Let be a real symmetric matrix of order with distinct eigenvalues, and let be such that and the inner product for every eigenvector of .
Let denote the space of polynomials of degree at most . Show that
defines an inner product on , where the expression on the right above is the Euclidean inner product in
Linearity of 1st Argument
We also need to show that if and only if .
Forward Direction (alt)
Suppose . It suffices to show . However, this a trivial consequence of the fact that (which is clear from the fact that for with degree less than and that does not lie in the orthogonal compliment of any of the distinct eigen vectors of ).
Claim: If , then .
where are the orthogonal eigenvectors of and all are non-zero
Notice that is a linear combination of , the coefficients of the polynomial , and the scaling coefficient of the eigenvector e.g.
Since and , this implies .
If , then
Consider the recurrence
where the and are scalars and . Show that , where is a polynomial of degree
where (respectively ) has degree (respectively ). Then for
which is as desired.
Suppose the scalars above are such that and is chosen so that . Use this to show that that the polynomials in part (b) are othogonal with respect to the inner product from part (a.
Since and , it is equivalent to show that for .
it is then sufficient to show that
Consider the n-panel trapezoid rule for calculating the integral of a continuous function ,
Find a piecewise linear function such that
for any continuous function such that is integrable over [0,1]. Hint: Find by applying the fundamental theorem of calculus to .
Rewrite given equation on specific interval
For a specific interval , we have from hypothesis
Distributing and rearranging terms gives
Starting with the hint and applying product rule, we get
Also, we know from the Fundamental Theorem of Calculus
Setting the above two equations equal to each other and solving for yields
Let . Therefore since is linear
By comparing equations (1) and (2) we see that
Plugging in either or into equation (3), we get that
Apply the previous result to , , to obtain a rate of convergence.
Let denote the set of all real-valued continuous functions defined on the closed interval be positive everywhere in . Let be a system of polynomials with for each , orthogonal with respect to the inner product
For a fixed integer , let be the distinct roots of in . Let
be polynomials of degree . Show that
Hint: Use orthogonality to simplify
Since is a polynomial of degree for all , is a polynomial of degree .
Notice that for where are the distinct roots of . Since is a polynomial of degree and takes on the value 1, distinct times