Example 8.  Compare the "discrete interpolation polynomial" and "Legendre series approximation," on the interval  [Graphics:Images/LegendrePolyMod_gr_323.gif].  
8 (b).  Use the function  [Graphics:Images/LegendrePolyMod_gr_325.gif].  

Solution 8 (b).

[Graphics:../Images/LegendrePolyMod_gr_336.gif]



[Graphics:../Images/LegendrePolyMod_gr_337.gif]

[Graphics:../Images/LegendrePolyMod_gr_338.gif]

 

 

Details for this computation.

First, construct the coefficients.

[Graphics:../Images/LegendrePolyMod_gr_339.gif]



[Graphics:../Images/LegendrePolyMod_gr_340.gif]

 

 

Construct the terms.

[Graphics:../Images/LegendrePolyMod_gr_341.gif]



[Graphics:../Images/LegendrePolyMod_gr_342.gif]

[Graphics:../Images/LegendrePolyMod_gr_343.gif]

 

 

Then add up the terms.

[Graphics:../Images/LegendrePolyMod_gr_344.gif]


[Graphics:../Images/LegendrePolyMod_gr_345.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005