zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the Hausdorff dimension of some graphs. (English) Zbl 0603.28003
Consider the functions $$ W\sb b(x)=\sum\sp{\infty}\sb{n=-\infty}b\sp{- \alpha n}[\Phi (b\sp nx+\theta\sb n)-\Phi (\theta\sb n)], $$ where $b>1$, $0<\alpha <1$, each $\theta\sb n$ is an arbitrary number, and $\Phi$ has period one. We show that there is a constant $C>0$ such that if b is large enough, then the Hausdorff dimension of the graph of $W\sb b$ is bounded below by $2-\alpha -(C/\ln b)$. We also show that if a function f is convex Lipschitz of order $\alpha$, then the graph of f has $\sigma$- finite measure with respect to Hausdorff’s measure in dimension $2- \alpha$. The convex Lipschitz functions of order $\alpha$ include Zygmund’s class $\Lambda\sb{\alpha}$. Our analysis shows that the graph of the classical van der Waerden-Takagi nowhere differentiable function has $\sigma$-finite measure with respect to $h(t)=t/\ln (1/t)$.

28A75Length, area, volume, other geometric measure theory
42A32Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
26A27Nondifferentiability of functions of one real variable; discontinuous derivatives
Full Text: DOI