Polynomial approximation: Chebyshev and Legendre

[ch3cooh]

Chebyshev polynomials and Legendre polynomials are both orthogonal polynomials for determining the least square approximation of a function. Aren't they supposed to give the same result for a given function?
I tried mathematica but the I didn't get the same answer :( Is this precision problem or something else?

In[79]:= Collect[
  Sum[Integrate[LegendreP[i, x]*Sin[2 x], {x, -1, 1}]/
    Integrate[LegendreP[i, x]^2, {x, -1, 1}]*LegendreP[i, x], {i, 0,
    3}], x] // N

Out[79]= 1.94378 x - 1.06264 x^3

In[80]:= Collect[
  Sum[Integrate[
     1/Sqrt[1 - x^2] ChebyshevT[i, x]*Sin[2 x], {x, -1, 1}]/
    Integrate[1/Sqrt[1 - x^2] ChebyshevT[i, x]^2, {x, -1, 1}]*
    ChebyshevT[i, x], {i, 0, 3}], x] // N

Out[80]= 1.92711 x - 1.03155 x^3

 

[Stephen Tashi]

Aren't they supposed to give the same result for a given function?

You'll probably get more help if you explain what result you are talking about in standard mathematical notation.

 

[LCKurtz]

Chebyshev polynomials and Legendre polynomials are both orthogonal polynomials for determining the least square approximation of a function. Aren't they supposed to give the same result for a given function?

I haven't looked at that stuff in years, but I think the answer is "no". They have different weight functions which give different inner products and consequently different norms in which to measure the least squares fit.

 

[Stephen Tashi]

I think "least squares fit" makes it clear what norm is used to measure the fit. For a "nice" function (one that can be expressed as an infinite power series) , if you express it as an infinite series of Chebyshev polynomials you get an exact fit. You also get an exact fit if you express it as an infinite series of some other family of othrogonal polynomials. So perhaps the question has to do with approximating the function as a finite series.

I find the Mathematica code in the original post puzzling. It looks like its computing a ratio between a function and the approximating function.

 

[CellCoree]

I would first like to clarify that while Chebyshev and Legendre polynomials are both useful for polynomial approximation, they are not necessarily expected to give the same result for a given function. This is because they have different properties and are optimized for different types of functions.

Chebyshev polynomials are typically used for approximating functions over a specific interval, such as the interval [-1, 1]. They have the property of minimizing the maximum error over this interval, making them useful for applications such as numerical integration and solving differential equations.

On the other hand, Legendre polynomials are more commonly used for approximating smooth functions over a wider interval, such as [-∞, ∞]. They have the property of minimizing the mean squared error over the entire interval, making them useful for applications such as signal processing and data fitting.

In regards to the Mathematica results, it is possible that there could be a precision issue or some other technical issue causing the discrepancy. However, it is also important to note that the two methods are not equivalent and may give slightly different results. It would be helpful to further investigate and compare the properties and limitations of both methods to determine the best approach for a specific function.