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Extra resources for A note on closed geodesics for a class of non-compact Riemannian manifolds

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9) unmittelbar aus den N + 1 Momenten s0 , . . , sN ergeben. Diese N + 1 Momente gen¨ugen den N − 1 Bedingungen dieses Lemmas, womit also zwei Freiheitsgrade vorliegen. 7 werden noch drei M¨oglichkeiten diskutiert, wof¨ur abk¨urzend s0 := s ( x0 ), sN := s ( xN ) gesetzt wird: Naturliche ¨ Randbedingungen : s0 = sN = 0; Vollst¨andige Randbedingungen : s0 = f0 , sN = fN Periodische Randbedingungen : s0 = s N , s0 = sN . 8 gerechtfertigt. 15) bei der die linke Seite eine Approximation an sj und die rechte Seite eine Differenzenapproximation an f ( xj ) darstellt.

Zun¨achst werden die Approximationseigenschaften anhand des folgenden Beispiels dargestellt. 2 ✻ 1/2 ✻ ........ ............. ... .......... ... .... ... .... .... .... ....... .... . ... ...... . .... .. ... . ... . . ......... . ..... . . ...... . . ....... . . .. .... ... . ... ... . ... . . .... ..... . ... ..... .... . . .... ........ . ... .... . . .......... 1/2 f (x) 0 1/2 1 r( x ) . ......... .... .... ..... ..... . . .....

5 31 Fehlerabsch¨atzungen f¨ur interpolierende kubische Splines beziehungsweise in Vektorschreibweise ⎛ ⎜ ⎝ ⎞ f (x1 ) .. ⎟ ⎠ ⎛ = f (xN −1 ) ⎞ g1 .. ⎜ ⎝ ⎛ ⎟ ⎜ ⎠ − ⎝ gN −1 ⎞ R1 .. ⎛ δ1 .. ⎟ ⎜ ⎠ − ⎝ RN −1 ⎞ ⎟ ⎠. 21) f¨uhrt auf eine Fehlerdarstellung der Form ⎛ ⎜ B⎝ ⎞ f (x1 ) − s (x1 ) .. ⎛ ⎟ ⎠ ⎜ ⎝ = f (xN −1 ) − s (xN −1 ) ⎞ δ1 − δ1 .. ⎟ ⎠. 13 regul¨ar, und mehr noch erh¨alt man mit der Identit¨at hj hj+1 2 − − 3 3( hj + hj+1 ) 3( hj + hj+1 ) 1 , 3 = j = 1, 2, . . , N − 1, die Absch¨atzung max |f ( xj ) − s ( xj ) | ≤ 3 max |δ1 | + | δ1 |, .

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