A constant- coefficient operator A is a linear operator of the form
where D is the derivative operator and ak are real constants. If , then A is of order n. We can apply A to a function y defined on some interval J to obtain another function on J given by
In this section, we shall consider only functions the derivative of which exists for every order on the interval . The set of all such functions is denoted by and is called the class of infinitely differentiable functions. If , so is .
It is easy to show that a constant coefficient
operator is also a linear transformation, as defined in §4.1.
Conversely, given any polynomial of real coefficients, there is a corresponding operator A with the same constant coefficients.
Let A and B be constant coefficient operators with associated polynomials pA and pB, respectively. Let l be any real number, then
In other words, the association between operators and polynomials is 1-1 onto and they satisfy the same algebraic relations.
Parts (b)-(d) follow immediately from
Let , then they must have the same degree as well as coefficients. This means A and B have the same order and coefficients. Hence, . This proves the ‘if’ part of (a).
Next, given , we have . Let , where r is a constant. Then . Hence,
Since , we have , which proves the ‘only if’ part of (a).
According to Theorem 8.6, constant coefficient operators can be manipulated like polynomials. For example, if pA(r) can be factorized as
so can A:
[Note that the factors on the right sides
commute.] Now, the fundamental theorem
of algebra guarantees the factorization (8.10).
Thus, pA(r) can always be factorized into a product of linear and quadratic factors of real coefficients. Ditto A.
Let . Then and
Let . Then