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BIVARIATE SMOOTHING SPLINES AS TENSOR PRODUCTS
The Operators G and G
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_ m-l absolutely continuous function AC a,b AC^m apply Lemma assume bivariate bounded variation Bxy[f Chapter Committee Supervisory Committee compute differentiation is performed ds dt Fm a,b G and G G G f Gx[f GxGy[f i+j<m i<p c i=l j=l ID i,D implies interpolating spline j<q a Lemma l.3 Lemma l.l9 linear functionals mean value theorem nonsingular obtain P,q P,q positive definite matrix problem Ps(f pseudonorm Q.E.D. Lemma relax the interpolation s e a,b Salt Lake City Sard Sard's kernel theorem simple polynomial splines smoothing spline smooths the data solution space AC spline functions spline of interpolation spline of Theorem Supervisory Committee Supervisory symmetric system of equations Taylor formula tensor products Theorem l.4 univariate splines University of Utah Utah Salt Lake values weighting matrix x-t)k_l xy xy xy y x y e c,d zero