Perfect spline

In the mathematical subfields function theory and numerical analysis, a univariate polynomial spline of order $m$ is called a perfect spline{{Cite book|last1=Powell|first1=M. J. D.|url=https://books.google.com/books?id=ODZ1OYR3w4cC&dq=perfect+spline+numerical+analysis&pg=PA290|title=Approximation Theory and Methods|last2=Powell|first2=Professor of Applied Numerical Analysis M. J. D.|date=1981-03-31|publisher=Cambridge University Press|isbn=978-0-521-29514-7|pages=290|language=en}}{{Cite book|last=Ga.)|first=Short Course on Numerical Analysis (1978, Atlanta|url=https://books.google.com/books?id=1XbHCQAAQBAJ&dq=perfect+spline+numerical+analysis&pg=PA67|title=Numerical Analysis|date=1978|publisher=American Mathematical Soc.|isbn=978-0-8218-0122-2|pages=67|language=en}}{{Cite book|last=Watson|first=G. A.|url=https://books.google.com/books?id=kjJ8CwAAQBAJ&dq=perfect+spline+numerical+analysis&pg=PA92|title=Numerical Analysis: Proceedings of the Dundee Conference on Numerical Analysis, 1975|date=2006-11-14|publisher=Springer|isbn=978-3-540-38129-7|pages=92|language=en}} if its $m$-th derivative is equal to $+1$ or $-1$ between knots and changes its sign at every knot.

The term was coined by Isaac Jacob Schoenberg.

Perfect splines often give solutions to various extremal problems in mathematics. For example, norms of periodic perfect splines (they are sometimes called Euler perfect splines) are equal to Favard's constants.