(a) Give an example of an interpolating polynomial through...

(a) Give an example of an interpolating polynomial through $n+1$ points that has degree $<n$. (b) Can you explain, without referring to the explicit formulas, why all the Lagrange interpolating polynomials based on $n+1$ points must have degree equal to $n$ ?

Key Concepts

Interpolating Polynomial

An interpolating polynomial is a polynomial constructed to pass through a set of given points. In the context of numerical methods and approximation theory, it is used to estimate functions when only a finite set of data is available. The uniqueness and degree of the interpolating polynomial depend on the number of points and the configuration of those points.

Degree of a Polynomial

The degree of a polynomial is the highest exponent in its expression when written in standard form. For interpolation, a set of n+1 points generally leads to a unique polynomial of degree at most n. However, if the points have a special structure (such as lying on a lower-degree curve), the resulting interpolating polynomial may have a degree lower than n.

Lagrange Interpolation

Lagrange interpolation is a method to construct an interpolating polynomial using a linear combination of specially designed basis polynomials. Each basis polynomial is constructed in such a way that it is equal to one at one specific interpolation point and zero at all others. This method guarantees that when the basis polynomials are combined with the corresponding function values, the resulting polynomial matches the given data exactly.

Uniqueness of Polynomial Interpolation

The theorem of uniqueness in polynomial interpolation states that for a given set of n+1 distinct points, there exists exactly one polynomial of degree at most n that interpolates the points. This concept underpins many interpolation methods, including Lagrange's, and assures us that unless the data points are degenerate (i.e., do not uniquely determine the polynomial), the degree of the interpolating polynomial will be exactly n.

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