Multiscale calculus is a discrete counterpart to continuous calculus as originally developed by Isaac Newton. It is a computer-friendly adaption to the laws of calculus.

Contents

1 How it works 2 The multiscale operators 3 Description of the operators 4 Computer implementation 5 Example: the differential operator 6 Function isomorphism

How it works

In discrete calculus the function values can be only be accessed at discrete positions and through three multiscale operators.

The multiscale operators

大 :

T :

Z :

Description of the operators

1. The evaluation operator 大 (pronounced "da"), returns the functions value at the origin.

2. The translation operator T shifts the function by integer value. Together with 大, evaluations at integer positions are possible.

3. The dilatation operator Z allows a zoom into the function by a factor of 2.

Computer implementation

 // Licensed under the GNU General Public License
 interface Dadim {
    public Object da();
    public void trans(int dir, int length);
    public Dadim zoom(int dir);
 }

For accelerated access to double values in the context of numerical evaluation there is an important subclass DDadim.

 interface DDadim extends Dadim {
    public double doubleValue();
 }

Example: the differential operator

The differential operator d has a discrete counter part Δ. To define it in multiscale calculus, the three operators have to be implemented:

大 Δ = 大 T - 大

T Δ = Δ T

Z Δ = Δ Z

In order to evaluate the differential of f(x) we apply Z to zoom into the desired precision, then T to shift the argument and then 大 for evaluation.

大 Δ f(x) = f(1) - f(0)

Function isomorphism

Continuous functions can be directly interpreted in the context of multiscale calculus just be restricting the access methods. Vice versa, multiscale functions can be returned into continuous functions by limiting the resolution to infinity.

大 f