I know that a spline basically involves joining several functions together. I am hoping to get a more precise definition so I know exactly when to use the word "spline". I have a few questions regarding splines.
- Does every piecewise function have to be the same form? For example, if one function is a cubic polynomial, does every other function have to be a cubic polynomial as well? Or can we join different functions like a quartic and a quadratic?
- Do the points where they join have to be both continuous AND continuous on the 1st and 2nd derivative? Or can it just be continuous even if the derivatives are not continuous?
Thanks!
A spline is simply a piecewise continuous curve. That is all. There is no restriction on whether each "piece" should be a polynomial function. Exponential splines and trignometric splines do exist as alternatives to the traditional polynomial splines.
For polynomial splines, it is common to use the same degree for every piecewise function. But this is not a requirement for it to be called a spline. I have seen papers advocating splines created by joining polynomials of different degree together.
For continuity, $C^0$ continuity is typically the only requirement. But in some scientific fields, this requirement might even be relaxed.