Example 8.  Compare the "discrete interpolation polynomial" and "Legendre series approximation," on the interval  [Graphics:Images/LegendrePolyMod_gr_323.gif].  
8 (a).  Use the function  [Graphics:Images/LegendrePolyMod_gr_324.gif].   

Solution 8 (a).

[Graphics:../Images/LegendrePolyMod_gr_326.gif]



[Graphics:../Images/LegendrePolyMod_gr_327.gif]

[Graphics:../Images/LegendrePolyMod_gr_328.gif]

 

 

Details for this computation.

First, construct the coefficients.

[Graphics:../Images/LegendrePolyMod_gr_329.gif]



[Graphics:../Images/LegendrePolyMod_gr_330.gif]

 

 

Construct the terms.

[Graphics:../Images/LegendrePolyMod_gr_331.gif]



[Graphics:../Images/LegendrePolyMod_gr_332.gif]

[Graphics:../Images/LegendrePolyMod_gr_333.gif]

 

 

Then add up the terms.

[Graphics:../Images/LegendrePolyMod_gr_334.gif]


[Graphics:../Images/LegendrePolyMod_gr_335.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005