Example 9.  Compare the "discrete interpolation polynomial" and "shifted Legendre series approximation"
9 (a).  Use the function  [Graphics:Images/LegendrePolyMod_gr_369.gif],  on the interval  [Graphics:Images/LegendrePolyMod_gr_370.gif].  

Solution 9 (a).

[Graphics:../Images/LegendrePolyMod_gr_373.gif]



[Graphics:../Images/LegendrePolyMod_gr_374.gif]

[Graphics:../Images/LegendrePolyMod_gr_375.gif]

 

 

Details for this computation.

First, construct the coefficients.

[Graphics:../Images/LegendrePolyMod_gr_376.gif]



[Graphics:../Images/LegendrePolyMod_gr_377.gif]

 

 

Construct the terms.

[Graphics:../Images/LegendrePolyMod_gr_378.gif]



[Graphics:../Images/LegendrePolyMod_gr_379.gif]

[Graphics:../Images/LegendrePolyMod_gr_380.gif]

 

 

Then add up the terms.

[Graphics:../Images/LegendrePolyMod_gr_381.gif]


[Graphics:../Images/LegendrePolyMod_gr_382.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005