interpolation

An already existing linear relationship, such as the expansion of a polynomial of n^th degree in terms of powers xⁱ of degrees zero through n,

$$\sum_{i=0}^na_ix^i,$$ p(x) = (211)

can be written as the inner product of two vectors, say a row of coefficients $\{a_0, a_1, \ldots, a_n\}$ and a column of powers, $\{1, x, \ldots, x^n\}$. Any change of basis, such as by replacing powers xⁱ with Newton polynomials $n^{(i)}(x)=x(x-1)\cdots(x-i+1)$, gotten by linear transformation, would be represented by a matrix whose inverse transforms the coeffients, keeping the representation intact.

If the expansion is not available, but it is desired to have a certain basis together with a family of linear functionals from which the coefficients are obtained, there is a determinant which can be used. For example, the combination of powers $\{x^i\}$ and values at a set of points $(x_0,x_1, \ldots, x_n\}$ will produce the Lagrange interpolation polynomials via the Vandermonde determinant. Write

$$\left\vert \begin{array}{ccccc} 1 & 1 & 1 & \cdots & 1 \\ x_1 & ... ...& x_3^n & \cdots & x^n \\ y_1 & y_2 & y_3 & \cdots & y \end{array} \right\vert$$ = 0. (212)

The rows correspond to basis elements, the columns to linear functionals; the vanishing of the determinant results from duplicating a column whenever the value of the functional at the additional point matches the interpolation. Assigning suitable names to the pieces resulting from partitioning the determinant according to its last row and column, and the corresponding Laplace development, the equation reads

y = Y^TM^-1X. (213)

This time there is a metric matrix which has to be calculated by inverting the matrix of mutual interactions, which will be possible if the two sets of constituents are actually linearly independent and form bases. The resulting representation formula will always work, but the question of its convergence as the number of reference points increases is usually fairly complicated. With power or polynomial bases, the degree of the polynomials is correspondingly high leaving the interior of the convex hull of the points as the only region in which a useful approximation should be expected. Whether that expectation can be realized, and many other points of interest, are discussed in Walsh's Colloquium Publication [27].