Surface Approximation and Data Smoothing Using Generalized Spline Functions |
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Page 10
... implies that Because W is positive definite , the second equation of ( 1.6 ) implies L ( 1-2 ) = 0 , i = 1 , ... , N. Assumption ( 2 ) now im- i ( 1.6 ) plies that 1 2 and assumption ( 1 ) then implies that the solutions = of ( 1.4 ) ...
... implies that Because W is positive definite , the second equation of ( 1.6 ) implies L ( 1-2 ) = 0 , i = 1 , ... , N. Assumption ( 2 ) now im- i ( 1.6 ) plies that 1 2 and assumption ( 1 ) then implies that the solutions = of ( 1.4 ) ...
Page 69
... implies that minimizes σ . By way of contradiction , let us assume that the system of equations ( 3.12 ) has two ... implies 1 - 2 επ - m - 1 ° $ 1 - 2 the first equation Since w is positive definite , the second equation of ( 3.16 ) ...
... implies that minimizes σ . By way of contradiction , let us assume that the system of equations ( 3.12 ) has two ... implies 1 - 2 επ - m - 1 ° $ 1 - 2 the first equation Since w is positive definite , the second equation of ( 3.16 ) ...
Page 80
... implies ( 2m - j - 1 ) ( 2m - j - 1 ) ( ih + ) ― ( ih- ) = X .. ! ( 2m - 1 ) ( -1 ) ij j j = 0 , m - 1 .... i = 0 , N .... ( A.3 ) ( 2m - j - 1 ) ( 2m - j - 1 ) ( 0+ ) = ( 1- ) = 0 If we j = 0 , m - 1 combine equations ...
... implies ( 2m - j - 1 ) ( 2m - j - 1 ) ( ih + ) ― ( ih- ) = X .. ! ( 2m - 1 ) ( -1 ) ij j j = 0 , m - 1 .... i = 0 , N .... ( A.3 ) ( 2m - j - 1 ) ( 2m - j - 1 ) ( 0+ ) = ( 1- ) = 0 If we j = 0 , m - 1 combine equations ...
Contents
UNIVARIATE SMOOTHING SPLINES | 3 |
CHAPTER II | 38 |
BIVARIATE SMOOTHING SPLINES | 58 |
2 other sections not shown
Common terms and phrases
absolutely continuous apply Lemma approximation bivariate bounded variation CALI CALIFO CALIFORNIA UNIVERSITY Chapter coefficients defined DIEGO NVS DIEGO THE LIBRARY ERSITY LIBRARY FORNIA GG f i+j<m I₁ implies interpolating spline kernel theorem L₁ L₂ Lemma LIBRA LIBRARY CALIFORNIA LIBRARY DIEGO LIBRARY ERSITY LIBRARY LIBRARY LIBRARY SAN DIEGO LIBRARY THE UNIVE LIBRARY THE UNIVERSITY LIBRARY UNIVERSITY linear functionals M₂ minimizes obtain polynomial positive definite positive definite matrix Proof RNIA RSITY OF SAN Salt Lake City SAN DIEGO CALIFORNIA SAN DIEGO ERSITY SAN DIEGO LIBRARY SAN DIEGO UNIVERSITY SAN ERSITY smoothing spline space AC Taylor formula tensor products Theorem 1.4 UNIV LIBRARY UNIV RSITY univariate UNIVERSITY CALIFORNIA UNIVERSITY DIEGO UNIVERSITY LIBRARY UNIVERSITY OF CALIFORNIA UNIVERSITY OF SAN University of Utah UNIVERSITY SAN DIEGO UNIVERSITY UNIVERSITY Utah Salt Lake xy f Σ λ Σ Σ ψε ху