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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
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
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)?
integral from 0 to 1 of ((2/pi)*cos((pi*x)/2)-p(x))^2 dx = the magnitude squared of (2/pi)*cos((pi*x)/2)-p(x)
I'm really quite lost as to what this question in asking. Any help would be greatly apreciated : )