Does spline function contains normal function Usually we say a spline is a function that is piecewise defined by polynomial functions with high degree of smoothness. Thus the `usual' spline function has several polynomial segments. I am wondering a `normal' function, which has only one segment, say x^2, can also be considered as a spline ?

> I am wondering a `normal' function, which has only one segment, say x^2, can also be considered as a spline?

I will interpret your "normal function" as an infinitely smooth function. Next, by "spline" I mean the well-known tensor-product splines (so that its spline space is "complete").

The answers then are:

1) **Any polynomial is a spline**. This is due to the fact (see Reference (R) below) that any piecewise polynomial with sufficient smoothness at the spline knots (break points) can be represented as a linear combination of B-splines. Polynomials are special cases where the pieces join $C^\infty$ at any set of discrete knots.

Reference: (R) Corollary 2 at the bottom of page 11 of this online doc: <

2) **Any infinitely smooth but non-polynomial function is obviously NOT a spline**.