Taylor series in terms of discrete derivative

In summary, the Taylor series for a function f(x) with derivative given by Hilger's equation is: f(t) = \sum_{k=0}^{n-1}f^{\Delta^k}(a)\, h_{k}(t, a) \, + R.
  • #1
Jhenrique
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All analitic function can be express how: [tex]f(x) = \frac{1}{0!} \frac{d^0f}{dx^0}(x_0) (x - x_0)^0 + \frac{1}{1!} \frac{d^1 f}{dx^1}(x_0) (x - x_0)^1 + \frac{1}{2!} \frac{d^2f}{dx^2}(x_0) (x - x_0)^2 + \frac{1}{3!} \frac{d^3f}{dx^3}(x_0) (x - x_0)^3 + ...[/tex] that is the taylor series of the function f(x).

Analogously, should exist a taylor series that use the discrete derivative ##\frac{\Delta f}{\Delta x} = f[x+1] - f[x]##. I found this
NumberedEquation4.gif
in wolframpage http://mathworld.wolfram.com/ForwardDifference.html but appears strange. I didn't undertood what such series means. Someone can explain me?
 
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  • #2
I'm going to use notation from time scale calculus, which is a unification of continuous and discrete calculus. It makes my message more obvious.

A time scale is a non-empty closed subset ##\mathbb{T}\subset\mathbb{R}##.
Given a function ##f : \mathbb{T}\to\mathbb{R}##, we define the (Hilger) derivative ##f^{\Delta}## in such a way such that
  • if ##\mathbb{T} = \mathbb{R}## then Hilger derivative is equal to ##f^{\Delta}(t) = f'(t)## and
  • if ##\mathbb{T} = \mathbb{Z}## then Hilger derivative is equal to ##f^{\Delta}(t) = \Delta f(t)##.

To study the Taylor theorem, we also have "h-polynomials" which are defined
##h_{0}(t,s) = 1## and
##h_{k+1}(t,s) = \int_s^t h_k(\tau, s) \, \Delta \tau##.
In words "start with the constant and keep integrating over and over".
  • if ##\mathbb{T} = \mathbb{R}## then ##h_k(t,s) = \frac{(t-s)^k}{k!}## and
  • if ##\mathbb{T} = \mathbb{Z}## then ##h_k(t,s) = \frac{(t-s)(t-s-1)(t-s-2)\ldots(t-s-k+1)}{k!}##

The Taylor theorem itself is
##f(t) = \sum_{k=0}^{n-1}f^{\Delta^k}(a)\, h_{k}(t, a) \, + R##
where R is the remainder (I'm not going to write down the remainder term).
  • if ##\mathbb{T} = \mathbb{R}## then
    ##f(t) = f(a) + f'(a) \frac{t-s}{1!} + f''(a) \frac{(t-s)^2}{2!}+\ldots+f^{(k)}(a) \frac{(t-s)^k}{k!} + \ldots## and
  • if ##\mathbb{T} = \mathbb{Z}## then
    ##f(t) = f(a) + \Delta f(a) \frac{t-s}{1!} + \Delta^2 f(a) \frac{(t-s)(t-s-1)}{2!}+\ldots+\Delta^{k}f(a) \frac{(t-s)(t-s-1)\ldots(t-s-k+1)}{k!} + \ldots##.


The important thing is that in discrete calculus these ##\frac{(t-s)(t-s-1)(t-s-2)\ldots(t-s-k+1)}{k!}## serve the same role as ##\frac{(t-s)^k}{k!}## does in the continuous calculus.
 
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  • #3
good answer!
 

1. What is a Taylor series in terms of discrete derivative?

A Taylor series in terms of discrete derivative is a mathematical representation of a function using a sum of its discrete derivatives. It is a generalization of the Taylor series, which uses continuous derivatives.

2. How is a Taylor series in terms of discrete derivative calculated?

A Taylor series in terms of discrete derivative is calculated by taking the discrete derivatives of a function at a specific point and using them to form a series. The series can then be used to approximate the value of the function at nearby points.

3. What is the significance of using discrete derivatives in a Taylor series?

The use of discrete derivatives in a Taylor series allows for a more accurate representation of a function, especially for functions that are not smooth or continuous. It also allows for the approximation of functions that cannot be represented using a traditional Taylor series.

4. How does the order of the Taylor series in terms of discrete derivative affect its accuracy?

The higher the order of the Taylor series in terms of discrete derivative, the more accurate the approximation of the function will be. However, as the order increases, the complexity of the series also increases, making it more difficult to calculate.

5. In what fields is the use of Taylor series in terms of discrete derivative most common?

Taylor series in terms of discrete derivative are commonly used in fields such as physics, engineering, and computer science for approximating and analyzing functions that are discrete and non-smooth. They are also used in numerical analysis and optimization algorithms.

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