2Orthogonal polynomials

IB Numerical Analysis

2.1 Scalar product

The scalar products we are interested in would be generalization of the usual

scalar product on Euclidean space,

hx, yi =

n

X

i=1

x

i

y

i

.

We want to generalize this to vector spaces of functions and polynomials. We

will not provide a formal definition of vector spaces and scalar products on an

abstract vector space. Instead, we will just provide some examples of commonly

used ones.

Example.

(i)

Let

V

=

C

s

[

a, b

], where [

a, b

] is a finite interval and

s ≥

0. Pick a weight

function

w

(

x

)

∈ C

(

a, b

) such that

w

(

x

)

>

0 for all

x ∈

(

a, b

), and

w

is

integrable over [

a, b

]. In particular, we allow

w

to vanish at the end points,

or blow up mildly such that it is still integrable.

We then define the inner product to be

hf, gi =

Z

b

a

w(x)f(x)d(x) dx.

(ii)

We can allow [

a, b

] to be infinite, e.g. [0

, ∞

) or even (

−∞, ∞

), but we have

to be more careful. We first define

hf, gi =

Z

b

a

w(x)f(x)g(x) dx

as before, but we now need more conditions. We require that

R

b

a

w

(

x

)

x

n

d

x

to exist for all

n ≥

0, since we want to allow polynomials in our vector

space. For example,

w

(

x

) =

e

−x

on [0

, ∞

), works, or

w

(

x

) =

e

−x

2

on

(

−∞, ∞

). These are scalar products for

P

n

[

x

] for

n ≥

0, but we cannot

extend this definition to all smooth functions since they might blow up too

fast at infinity. We will not go into the technical details, since we are only

interested in polynomials, and knowing it works for polynomials suffices.

(iii) We can also have a discrete inner product, defined by

hf, gi =

m

X

j=1

w

j

f(ξ

j

)g(ξ

j

)

with

{ξ

j

}

m

j=1

distinct points and

{w

j

}

m

j=1

>

0. Now we have to restrict

ourselves a lot. This is a scalar product for

V

=

P

m−1

[

x

], but not for

higher degrees, since a scalar product should satisfy

hf, fi >

0 for

f 6

= 0.

In particular, we cannot extend this to all smooth functions.

With an inner product, we can define orthogonality.

Definition

(Orthogonalilty)

.

Given a vector space

V

and an inner product

h·, ·i, two vectors f, g ∈ V are orthogonal if hf, gi = 0.