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Approximation by Orthogonal Polynomials

​
function
sin(πx)
-x
θ(x)
xθ(x)
(x+0.01)θ(x)
degree
1
2
3
4
5
interpolation
interpolation error
interpolation polynomials
degree
normalized Legendre polynomial
0
1
2
1
3
2
x
2
3
2
5
2
2
x
-
5
2
2
3
5
2
7
2
3
x
-
3
2
7
2
x
4
105
4
x
8
2
-
45
2
x
4
2
+
9
8
2
5
63
8
11
2
5
x
-
35
4
11
2
3
x
+
15
8
11
2
x
The Gram–Schmidt process is used to obtain orthonormal polynomials with respect to the
2
L
norm. These polynomials are then used to interpolate a number of functions; a logarithmic error plot is provided.
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