# AN ELEMENTARY PROOF OF THE OPTIMAL RECOVERY OF THE THIN PLATE SPLINE RADIAL BASIS FUNCTION

• KIM, MORAN (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY) ;
• MIN, CHOHONG (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY)
• Received : 2015.10.30
• Accepted : 2015.12.15
• Published : 2015.12.25

#### Abstract

In many practical applications, we face the problem of reconstruction of an unknown function sampled at some data points. Among infinitely many possible reconstructions, the thin plate spline interpolation is known to be the least oscillatory one in the Beppo-Levi semi norm, when the data points are sampled in $\mathbb{R}^2$. The traditional proofs supporting the argument are quite lengthy and complicated, keeping students and researchers off its understanding. In this article, we introduce a simple and short proof for the optimal reconstruction. Our proof is unique and reguires only elementary mathematical background.

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

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