orthogonal polynomials

Question 1

R[x]

Answer 1

the ∞ dimensional vector space of all real polynomials in x

Question 2

R[x]n

Answer 2

the n+1 dimensional space of all real polynomials of degree at most n

Question 3

inner product

Answer 3

(f,g) = ∫ f(x)g(x)k(x) dx from a to b
where k(x) is a fixed continuous function st k(x) >0 for all x in [a,b]

Question 4

symmetric differential operator

Answer 4

if V is a real vector space with inner product ( , ) and L : V->V is a linear operator then L is symmetric/self adjoint with respect to the inner product if (Lv,w) = (v,Lw)

Question 5

differential operator symmetric and its matrix

Answer 5

if V = R^n with standard inner product and L is represented by an nxn matrix M then L is symmetric (self adjoint) iff M is symmetric (M=M^t)

Question 6

symmetric linear operator and eigenfunction

Answer 6

let {P0,P1…} be an orthonormal set of polynomials with respect to ( , ) Pj is of degree j and L is symmetric linear operator st LPj has degree ≤ j then Pj is an eigenfunction of L
LPj = λPj

Question 7

how to write matrix representing L

Answer 7

do L on the basis and turn resulting polynomial into vectors
put vectors together to make M
M is upper triangular as when acting on x^n gives degree ≤n
eigenvalues on diagonal
can find eigenvectors then from this get eigenfunctions

Question 8

hermitian

Answer 8

let V be a complex vector space
a linear operator L is said to be hermitian (self adjoint) if <Lu,v> = <u,Lv>
also La anti hermitian if <Lu,v> = - (<u,Lv>)*

Question 9

powers of hermitian operators

Answer 9

if L hermitian operator L^n also hermitian

Question 10

how to rewrite anti hermitian operator

Answer 10

if La anti hermitian La = iL for L hermitian

Question 11

matrix and linear operator hermitian

Answer 11

if C^n with standard basis and standard inner product, a linear operator L is represented by a matrix M
M is hermitian (M=M*) iff L is hermitian

Question 12

L hermitian and inner product

Answer 12

if L hermitian then <Lv,v> is real for all v

Question 13

eigen values of hermitian operator

Answer 13

eigenvalues of hermitian operator are all real

Question 14

La anti hermitian and inner product

Answer 14

if La is anti hermitian (La v,v) is purely imaginary for all v in V

Question 15

2 hermitian operators on v

Answer 15

let L and m be hermitian operators on V then
4 |<Lmv,v>|^2 ≥ |<(Lm-mL)v,v|^2