Vector—valued fractal interpolation functions and their box dimension

1991 ◽  
Vol 42 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Peter R. Massopust
Keyword(s):  
Box Dimension ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Vector Valued ◽  
Interpolation Functions

EXISTENCE AND BOX DIMENSION OF GENERAL RECURRENT FRACTAL INTERPOLATION FUNCTIONS

2020 ◽  
pp. 1-13
Author(s):  
HUO-JUN RUAN ◽  
JIAN-CI XIAO ◽  
BING YANG
Keyword(s):  
Iterated Function System ◽  
General Setting ◽  
Iterated Function Systems ◽  
Box Dimension ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Recurrent System ◽  
Definition Of ◽  
Interpolation Functions

Abstract The notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley et al. [‘Recurrent iterated function systems’, Constr. Approx.5 (1989), 362–378]. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on $\mathbb {R}^2$ . We generalise the definition of RFIFs so that iterated functions in the recurrent system need not be contractive with respect to the first variable. We obtain the box dimensions of all self-affine RFIFs in this general setting.

BOX DIMENSION OF WEYL–MARCHAUD FRACTIONAL DERIVATIVE OF LINEAR FRACTAL INTERPOLATION FUNCTIONS

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950058 ◽  
Author(s):  
WEN LIANG PENG ◽  
KUI YAO ◽  
XIA ZHANG ◽  
JIA YAO
Keyword(s):  
Fractional Derivative ◽  
Linear Relationship ◽  
Box Dimension ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

This paper mainly explores Weyl–Marchaud fractional derivative of linear fractal interpolation functions (FIFs). We prove that Weyl–Marchaud fractional derivative of a linear FIF is still a linear FIFs. More generally, we get a conclusion that there exists some linear relationship between the order of Weyl–Marchaud fractional derivative and box dimension of linear FIFs.

HIDDEN VARIABLE VECTOR VALUED FRACTAL INTERPOLATION FUNCTIONS

Fractals ◽  
2005 ◽  
Vol 13 (03) ◽  
pp. 227-232 ◽  
Author(s):  
P. BOUBOULIS ◽  
L. DALLA
Keyword(s):  
Fractal Interpolation ◽  
Random Grids ◽  
Hidden Variable ◽  
Fractal Interpolation Functions ◽  
Vector Valued ◽  
Interpolation Functions

We present a method of construction of vector valued bivariate fractal interpolation functions on random grids in ℝ2. Examples and applications are also included.

A TYPE OF FRACTAL INTERPOLATION FUNCTIONS AND THEIR FRACTIONAL CALCULUS

Fractals ◽  
2016 ◽  
Vol 24 (02) ◽  
pp. 1650026 ◽  
Author(s):  
YONG-SHUN LIANG ◽  
QI ZHANG
Keyword(s):  
Fractional Calculus ◽  
Local Structure ◽  
Continuous Functions ◽  
Box Dimension ◽  
Fractal Interpolation ◽  
Chebyshev Systems ◽  
Fractal Interpolation Functions ◽  
The Relationship ◽  
Interpolation Functions

Combine Chebyshev systems with fractal interpolation, certain continuous functions have been approximated by fractal interpolation functions unanimously. Local structure of these fractal interpolation functions (FIF) has been discussed. The relationship between order of Riemann–Liouville fractional calculus and Box dimension of FIF has been investigated.

A GENERALIZATION OF FRACTAL INTERPOLATION STOCHASTIC PROCESSES TO HIGHER DIMENSIONS

Fractals ◽  
2001 ◽  
Vol 09 (04) ◽  
pp. 415-428 ◽  
Author(s):  
ROBERT MAŁYSZ
Keyword(s):  
Stochastic Processes ◽  
Higher Dimensions ◽  
Upper Bounds ◽  
Fractal Interpolation ◽  
Original Process ◽  
Stationary Increments ◽  
Fractal Interpolation Functions ◽  
Self Similar ◽  
Vector Valued ◽  
Interpolation Functions

We generalize the notion of fractal interpolation functions (FIFs) to stochastic processes. We prove that the Minkowski dimension of trajectories of such interpolations for self-similar processes with stationary increments converges to 2-α. We generalize the notion of vector-valued FIFs to stochastic processes. Trajectories of such interpolations based on an equally spaced sample of size n on the interval [0,1] converge to the trajectory of the original process. Moreover, for fractional Brownian motion and, more generally, for self-similar processes with stationary increments (α-sssi) processes, upper bounds of the Minkowski dimensions of the image and the graph converge to the Hausdorff dimension of the image and the graph of the original process, respectively.

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