By P. Fitzpatrick
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Additional info for Advanced Calculus [Int'l Student edn]
The preceding problem has been reduced to one in matrix theory, and eigenvalue and eigenvector considerations are of prime importance in that theory. It is helpful to study their analogues for integral operators. We consider the equation where K is still given by Eq. 1). 3, any function (j> satisfying Eq. 5) must have the form Again we compute: with Using arguments identical to those of the previous section we find where The matrix system Eq. 7) has n eigenvalues and n eigenvectors. from K. In particular, if it should happen that aj(x) = Pj(x), j = 1,2, • • - , « , then K(x,y) = K(y,z).
7) with all admissible /'s, then we call MI the onenorm of the integral operator. Often one can demonstrate that a norm exists, but the determination of its actual value is usually quite difficult. Fortunately, in much of our work that value is not of great importance. The use of the term norm applied to both functions or vectors and certain operators may be confusing initially. However, observe that the norm as newly defined is still a measure of size—it measures the size of the process of integration.
5) leads immediately to Thus, g has a one-norm and we have estimated its size. Finally, let us consider two different functions f\ and /2 that are close together. More accurately, we can now say where e is small. Then, after calculating a little further, we find for the corresponding g's as defined by Eq. 4) that This confirms our notion that integration is a smoothing operation. A small change in / results in a small change in g, but what intuition tells us is now expressed mathematically. The number MO occurring in Eq.
Advanced Calculus [Int'l Student edn] by P. Fitzpatrick