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In the mathematical subfields function theory and numerical analysis, a univariate polynomial spline of order $m$ is called a perfect spline^[1]^[2]^[3] 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.

References

^ Powell, M. J. D.; Powell, Professor of Applied Numerical Analysis M. J. D. (1981-03-31). Approximation Theory and Methods. Cambridge University Press. p. 290. ISBN 978-0-521-29514-7.
^ Ga.), Short Course on Numerical Analysis (1978, Atlanta (1978). Numerical Analysis. American Mathematical Soc. p. 67. ISBN 978-0-8218-0122-2.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
^ Watson, G. A. (2006-11-14). Numerical Analysis: Proceedings of the Dundee Conference on Numerical Analysis, 1975. Springer. p. 92. ISBN 978-3-540-38129-7.

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[1] Powell, M. J. D.; Powell, Professor of Applied Numerical Analysis M. J. D. (1981-03-31). Approximation Theory and Methods. Cambridge University Press. p. 290. ISBN 978-0-521-29514-7.

[2] Ga.), Short Course on Numerical Analysis (1978, Atlanta (1978). Numerical Analysis. American Mathematical Soc. p. 67. ISBN 978-0-8218-0122-2.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)

[3] Watson, G. A. (2006-11-14). Numerical Analysis: Proceedings of the Dundee Conference on Numerical Analysis, 1975. Springer. p. 92. ISBN 978-3-540-38129-7.

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