Example 7.  Start with  [Graphics:Images/LegendrePolyMod_gr_276.gif]  over the interval [Graphics:Images/LegendrePolyMod_gr_277.gif].  Use the Gram-Schmidt orthogonalization to construct a few of the orthogonal polynomials [Graphics:Images/LegendrePolyMod_gr_278.gif]  over the interval [Graphics:Images/LegendrePolyMod_gr_279.gif].  Construct the corresponding Legendre polynomials  [Graphics:Images/LegendrePolyMod_gr_280.gif].

Solution 7.

We can let Mathematica do it all using recursion as follows.  First define the inner product and the recursive formulas.

[Graphics:../Images/LegendrePolyMod_gr_281.gif]

The polynomials are generated using the above recursive formulas.  

[Graphics:../Images/LegendrePolyMod_gr_282.gif]



[Graphics:../Images/LegendrePolyMod_gr_283.gif]

 

 

Details

Set up the inner product function.  

[Graphics:../Images/LegendrePolyMod_gr_284.gif]

Construct the first few orthogonal polynomials.  

[Graphics:../Images/LegendrePolyMod_gr_285.gif]



[Graphics:../Images/LegendrePolyMod_gr_286.gif]

 

 

[Graphics:../Images/LegendrePolyMod_gr_287.gif]




[Graphics:../Images/LegendrePolyMod_gr_288.gif]

[Graphics:../Images/LegendrePolyMod_gr_289.gif]

 

 

 

[Graphics:../Images/LegendrePolyMod_gr_290.gif]

[Graphics:../Images/LegendrePolyMod_gr_291.gif]

 

 

 

[Graphics:../Images/LegendrePolyMod_gr_292.gif]

[Graphics:../Images/LegendrePolyMod_gr_293.gif]

 

 

 

[Graphics:../Images/LegendrePolyMod_gr_294.gif]

[Graphics:../Images/LegendrePolyMod_gr_295.gif]

 

 

These orthogonal polynomials are related to the Legendre polynomials, all we need to do is normalize them so that  [Graphics:../Images/LegendrePolyMod_gr_296.gif].

[Graphics:../Images/LegendrePolyMod_gr_297.gif]



[Graphics:../Images/LegendrePolyMod_gr_298.gif]

[Graphics:../Images/LegendrePolyMod_gr_299.gif]

 

 

 

[Graphics:../Images/LegendrePolyMod_gr_300.gif]

[Graphics:../Images/LegendrePolyMod_gr_301.gif]

[Graphics:../Images/LegendrePolyMod_gr_302.gif]

 

 

 

[Graphics:../Images/LegendrePolyMod_gr_303.gif]



[Graphics:../Images/LegendrePolyMod_gr_304.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005