Prerequisites: Properties of Resolvent Kernels

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

There are three properties listed here:

  1. \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
  2. \Gamma(s,t;\lambda)=K(s,t)+\lambda\int K(s,x)\Gamma(x,t;\lambda)dx
  3. It satisfies integrodifferential equations {\partial\Gamma(s,t;\lambda)\over\partial\lambda}=\int\Gamma(s,x;\lambda) \Gamma(x,t;\lambda)dx


Property 1

  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}
  2. Two Assumptions: 1. |\lambda|<B^{-1}, 2. \int |K(s,t)|^2ds<E^2, E is a constant.
  3. \int|K_m(s,t)|^2 dt \le C_m^2 \le C_1^2 B^{2m-2}
  4. Schwartz Inequality (Wikipedia): Fot two square integrable functions, |\int f(x)g(x)dx|^2\le\int|f(x)|^2dx\int|g(x)|^2dx
  5. 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

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

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