (i.e. for any x₀∈IntI there exists an affine function such that a(x₀)=f(x₀) and a?f on I). Convex functions of higher order (precisely of an odd order) have a similar property: they are supported by the polynomials of degree no greater than the order of convexity. In this paper the attaching method is developed. It is applied to obtain the general result—Theorem 2, from which the mentioned above support theorem and some related properties of convex functions of higher (both odd and even) order are derived. They are applied to obtain some known and new Hadamard-type inequalities between the quadrature operators and the integral approximated by them. It is also shown that the error bounds of quadrature rules follow by inequalities of this kind.     

Necessary optimality conditions and saddle points for approximate optimization in banach spaces

In this article we study approximate optimality in the setting of a Banach space. We study various solution concepts existing in the literature and develop very general necessary optimality conditions in terms of limiting subdifferentials. We also study saddle point conditions and relate them to various solution concepts. Part of this research was carried out when the author was a post-doctoral fellow at UAB, Barcelona by the Grant No. SB99-B0771103B of the Spanish Ministry of Education and Culture. The hospitality and the facilities provided at CODE, UAB is gratefully acknowledged.     

Weak tangent and level sets of Takagi functions

In this paper, we study some properties of Takagi functions and their level sets. We show that for Takagi functions $$T_{a,b}$$ with parameters a, b such that ab is a root of a Littlewood polynomial, there exist large level sets. As a consequence, we show that for some parameters a, b, the Assouad dimension of graphs of $$T_{a,b}$$ is strictly larger than their upper box dimension. In particular, we can find weak tangents of those graphs with large Hausdorff dimension, larger than the upper box dimension of the graphs.     

Strongly hyperbolically convex functions

Let C(w₁,w₂,w₃) denote the circle in through w₁,w₂,w₃ and let denote one of the two arcs between w₁,w₂ belonging to C(w₁,w₂,w₃). We prove that a domain Ω in the Riemann sphere, with no antipodal points, is spherically convex if and only if for any w₁,w₂,w₃∈Ω, with w₁≠w₂, the arc of the circle which does not contain lies in Ω. Based on this characterization we call a domain G in the unit disk D, strongly hyperbolically convex if for any w₁,w₂,w₃∈G, with w₁≠w₂, the arc in D of the circle is also contained in G. A number of results on conformal maps onto strongly hyperbolically convex domains are obtained.