Fractals and Fourier Transforms

[David Makin]

I'm posting this for those who may have the Math to follow it - I don't, so I haven't even looked.
To be honest I've never even messed with ordinary FFTs though I've often thought I ought to spend the time to understand them !!

If you're on LinkedIn this is from a post in the Fractals group....

http://worldsciencepublisher.org/journals/index.php/AITS/article/view/274

Of course it could be a load of bull, but then again....I'd appreciate it if anyone interested who checks it out lets me know if it's of any use or not

[kram1032]

I see...
This doesn't have much to do with fractals as you probably have expected them.
It's about fractional derivatives. E.g. a "halfth order" derivative or a "four thirds order" one, etc.
Still a very interesting topic. Though that's about all I get from that. Lots of formulas. REALLY short.
It looks like a detail of a bigger puzzle, rather than a "complete finding", if that makes any sense. (Then, I guess, maths wont ever be complete...)

[hobold]

(Then, I guess, maths wont ever be complete...)

I know what Kurt Gödel would answer to that!

SCNR

[Feline]

...and just in case you wanted to know a little more about how fractional differentials apply to approximation theory and inequalities, this book http://books.google.com/books/about/?id=gAe7DXzOSewC is not only a pretty complete introduction, but features cover art by yours truly (which I'm not entirely sure whether that recommends it).

[Apophyster]

Hi Feline,
I think the cover art is rather nice.  At least the two outside images.
Are you feline by (astrological) sign or by nature?
I'm double feline by sign. 

Fred

[xueer]

Theory and Applications of Local Fractional Fourier Analysis

http://works.bepress.com/yang_xiaojun/41/


Local fractional Fourier analysis is a generalized Fourier analysis in fractal space. The local fractional calculus is one of useful tools to process the local fractional continuously non-differentiable functions (fractal functions). Based on the local fractional derivative and integration, the present work is devoted to the theory and applications of local fractional Fourier analysis in generalized Hilbert space. We investigate the local fractional Fourier series, the Yang-Fourier transform, the generalized Yang-Fourier transform, the discrete Yang-Fourier transform and fast Yang-Fourier transform.


Yang Xiaojun. "Theory and Applications of Local Fractional Fourier Analysis" Advances in Mechanical Engineering and its Applications 1.4 (2012): 70-85.
Available at: http://works.bepress.com/yang_xiaojun/41


Advanced Local Fractional Calculus and Its ApplicationsYang Xiaojun, China University of Mining & Technology

Article comments
This book is coprighted by Xiao-Jun Yang.
Abstract
This book is the first international book to study theory and applications of local fractional calculus (LFC). It is an invitation both to the interested scientists and the engineers. It presents a thorough introduction to the recent results of local fractional calculus. It is also devoted to the application of advanced local fractional calculus on the mathematics science and engineering problems. The author focuses on multivariable local fractional calculus providing the general framework. It leads to new challenging insights and surprising correlations between fractal and fractional calculus. Keywords: Fractals - Mathematical complexity book - Local fractional calculus- Local fractional partial derivatives - Fractal Lagrange multipliers method - Multiple local fractional integrals - Local fractional gradient-Local fractional divergence theorem- Local fractional Stokes theorem- Green's first theorem in fractal domain- Green's second theorem in fractal domain - Fractal Riemann space - Geometry in fractal Riemann space - Fractal orthogonal coordinate tensors- Local fractional Euler–Lagrange equations - Principle of minimum potential energy in fractal medium- Principle of minimum potential energy in fractal medium - Principle of minimum complementary energy in fractal medium - J-integral formula in fractal fracture mechanics - Local fractional heat conduction equations Related subjects: Fractals- Complexity-Local Fractional Calculus And Applications- Fractal fracture mechanics- Fractal elasticity- Fractal heat conduction- Conservation of mass in fractal domain-Fractal transport equation- Fractal electromagnetic Table of contents  Preliminary Results  Local Fractional Calculus of One-variable Function  Local Fractional Partial Derivatives and Fractal Lagrange Multipliers Method  Multiple Local Fractional Integrals of Functions on Cantor Set  Local Fractional Line Integrals, Surface Integrals and Tensors  Local Fractional Calculus of Variations  A New Optimization Method for Functions on Cantor Set  Local Fractional Euler–Lagrange Equations  Applications of Local Fractional Calculus to Mechanics
Suggested Citation
X. J. Yang, Advanced Local Fractional Calculus and Its Applications (1 ed). New York: World Science Publisher, 2012.

I'm posting this for those who may have the Math to follow it - I don't, so I haven't even looked.
To be honest I've never even messed with ordinary FFTs though I've often thought I ought to spend the time to understand them !!

If you're on LinkedIn this is from a post in the Fractals group....

http://worldsciencepublisher.org/journals/index.php/AITS/article/view/274

Of course it could be a load of bull, but then again....I'd appreciate it if anyone interested who checks it out lets me know if it's of any use or not