Polynomial Approximation from Chebyshev coefficients

Question

I would like to expand a function $f(r)$ in the domain $[0,R]$, around the points $r =0$, and $r = R$ in the following manner

$f(r = 0) = \Sigma_{i=0,i = even}^{imax} f_i (r/R)^i$

and

$f(r = R) = \Sigma_{k=0}^{kmax} f_k (1 - r/R)^k$

I would like to get the coefficients $f_i$ and $f_k$ by first decomposing $f(r)$ into Chebyshev's and then turn that into the power series. Here is what I have so far for a random function :

g[x_] = 0.17768*x^5 + 0.115594*x^4 - 0.049490*x^3 - 0.659085*x^2 - 0.2254209;

The function looks like

Now to expand $g[x]$ in terms of Chebyshev I do this

In[8]:= GS = 6; (*gridsize*)

In[9]:= Clear[A, a]

In[10]:= A = Array[a, GS + 1, 0]

Out[10]= {a[0], a[1], a[2], a[3], a[4], a[5], a[6]}

In[11]:= For[i = 1, i <= GS + 1, i++, a[i] = 2/\[Pi]*   
 (Integrate[g[y]*ChebyshevT[i, y] 1/Sqrt[1 - y^2], {y, -1, 1}])]

In[12]:= a[0] = 1/\[Pi]* (Integrate[g[y]*ChebyshevT[0, y] 1/Sqrt[1 - y^2],    
 {y, -1, 1}])

Out[12]= -0.511616

In[13]:= A

Out[13]= {-0.511616, 0.0739325, -0.271746, 0.0431525, 0.0144492, 0.011105, -1.00221*10^-15}

In[14]:= t1 = Table[a[i]*ChebyshevT[i, y]  , {i, 0, GS}]

Out[14]= {-0.511616, 0.0739325 y, -0.271746 (-1 + 2 y^2), 0.0431525 
(-3 y + 4 y^3), 0.0144492 (1 - 8 y^2 + 8 y^4), 0.011105 (5 y - 20 y^3 + 16    
y^5), -1.00221*10^-15 (-1 + 18 y^2 - 48 y^4 + 32 y^6)}

In[15]:= g1[y_] = Sum[t1[[i]], {i, 1, GS + 1}]  

Out[15]= -0.511616 + 0.0739325 y - 0.271746 (-1 + 2 y^2) 
+ 0.0431525 (-3 y + 4 y^3) + 0.0144492 (1 - 8 y^2 + 8 y^4) 
+ 0.011105 (5 y - 20 y^3 + 16 y^5) -1.00221*10^-15 (-1 + 18 y^2 - 48 y^4 +      
32 y^6)

Now I plot the function and the Chebyshev approximation $g1[x]$.

and you can not really see the difference between the two plots, so this makes me think I am doing the Chebyshev part correctly.

After this I attempt to get the power series approximation using (Numerical Recipes sectinon 5.10, Clenshaw recurrence)

In[17]:= Clear[D1, d]

In[18]:= D1  = Array[d, GS]

Out[18]= {d[1], d[2], d[3], d[4], d[5], d[6]}

In[19]:= d[GS + 1] = 0

Out[19]= 0

In[20]:= d[GS] = 0

Out[20]= 0

In[23]:= For[i = GS - 1, i >= 1, i--, d[i] = 2*x*d[i + 1] - d[i + 2] + a[i]]

In[24]:= D1

Out[24]= {0.041885 - 2 (0.0144492 + 0.02221 x) x 

+ 2 x (-0.286195 - 0.02221 x + 2 x (0.0320475 + 2 (0.0144492 + 0.02221 x) x)), -0.286195 - 0.02221 x + 2 x (0.0320475 + 2 (0.0144492 + 0.02221 x) x), 0.0320475 + 2 (0.0144492 + 0.02221 x) x, 0.0144492 + 0.02221 x, 0.011105, 0}

In[27]:= d[0] = x*d[1] - d[2] + a[0]

Out[27]= -0.225421 + 0.02221 x - 2 x (0.0320475 + 2 (0.0144492 + 0.02221 x) x) + x (0.041885 - 2 (0.0144492 + 0.02221 x) x + 2 x (-0.286195 - 0.02221 x + 

2 x (0.0320475 + 2 (0.0144492 + 0.02221 x) x)))

In[28]:= f[x_] = Simplify[d[0]]

Out[28]= -0.225421 + 5.48173*10^-16 x - 0.659085 x^2 - 0.04949 x^3 + 0.115594 x^4 + 0.17768 x^5

Now I plot all there versions

and again I can see no difference between different versions.

I take this to mean that I have used the Clenshaw recurrence correctly.

Question 1: Did I really use it correctly or did I just get lucky?

Question 2: How would I change my code to deal with the function $f(r)$ around the origin which has to be normalized in its domain?

Question 3: Where do I even begin with $f(r)$ around $r =R$ since that does not have the normal power series form?

Answer 1

I am going to go out on a limb here as my mathematics is not as solid as I would like in this area. But, the primary difference between a Taylor series and expansion in terms of Chebyshev polynomials is the Chebyshev expansion is global while the Taylor series is not. Hence, the phrase "expanding around a point" is not applicable in the Chebyshev case. Instead, you are changing the basis your function is expressed in. Quite literally, the integral

$$f_i = \int^1_{-1} \frac{T_i(x) f(x)}{\sqrt{1 - x^2}} dx$$

is the projection of $f(x)$ onto the polynomial $T_i(x)$. Fourier series operates the same way.

As to your first question of whether you got lucky: no, my own experimentation with this several years ago reveals the same behavior. For instance, here's a comparison of the absolute errors ($\operatorname{abs}(f(x) - \sum_i f_i(x))$) for a 14 term Chebyshev vs. a 14 term Taylor expansion about 0 for $\sin x$:

The Taylor expansion does quite well for a broad range, but the Chebyshev is consistently at the $10^{-18}$ level which is on par with machine precision. In fact, to plot it correctly, I had to increase the WorkingPrecision to see anything at all!

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To turn this into a power series, simply substitute in the Chebyshev polynomials into the expansion

$$f(x) \approx \sum^N_i f_i T_i(x)$$

and rearrange. For the first three terms of $\sin(x)$, the Chebyshev expansion is

0.999979 x - 0.166497 x^3 + 0.00799225 x^5

which the Taylor series closely resembles

x - 0.166667 x^3 + 0.00833333 x^5

Obviously, they diverge more extensively as the number of terms increases.