By Ingold L.

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**Example text**

Cos 2 ! ” cos 2 ! D 0 , ) ! D ` 8` 2 f0; 1; : : : ; L sin 2 ! D 0 ! 184) 1; Lg : 1; Lg concludes the proof. How can we setup a linear model for Fourier analysis? 15. Here we shall assume that the observed data have been made available on an equidistant angular grid, in short equidistant lattice of the unit circle parameterized by . 1 ; : : : ; n /. i 1/ 2I 8i 2 f1; : : : ; I g is “D-optimal”. 15 contains four examples for such a lattice. 9 shows such a lattice. In summary, the finite dimensional observation space Y, dimY D n, n D I , has integer dimension I .

B, b 7 ! 101, right). Indeed, the transformations x 7 ! x , a 7 ! ex , and y 7 ! y , b 7 ! ey are dual or inverse. Fourth, let us begin the eigenspace analysis versus 1-2 Minimum Norm Solution (MINOS) 27 eigenspace synthesis of the rectangular matrix, A 2 Rn m , r WD rkA D n; n < m. Indeed, the eigenspace of the rectangular matrix looks differently when compared to the eigenspace of the quadratic, symmetric, positive-definite matrix Gx 2 Rm m , rkGx D m and Gy 2 Rn n , rkGy D n of the left and the right metric.

K C 1/; l 2 for integer values k 2 f0; 1; : : : ; 1g, l 2 f k; k C 1; : : : ; 1; 0; 1; : : : ; k 1; kg. Thanks to such a structure of the infinite-dimensional eigenspace of the Laplace–Beltrami operator we discuss the solutions of the underdetermined regression problem (linear algebraic regression) in the context of “canonical MINOS”. We solve the system of linear equations fAx D yjA 2 Rn m ; rkA D n; n mg by singular value decomposition as shortly outlined in Appendix A. 1-31 Fourier Series Fourier series represent the periodic behavior of a function x.