3Approximation of linear functionals

IB Numerical Analysis

3.1 Linear functionals

In this chapter, we are going to study approximations of linear functions. Before

we start, it is helpful to define what a linear functional is, and look at certain

examples of these.

Definition

(Linear functional)

.

A linear functional is a linear mapping

L

:

V →

R, where V is a real vector space of functions.

In generally, a linear functional is a linear mapping from a vector space to

its underlying field of scalars, but for the purposes of this course, we will restrict

to this special case.

We usually don’t put so much emphasis on the actual vector space

V

. Instead,

we provide a formula for

L

, and take

V

to be the vector space of functions for

which the formula makes sense.

Example.

(i) We can choose some fixed ξ ∈ R, and define a linear functional by

L(f) = f(ξ).

(ii) Alternatively, for fixed η ∈ R we can define our functional by

L(f) = f

0

(η).

In this case, we need to pick a vector space in which this makes sense, e.g.

the space of continuously differentiable functions.

(iii) We can define

L(f) =

Z

b

a

f(x) dx.

The set of continuous (or even just integrable) functions defined on [

a, b

]

will be a sensible domain for this linear functional.

(iv)

Any linear combination of these linear functions are also linear functionals.

For example, we can pick some fixed α, β ∈ R, and define

L(f) = f(β) − f(α) −

β − α

2

[f

0

(β) + f

0

(α)].

The objective of this chapter is to construct approximations to more compli-

cated linear functionals (usually integrals, possibly derivatives point values) in

terms of simpler linear functionals (usually point values of f itself).

For example, we might produce an approximation of the form

L(f) ≈

N

X

i=0

a

i

f(x

i

),

where V = C

p

[a, b], p ≥ 0, and {x

i

}

N

i=0

⊆ [a, b] are distinct points.

How can we choose the coefficients

a

i

and the points

x

i

so that our approxi-

mation is “good”?

We notice that most of our functionals can be easily evaluated exactly when

f

is a polynomial. So we might approximate our function

f

by a polynomial,

and then do it exactly for polynomials.

More precisely, we let

{x

i

}

N

i=0

⊆

[

a, b

] be arbitrary points. Then using the

Lagrange cardinal polynomials `

i

, we have

f(x) ≈

N

X

i=0

f(x

i

)`

i

(x).

Then using linearity, we can approximate

L(f) ≈ L

N

X

i=0

f(x

i

)`

i

(x)

!

=

N

X

i=0

L(`

i

)f(x

i

).

So we can pick

a

i

= L(`

i

).

Similar to polynomial interpolation, this formula is exact for

f ∈ P

N

[

x

]. But we

could do better. If we can freely choose

{a

i

}

N

i=0

and

{x

i

}

N

i=0

, then since we now

have 2

n

+ 2 free parameters, we might expect to find an approximation that is

exact for

f ∈ P

2N+1

[

x

]. This is not always possible, but there are cases when

we can. The most famous example is Gaussian quadrature.