Integral Equations

Prerequisites: Properties of Resolvent Kernels

Posted by jessepfrancis on Jan 21, 2020Jan 21, 2020

This is a bunch of prerequisites to understand section 3.5 of Ram P. Kanwal.

There are three properties listed here:

$\Gamma(s,t;\lambda)=\sum_{m=1}^\infty\lambda^{m-1}K_m(s,t)$ is absolutely and uniformly convergent for all values of s and t in the circle $|\lambda|<\frac1B$
$\Gamma(s,t;\lambda)=K(s,t)+\lambda\int K(s,x)\Gamma(x,t;\lambda)dx$
It satisfies integrodifferential equations ${\partial\Gamma(s,t;\lambda)\over\partial\lambda}=\int\Gamma(s,x;\lambda) \Gamma(x,t;\lambda)dx$

Prerequisites

Property 1

Direct Comparison Test (Wikipedia): If $\sum b_n<\infty$ and $|a_n|<|b_n|$ then $\sum a_n$ is also absolutely convergent. (Make note of variants in the Wikipedia Page in NET Notebook.) Here, $a_n=\lambda^{m-1}K_m(s,t),b_n=C_1E(\lambda B)^{m-1}$
Two Assumptions: 1. $|\lambda|<B^{-1}$ , 2. $\int |K(s,t)|^2ds<E^2, E$ is a constant.
$\int|K_m(s,t)|^2 dt \le C_m^2 \le C_1^2 B^{2m-2}$
Schwartz Inequality (Wikipedia): Fot two square integrable functions, $|\int f(x)g(x)dx|^2\le\int|f(x)|^2dx\int|g(x)|^2dx$
Convergence of Geometric Series (Wikipedia): If $r\le1, \sum_{n=1}^\infty ar^k={a\over 1-r}$

Property 2

In step 2, they are changing limits from m=2 to m=1. If it is confusing, you can try substituting p = m-1.

Property 3

If a series is uniformly convergent, summation and integral can be interchanged (Proof – Math.SE).

Some Extra Links

Compilation of Series Convergence Tests (Wikipedia)
Some important sequences and series (Blog)
Generalisation of Schwartz Inequality: Hölder inequality (Wikipedia)
Also worth reading about application of Schwartz Inequality – used in Analysis, Algebra, Statistics, and a variety of fields!
A video discussing shifting limits of an infinite series (YouTube)

Published by jessepfrancis

Christian. Artist. Mathematician. Programmer. Teacher. Visit https://mathematicos.in/aboutjesse/ for my full profile. View all posts by jessepfrancis

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Pingback: Iterated Kernels and Fredholm Theorems (IEL6) – Mathematics, Education and Technology

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