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In Schumaker, Larry L. "Computing bivariate splines in scattered data fitting and the finite-element method." Numerical Algorithms 48.1-3 (2008): 237-260 (Link to journal, link to author page for PDF), Section 5, the authors claim that

It follows from standard Hilbert space approximation results that there exists a unique DLSQ-spline $s_l$ which is characterized by the property $$<s_l−f, \psi>_A=0,\quad \forall \psi\in S(\bigtriangleup)$$

I Know that every Polynomial spline $s$ can be presented in $B$-form (Bezier-Bernestein Form) that exists a $\{c_\xi\}\subseteq \mathbb{R}$ such that $$s = \sum_{|\xi|=d} c_\xi B_\xi^d.$$ In this case is standard Hilbert space approximation results like Galerkin methods (link to Galerkin wiki entry) in numerical analysis? What is "standard Hilbert space approximation results" around this content?

Any guide is welcome.

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On section 4 of Von Golitschek, Manfred, Ming-Jun Lai, and Larry L. Schumaker. Error bounds for minimal energy bivariate polynomial splines. Numer. Math (1999), authors show that problem of finding a spline with minimal energy surface can formulated into a standard approximation problem in Hilbert space. For DLSQ splines also we can follow their attitude.

Thus it is not Galerkin method like and the expression "standard Hilbert space approximation results" refers to converted problem into Hilbert space literature.

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