GramSchmidt process for Taylor basis

In summary, the limits for the integral depend on the topological space and inner product used. In the example given, the space ##L_2([-1,1])## was considered, making the choice of the interval ##[-1,1]## logical since Taylor series are often developed around ##x=0##. Different spaces result in different bases, and the choice is determined by the space and algorithm used. The reason for choosing the interval ##[-1,1]## is not solely based on the desire for an orthonormal basis, as there are other examples where the integration is over the entire manifold. The allowed functions must also be taken into consideration, and the Hilbert space must be defined before discussing the algorithm and
  • #1
Somali_Physicist
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Why are the limits as so for the integral?
 

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  • #2
In the end it depends which topological space you consider and how the inner product is defined. Those two information set the conditions. As it is an example, we can simply consider ##L_2([-1,1])##. Taylor series are often developed around ##x=0##, so the choice of this interval makes some sense.
 
  • #3
fresh_42 said:
In the end it depends which topological space you consider and how the inner product is defined. Those two information set the conditions. As it is an example, we can simply consider ##L_2([-1,1])##. Taylor series are often developed around ##x=0##, so the choice of this interval makes some sense.
This arrives at the legendary basis so It is abit fishy that it is a “coincidence “.So I could pic any numbers around 0?
 
  • #4
Somali_Physicist said:
This arrives at the legendary basis so It is abit fishy that it is a “coincidence “.So I could pic any numbers around 0?
I haven't checked the algorithm. You might get additional scaling factors for your basis vectors by a different inner product, i.e. other spaces. Depends on whether you want to have a orthonormal or orthogonal basis.
 
  • #5
fresh_42 said:
I haven't checked the algorithm. You might get additional scaling factors for your basis vectors by a different inner product, i.e. other spaces. Depends on whether you want to have a orthonormal or orthogonal basis.
I don’t believe that matters here, I just want to clarify why the limits were used and can we change them ad hocly.
 
  • #6
Somali_Physicist said:
I don’t believe that matters here, I just want to clarify why the limits were used and can we change them ad hocly.
As I said, the limits are determined by the space you consider, which again is determined by the elements and the inner product. You must now reason backwards. First is the space, then algorithm and at the end the basis. Another space results in another basis.
 
  • #7
fresh_42 said:
As I said, the limits are determined by the space you consider, which again is determined by the elements and the inner product. You must now reason backwards. First is the space, then algorithm and at the end the basis. Another space results in another basis.
Ahh okay, well we want orthanormal bases, is it because of that we chose -1 to 1?
 
  • #8
Somali_Physicist said:
Ahh okay, well we want orthonormal bases, is it because of that we chose -1 to 1?
No. One possibility for an inner product for say, continuous functions on a real interval ##[a,b]## is e.g. ## \langle f,g \rangle = \int_a^b f(x)g(x)\,dx\,.## This determines how the algorithm is done and what basis finally drops out. You start with the Hilbert space which you want to consider, not the other way around. At the end of this article here https://www.physicsforums.com/insights/hilbert-spaces-relatives/ you can find some other examples. E.g. 5.10 is worth a look. There the integration is over the entire manifold. Of course in the above example we would get infinite integrals if we regarded ##\mathbb{R}## as our domain, so we need some restrictions on the allowed functions. But in general there is a variety of examples. As soon as you define and thus tell us, which Hilbert space you use, as soon can we talk about the algorithm and the borders of integration: different borders means different Hilbert spaces. So before the question even occurs, it has to be said, what we are talking about. I admit that this information isn't part of what you copied and in my opinion sloppy.
 

What is the Gram-Schmidt process for Taylor basis?

The Gram-Schmidt process for Taylor basis is a method for finding an orthonormal basis for a vector space of polynomials. It involves taking a set of linearly independent polynomials and applying a series of orthogonalization steps, resulting in a set of polynomials that are both orthogonal and normalized.

Why is the Gram-Schmidt process important in mathematics?

The Gram-Schmidt process is important because it allows us to construct an orthonormal basis for a vector space, which is useful in many areas of mathematics. This includes applications in linear algebra, functional analysis, and numerical analysis.

What are the steps involved in the Gram-Schmidt process for Taylor basis?

The steps involved in the Gram-Schmidt process for Taylor basis include finding a set of linearly independent polynomials, calculating the inner product of each polynomial with respect to the others, and using these inner products to modify the polynomials until they are orthogonal and normalized.

What are some applications of the Gram-Schmidt process for Taylor basis?

The Gram-Schmidt process for Taylor basis has numerous applications in mathematics, including solving systems of linear equations, approximating functions and solutions to differential equations, and constructing orthogonal polynomial expansions for numerical integration and interpolation.

Are there any limitations to the Gram-Schmidt process for Taylor basis?

While the Gram-Schmidt process for Taylor basis is a powerful tool, it does have some limitations. One limitation is that it can be computationally intensive for large sets of polynomials. Additionally, the process may not work for certain types of functions, such as those with singularities or discontinuities.

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