James V. Herod*

Page maintained by Evans M. Harrell, II, harrell@math.gatech.edu.

**SECTION 1.1. GEOMETRY AND INTEGRAL OPERATORS**

In this section, instead of working in the space R^{n}, we will work in a
space of functions defined on an interval. At an abstract level, many sets of
functions have the same properties as a vector space like R^{n}, and this
analogy will be extremely useful in this section. It will be developed rather
rapidly. If you would prefer a somewhat more detailed discussion of vector spaces,
read the first two sections of this
link before proceeding.
Most often, we will take the interval on which our functions are defined to be
[0,1]. Of course, we will not work in the class of *all *functions on
[0,1]; rather, in the spirit of the previous section, we ask that the linear
space should consist of

functions f for which

Then, we have an inner product space as we did in the previous section. This
space is called L^{2}( [0,1] ) . The dot product of two functions is
given by

and the norm of f is defined in terms of the dot product:

(Compare with the
norm
in R^{n}.)

It does not seem appropriate to study in detail the nature of
L^{2}[0,1] at this time. Rather, suffice it to say that the space is
large enough to contain all continuous functions - even functions which are
continuous except at a finite number of places. The interested student can
find what L^{2}[0,1] is by looking in standard books in Real
Analysis.

Having an inner product space, we can now decide if f and g in the space are perpendicular. The distance and the angle between f and g are given by the same formulas as we understood from the previous section: the distance from f to g is || f - g || and the angle [[alpha]] between f and g satisfies

cosine([[alpha]]) = < f, g >/||f|| ||g||

provided neither f nor g is zero.

Suppose { f_{p} } is a sequence of functions in L^{2}( [0,1]). It is
valuable to consider the possible meanings for lim_{p} f_{p}(x) = g(x). There are
three meanings.

The sequence {f_{p}} converges *point-wise* to g at each x in [0,1]
provided that for each x in [0,1],

lim_{p} f_{p}(x) = g(x).

The sequence converges to g *uniformly* on [0,1] provided that

lim_{p} sup_{x} |f_{p}(x) - g(x)| = 0.

And, the sequence converges to g *in norm* if

lim_{p} || f_{p} - g || = 0.

An understanding of these three modes of convergence should be sought. These are ideas that re-occur in mathematics.

(Compare with the notions of
convergence for sequences of vectors
in R^{n}.)

A type of integral equation will be studied in this section. For example, given a function called the kernel

K: [0,1]x[0,1] -> R

and a function f: [0,1] -> R, we seek a function y such that for each x in [0,1],

Such equations are called Fredholm equations of the second kind. An equation of the form

is a Fredholm equation of the first kind.

The requirements in this section on K and f will be that

These requirements are met if K and f are continuous.

For simplicity, we denote by **K** the linear function given by

Note that **K** has a domain large enough to contain all functions y which
are continuous on [0,1]. Also, if y is continuous then **K**(y) is a
function and its value at x is denoted **K**(y)(x). In spoken conversation,
it is not so easy to distinguish the number valued function K and the function
valued **K.** The bold character will be used in these notes to denoted the
latter.

It is well to note the resemblence of this function **K** to the
multiplication of a matrix A by a vector u:

This formula has the same form as that for **K** given above.

It is a historical accident that differential equations were understood before integral equations. Often an integral equation can be converted into a differential equation or vice versa, so many of the laws of nature which we think of as differential equations might just as well have been developed as integral equations initially. In some instances it is easier to differentiate than to integrate, but at other times integral operators are more tractable.

In this course integral operators will be called upon to solve differential equations, and this is one of their main uses. They have many other uses as well, most notably in the theory of filtering and signal processing. In most of these applications the integral and differential operators are linear transformations. The analogy between linear transformations and matrices is deep and useful.

Just as a matrix has an adjoint, the integral operator **K** has an adjoint,
denoted **K***. The adjoint plays an important role in the theory and use of
integral equations.
(Review the
adjoint
for matrices.)

In order to understand **K***, one must consider < **K**(f), g >
and seek **K*** such that < **K**f, g > = < f, **K***g
>.

An examination of these last equations leads one to guess that **K*** is
given by

or, keeping t as the variable of integration,

Those last equations verified that

< **K**(f), g > = < f, **K***(g) >.

Care had to be taken to watch whether the "variable of integration" is t or x in the integrals involved.

In summary, if K is the kernel associated with the linear operator **K**,
then the kernel associated with **K*** is given by K*(x,y) [[equivalence]]
K(y,x). It is of value to compare how to get **K*** from **K** with the
process of how to get A* from A:

A*p,q = Aq,p.

Consistent with the rather standard notation we have adopted above, it is clear that a briefer representation of the equation

is the concise equation y = **K**(y) + f, or (1 - **K** ) y = f.

**EXAMPLE:** Suppose that

To get K*, let's use other letters for the argument of K* and K to avoid
confusion. Suppose that 0 < u < v < 1. Then, K*(u,v) = K(v,u) = 0. In
a similar manner, K*(u,v) = (u-v)^{2} if 0 < v < u < 1. Note
that K* is not K.

The discussion of this example has been algebraic to this point. Consider this geometric notion that is suggested by the alternate name for "self-adjoint", namely, some call K "symmetric" if K(x,t) = K(t,x). The geometric name suggests a picture and the picture is the graph of K. The K of this example is not symmetric in x and t. Its graph is not symmetric about the line x = t. The function K is different from the function K*.

**Exercises.**

**I.1**. (a) Find the distance from sin([[pi]]x) to cos([[pi]]x) in
L^{2}[0,1] and

L^{2}[-1,1].

Ans: 1., [[Rho]](2)

(b) Find the angle between sin([[pi]]x) and cos([[pi]]x) in L^{2}[0,1]
and L^{2}[-1,1].

Ans: [[pi]]/2,[[pi]]/2.

**I.2**. Repeat 1. (a) and (b) for x and x^{2}.

Ans:1/R(30), 4/R(15),

Arccos(R(15)/4), [[pi]]/2

**I.3**. Suppose K(x,t) =1 + 2 x t^{2} on [0,1]x[0,1] and y(x) = 3 -
x. Compute **K**(y) and **K***(y). Ans:
(5+3x)/2, (15+14x^{2})/6

4. Suppose K(x,t) = BLC{( A(x t if 0 < x < t < 1,x t^{2} if 0
< t < x < 1)). For y(x) = 3 - x, compute **K**(y) and
**K***(y). Ans: **K**[y](x) = - F(x^{5},4) +
F(4x^{4},3) - F(3x^{3}, 2) + F(7x,6)

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