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Homework Statement
Let P2 be the vector space of real polynomials of degree less or equal than 2. Define the (nonlinear) function E : P2 to R as
E(p)=integral from 0 to 1 of ((2/pi)*cos((pi*x)/2)-p(x))^2 dx
where p=p(x) is a polynomial in P2. Find the point of minimun for E, i.e. find the polynomial q exists in P2 such that
E(q) is less than or equal to E(p) for all p exists in P2
TIP: Try to understand it geometrically (i.e. make a sketch with lines and points in R^2). Compare with the following: in the usual linear systems, how do you minimize |Ax - b| when b is not in R(A)?
P_1 is like Ax, and cos... is like a vector b outside the range of A, you cannot solve the equation, but you can minimize the distance between Ax and b. the way of doing this is with an orthogonal projection.
Homework Equations
-Both cos(x) and q(x) belong to the vector space C([0; 1]) of continuous functions on the interval [0; 1].
-The mapping (u, v) to the integral between 0 and one of (u(x)v(x)) dx defines a scalar product on C([0; 1]).
-The squared length of a vector u according to this scalar product would be
tha magnitude of u squared = (u, u) = the integral between 0 and one of (u(x))^2 dx
The Attempt at a Solution
Process:
-find a basis for P1 on the interval [0,1]
-use Gram-Schmid procedure to make this basis orthonormal
-compute the orthogonal projection of f=2/pi*cos(pi*x/2) on P_1, using the orthonormal basis and the scalar product
-minimize the integral with the information found above
Basis for P1 ( a subset of the vector space P(containing all polynomials)):
P1 = a+bx
Basis of P1 is [1,x]
GramSchmit:
orthogonal basis: [1, x - (<x,1>/<1,1>)*1] = [1,x-1/2]
normalized: [1,(1/12)(x-1/2)]
Projection:
if you have a space W spanned by an orthogonal set {x, y} and you want to project a vector v on it orthogonally, then you just compute the sum <v, x> x + <v, y> y so...
if v = 2/pi*cos(pi*x/2) and {x,y} = [1,(1/12)(x-1/2)] then:
<2/pi*cos(pi*x/2), 1> * 1 + < 2/pi*cos(pi*x/2), (1/12)*(x-1/2)> * (1/12)(x-1/2)
-> (4/pi^2) + (pi-4)/(6pi^3)