What does normalized (k + 1)th divided difference at distinct nodes mean?

First, we show that h^{(1)}(0)=0 by applying Rolle's theorem to h and its first derivative h^{(1)}(x) on the interval (-1/n,1/n) for each n. Then, assuming h^{(k)}(0)=0, we use the definition of the (k+1)th divided difference to show that h^{(k+1)}(0)=0. Therefore, by induction, h^{(k)}(0)=0 for all k \in N. In summary, we are using induction and Rolle's theorem to show that for an infinitely differentiable function h with h(1/n)=0 for all n \in N, all of its
  • #1
CantorSet
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Hi everyone,

This is not a homework question but a clarification on the following proof:

Suppose [itex]h[/itex] is an infinitely differentiable real-valued function defined on [itex]/Re[/itex] such that [itex]h(1/n)=0[/itex] for all [itex] n \in N [/itex]. Then prove [itex]h^{(k)}(0)=0[/itex] for all [itex] k \in [/itex].

Proof: Since h is infinitely differentiable in a neighborhood of 0, the kth derivative of h at 0 is the limit of its normalized (k+1)th divided difference at distinct nodes [itex]x_1,x_2,...,x_{k+1}[/itex] as they tend to 0: [tex] h^{(k)}(0)=k! \lim_{x_1,x_2,...,x_{k+1} \rightarrow 0 } \nabla (x_1,...,x_{k+1})h [/tex]

Now, choosing [itex] x_j := x_j(n) = \frac{1}{(n+j)}[/itex] and letting [itex] n \in \aleph [/itex] tend to infinity, we see that [itex] h^{k}(0) = 0 [/itex] for all [itex] k \in N [/itex].

I don't understand the proof above since I don't know what normalized (k+1)th divided difference at distinct nodes means. Does anyone know?
 
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  • #2
To rephrase the proof, we are using induction on k and Rolle's theorem.
 

FAQ: What does normalized (k + 1)th divided difference at distinct nodes mean?

What does normalized (k + 1)th divided difference at distinct nodes mean?

The normalized (k + 1)th divided difference at distinct nodes refers to a mathematical concept used in numerical analysis and interpolation. It is a way to measure the difference between two data points at a specific interval, and is commonly used to estimate values between known data points.

How is the normalized (k + 1)th divided difference calculated?

The normalized (k + 1)th divided difference is calculated by taking the difference between the (k + 1)th data point and the kth data point, and then dividing it by the difference between the kth and (k-1)th data points. This value can then be multiplied by a factor to scale it to a specific interval.

What is the significance of using distinct nodes in normalized divided differences?

Using distinct nodes in normalized divided differences ensures that the data points used are not repeated, which can lead to inaccurate results. This method also allows for a more efficient and accurate interpolation process.

How is the normalized (k + 1)th divided difference used in interpolation?

The normalized (k + 1)th divided difference is used in interpolation to estimate values between known data points. It is typically used in conjunction with other interpolation methods, such as Newton's divided differences, to improve the accuracy of the estimated values.

What are some common applications of normalized divided differences?

Normalized divided differences are commonly used in various fields, including engineering, physics, and computer science. They are used in data analysis, curve fitting, and interpolation to estimate values between known data points. They are also used in numerical methods for solving differential equations and optimization problems.

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