# On the Hausdorff dimension of some graphs

@article{Mauldin1986OnTH, title={On the Hausdorff dimension of some graphs}, author={R. Daniel Mauldin and S. C. Williams}, journal={Transactions of the American Mathematical Society}, year={1986}, volume={298}, pages={793-803} }

Consider the functions Wb(x)= b-cn[1(bnX + On)--1(0n)] n=-oo where b > 1, 0 0 such that if b is large enough, then the Hausdorff dimension of the graph of Wb is bounded below by 2a (C/ ln b). We also show that if a function f is convex Lipschitz of order a, then the graph of f has a-finite measure with respect to Hausdorff's measure in dimension 2 a. The convex Lipschitz functions of order a include Zygmund's class A,. Our analysis shows that the graph of the classical van der WaerdenTagaki…

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## References

SHOWING 1-10 OF 14 REFERENCES

On the Weierstrass-Mandelbrot fractal function

- MathematicsProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1980

The function W(t)≡∑n=−∞∞[(1−eiγnt)eiϕn]γ(2−D)n(11,ϕn=arbitraryphases) is continuous but non-differentiable and possesses no scale. The graph of ReW or Im W has Hausdorff-Besicovitch (fractal)…

The Lyapunov dimension of a nowhere differentiable attracting torus

- Mathematics, PhysicsErgodic Theory and Dynamical Systems
- 1984

Abstract The fractal dimension of an attracting torus Tk in × Tk is shown to be almost always equal to the Lyapunov dimension as predicted by a previous conjecture. The cases studied here can have…

The geometry of fractal sets

- Mathematics
- 1985

Preface Introduction Notation 1. Measure and dimension 2. Basic density properties 3. Structure of sets of integral dimension 4. Structure of sets of non-integral dimension 5. Comparable net measures…

Fractals: Form, Chance, and Dimension.

- Education
- 1978

Some people may be laughing when looking at you reading in your spare time. Some may be admired of you. And some may want be like you who have reading hobby. What about your own feel? Have you felt…

Partial Differential Equations

- HistoryNature
- 1907

THE appearance of these volumes marks the happy conclusion of a work undertaken, as the author reminds us in his preface, twenty-one years ago. Doubtless it would have been finished earlier had it…

Trigonometric series, Cambridge Univ

- Press, Cambridge,
- 1968

The geometry of fractal sets, Cambridge Tracts

- in Math.,
- 1985

Hausdorff measures, Cambridge Univ

- Press, Cambridge,
- 1970