Please see the attached file for the fully formatted problems.
Let V=L2[-1,1] be the Hilbert space of functions over the time interval [-1,1] with inner product
Let P5 V be the subspace of polynomials of order 4 or less, endowed with the inner product and norm of V, and let , be its natural basis. Define a linear transformation S as
Show that the subspace P5 is invariant under S. That is, verify that S maps members of P5 back into P5. Define T as the restriction of S to P5. (This part is already done. So I just send it for information.)
The matrix representation of T is below;
My question is:
And, if we define y(t) as
y(t)=1 + t + t2 + t3 + t4 what is the optimal value of that solves the approximation problem?
(Hint: T in min operator is not the matrix defined above. But, it is the operator defined above as Sx. So wherever we see S we can use T. (Sx=Tx) And U is just the space of polynomials.)© BrainMass Inc. brainmass.com March 4, 2021, 8:12 pm ad1c9bdddf
The explanations are in the attached pdf file.
As the course on polynomials requires the ability to calculate an inverse of a matrix as a prerequisite, and you have not ...
Linear Transformations, Hilbert Space and Inner Products are investigated.