Question

Let the points \(x_{0}, x_{1}, \ldots, x_{n}\) be fixed and consider the divided difference \(f\left[x_{0}, x_{1}, \ldots, x_{n}, x\right]\) as a function of \(x\). (This function appears as part of the expression for the error in polynomial interpolation.) Suppose next that \(f(x)\) is a polynomial of degree \(m\). Show that \- if \(m \leq n\), then \(f\left[x_{0}, x_{1}, \ldots, x_{n}, x\right] \equiv 0\); \- otherwise \(f\left[x_{0}, x_{1}, \ldots, x_{n}, x\right]\) is a polynomial of degree \(m-n-1\). \begin{aligned} &\text { [Hint: If } m>n, \text { show it first for the case } n=0 . \text { Then proceed by induction, examining the }\\\ &\text { function } \left.g(x)=f\left[x_{1}, \ldots, x_{n}, x\right] .\right] \end{aligned}

Short answer

Answer: Divided difference is a key concept in interpolation, specifically in Newton's divided difference interpolation formula. In the given problem, the divided difference helps us understand the behavior of the polynomial function, and how it is affected by its degree and the number of data points provided. Proving the two statements about divided difference for the given function provides insight into the structure of the polynomial and its derivatives as the number of data points increases.

Step-by-step solution

Definition of divided difference

Recall the definition of divided difference for the function \(f(x)\): \(f[x_0,x_1,\ldots,x_{n-1},x_n,x]=(f[x_1,\ldots,x_n,x]-f[x_0,\ldots,x_{n-1},x])/(x_n - x_{n-1}).\)

Base case for induction \(n=0\)

We are given that \(f(x)\) is a polynomial of degree \(m\). Let's first consider the case when \(n = 0\): \(f[x_0, x] = \frac{f(x) - f(x_0)}{x - x_0}\) Since \(f(x)\) is a polynomial of degree \(m\), when \(m > 0\), we get \(f[x_0, x]\) as a polynomial of degree \((m-1)\) and when \(m=0\), we get \(f[x_0, x] \equiv 0\). This establishes the base case of induction.

Inductive step

Assume that the given statements hold true for some positive integer \(n\). Now consider the divided difference \(f[x_1, \ldots, x_{n},x] = g(x)\), a polynomial of degree \((m-n-1)\). We will now show that the statements hold true for \(n+1\).

If \(m \leq (n+1)\), then \(f[x_0, x_1, \ldots, x_{n+1}, x] \equiv 0\)

Using the definition of divided difference, we can write: \(f[x_0, x_1, \ldots, x_{n+1}, x] = \frac{g(x) - g(x_0)}{x_{n+1} - x_0}\) If \(m \leq (n+1) \Rightarrow m \leq n\), from our induction assumption, we know that \(g(x) \equiv 0\). Therefore, \(f[x_0, x_1, \ldots, x_{n+1}, x] \equiv 0\).

Otherwise, \(f[x_0, x_1, \ldots, x_{n+1}, x]\) is a polynomial of degree \(m-(n+1)-1\)

If \(m > (n+1) \Rightarrow m > n\), then from our induction assumption, we know that \(g(x)\) is a polynomial of degree \((m-n-1)\). Thus, we can write: \(f[x_0, x_1, \ldots, x_{n+1}, x] = \frac{g(x) - g(x_0)}{x_{n+1} - x_0}\) As \(g(x)\) is a polynomial of degree \((m-n-1)\), this implies that \(f[x_0, x_1, \ldots, x_{n+1}, x]\) is a polynomial of degree \([m-n-1 - (m-n)] = (m-(n+1)-1)\). This completes the inductive step and proofs the given statements for all positive integer values of \(n\).

Key concepts

Divided Differences

Divided differences play a crucial role in polynomial interpolation. They are essentially an extension of the concept of a derivative, used to construct interpolation polynomials. The divided difference for a function \(f(x)\) given points \(x_0, x_1, \ldots, x_n\) is defined as:

• \(f[x_0, x_1, \ldots, x_n] = \frac{f[x_1, \ldots, x_n] - f[x_0, \ldots, x_{n-1}]}{x_n - x_0}\)

For example, for two points \(x_0\) and \(x_1\), the divided difference is an approximation of the derivative:

• \(f[x_0, x_1] = \frac{f(x_1) - f(x_0)}{x_1 - x_0}\)

This iterative calculation over more points helps in computing the coefficients of Newton’s interpolation polynomial.
When working with polynomials, divided differences simplify error terms in interpolation. Understanding these differences is key to grasping polynomial interpolation at a deeper level. They not only give insight into the nature of functions but also show how we can approximate complex functions with simpler polynomial forms.

Induction Proof

Induction is a powerful proof technique, often used in mathematics to prove a statement for all natural numbers. In the context of polynomial interpolation, it helps establish that a statement holds for any number \(n\). The exercise uses mathematical induction:

• Base Case: Start by proving the statement for the smallest value, usually \(n=0\) or \(n=1\).
• Inductive Step: Assume the statement is true for \(n = k\), and then prove it for \(n = k+1\).

In our problem, we begin by examining the simplest polynomial degree settings. For \(n=0\), the statement is assessed with a simple form of divided differences.
Once the base case is solid, we assume the property holds for some \(n\) and use this assumption to prove it for \(n+1\). The exercise demonstrates how, under the assumption that the statement holds for an \(n\)-degree polynomial, it can be expanded to hold for any \(n+1\) degree using divided differences. This step-by-step process strengthens the proof, ensuring the statement is valid for all relevant values of \(n\).

Polynomial Degree

The degree of a polynomial is the highest power of the variable within that polynomial. Understanding polynomial degree is vital in polynomial interpolation as it determines how closely a polynomial can fit a set of data points.
In the exercise, we see two scenarios:

• If the polynomial degree \(m\) is less than or equal to \(n\) (the number of points minus one), then the divided difference results in zero. This means the polynomial representation perfectly fits within those constraints.
• If \(m\) is greater than \(n\), the divided difference results in a new polynomial of reduced degree \(m-n-1\). This remaining degree indicates the "excess" capacity of the polynomial to interpolate, reflecting how much "more complex" the polynomial could be given fewer constraints.

Thus, polynomial degree offers a lens through which we can view the interpolation’s accuracy and potential, influencing both the polynomial's construction and its application to data fitting. Understanding this helps us better navigate how polynomials can be tailored to specific interpolation needs.