For arbitrary f(x), I have no reason to weight the tangents differently, so let k1 = k2. Now, among my assumptions was the following:
f(x) is a function of x. In other words, x is the linear scale of measure.
Suppose I assume that my vector function should have x(t) linear in t. Setting , I get , and the approximation reduces to a polynomial in ξ.
Result: The approximating spline has been reduced to an ordinary polynomial. This result is a practical one. Free to arrange the spline however I choose, I find the straight polynomial to be, in general, an excellent fit (see previous worksheet).
It can be shown that , as defined in the previous worksheet, and with k1=k2=1/3 is identical to the order 3 polynomial p(x) satisfying:
This is simply a confession of ignorance. Without specific knowledge of the behavior of the function, the complexity of the vector spline offers no advantage over ordinary polynomial approximation.
But I happen to know that polynomial approximation is sometimes a very bad idea. So what conditions can I identify to alert me? How can I search for these conditions when I am given only a set of data points? In other words, how can I use the data (and the problem they represent) to make a more informed choice?
One good way to answer a question is to begin with an assertion, and then to find counterexamples.
Onward. http://www.geogebratube.org/material/show/id/148508