- - - - - figure contour plots for the prediction function (leftand the peaks function given in ( (rightorder derivatives consider functions [ that are twice differentiable and define kg : ( ( )) dx as measure of the size of the second derivative example (behavior of kg intuitivelythe larger kg isthe more "wigglythe function will be as an explicit exampleconsider (xsin(oxfor [ ]where is free parameter we can explicitly compute ( - sin(ox)and consequently ( sinc( )kg sin (oxdx as |othe frequency of increases and we have kg nowin the context of data fittingconsider the following penalized least-squares optimization problem on [ ] (yi (xi )) kg gg = min ( where we will specify in what follows in order to apply the kernel machinerywe want to write this in the form ( )for some rkhs and null space clearlythe norm on should be of the form kgkh kg and should be well-defined ( finite and ensuring and are absolutely continuousthis suggests that we take { [ kg gg absolutely continuousg( ( }with inner product fgih : (xg (xdx one rationale for imposing the boundary conditions ( ( is as followswhen expanding about the point taylor' theorem (with integral remainder termstates that (xg( ( ( ( sds