Math 2.0

Approximations and expansions Splines Theorem

Minimal curvature property of natural cubic splines

Builds on Cubic splineSpline interpolationRiemann integralIntegration by parts

Statement

Let k≥1k \ge 1, let a=x0<x1<⋯<xk=ba = x_0 < x_1 < \cdots < x_k = b, and let y0,…,yk∈ℝy_0, \dots, y_k \in \R. Let ss be the natural cubic spline with knots x0,…,xkx_0, \dots, x_k such that s(xi)=yis(x_i) = y_i for all ii. Let g ⁣:[a,b]→ℝg \colon [a, b] \to \R be twice continuously differentiable with g(xi)=yig(x_i) = y_i for all ii. Then ∫abs′′(x)2 dx≤∫abg′′(x)2 dx,\int_a^b s''(x)^2 \, dx \le \int_a^b g''(x)^2 \, dx, with equality only if g=sg = s.

Remarks

Let h=g−sh = g - s, which vanishes at every knot. Integration by parts gives ∫abs′′h′′ dx=[s′′h′]ab−∫abs′′′h′ dx\int_a^b s'' h'' \, dx = \bigl[s'' h'\bigr]_a^b - \int_a^b s''' h' \, dx. The boundary term is 00 because s′′(a)=s′′(b)=0s''(a) = s''(b) = 0. On each [xi−1,xi][x_{i-1}, x_i] the function s′′′s''' is a constant cic_i, so the last integral is ∑ici(h(xi)−h(xi−1))=0\sum_i c_i \bigl(h(x_i) - h(x_{i-1})\bigr) = 0. Hence ∫g′′2=∫s′′2+∫h′′2\int g''^2 = \int s''^2 + \int h''^2, and equality forces h′′=0h'' = 0, so hh is linear with two zeros and h=0h = 0.

Example: for two data points (k=1k = 1) the natural cubic spline is the straight line through them. The statement says that every other C2C^2 curve through the two points has ∫g′′2>0\int g''^2 > 0.

The end conditions matter. For other cubic spline interpolants, such as the clamped one, the inequality holds only among functions gg that satisfy the same end conditions.

Going deeper

For a student

The theorem gives a variational characterisation of the natural cubic spline interpolant. It minimises ∫abg′′2\int_a^b g''^2 among the C2C^2 functions gg that satisfy the interpolation conditions. The same holds in the larger Sobolev space H2H^2. The proof shows an orthogonality: s′′s'' is orthogonal to h′′h'' for every hh vanishing at the knots, and this orthogonality is what makes ss the minimiser. Replacing the interpolation conditions by a penalised fit leads to smoothing splines, and the same idea in two variables leads to thin plate splines.

Reading path

To understand this page from the ground up, read these 38 pages in order. The foundations come first.

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Cubic spline, concept, no evidence levelCubic splineconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionRiemann integral, concept, no evidence levelRiemann integralconceptSequence, definition, no evidence levelSequencedefinitionSpline interpolation, concept, no evidence levelSpline interpolationconceptLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementDifferentiable functions are continuous, statement, no evidence levelDifferentiable functi…statementDerivative function, definition, no evidence levelDerivative functiondefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementProduct rule, statement, no evidence levelProduct rulestatementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementHigher derivative, definition, no evidence levelHigher derivativedefinitionExtreme value theorem, statement, no evidence levelExtreme value theoremstatementContinuously differentiable function, definition, no evidence levelContinuously differen…definitionRolle's theorem, statement, no evidence levelRolle's theoremstatementMean value theorem, statement, no evidence levelMean value theoremstatementSecond fundamental theorem of calculus, statement, no evidence levelSecond fundamental th…statementIntegration by parts, statement, no evidence levelIntegration by partsstatementMinimal curvature property of natural cubic splines, statement, no evidence levelMinimal curvature pro…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cubic spline concept
  2. Limit point of a set definition
  3. Local maximum definition
  4. Partition of an interval definition
  5. Riemann integral concept
  6. Sequence definition
  7. Spline interpolation concept
  8. Limit of a function definition
  9. Local minimum definition
  10. Mesh of a partition definition
  11. Riemann sum definition
  12. Real sequence definition
  13. Subsequence definition
  14. Continuity at a point definition
  15. Derivative definition
  16. Riemann integrable function definition
  17. Bounded sequence definition
  18. Limit of a sequence definition
  19. Monotone sequence definition
  20. Continuous function definition
  21. Differentiable function definition
  22. Fermat's theorem on stationary points statement
  23. Convergent sequence definition
  24. Monotone convergence theorem for sequences statement
  25. Every sequence has a monotone subsequence statement
  26. Differentiable functions are continuous statement
  27. Derivative function definition
  28. Bolzano–Weierstrass theorem statement
  29. Product rule statement
  30. Boundedness theorem statement
  31. Higher derivative definition
  32. Extreme value theorem statement
  33. Continuously differentiable function definition
  34. Rolle's theorem statement
  35. Mean value theorem statement
  36. Second fundamental theorem of calculus statement
  37. Integration by parts statement
  38. Minimal curvature property of natural cubic splines statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Spline_interpolation
About this page
Tag28C5WWGX9J
URIm2://28C5WWGX9J
Kindstatement, theorem
WritingW0 stub
MSC41A15, 65D07, 41A05
StatusThis copy of the site has no build data. Run m2 status 28C5WWGX9J.
AboutCubic spline
UsesCubic spline, Spline interpolation, Riemann integral, Integration by parts

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