This is a bunch of prerequisites to understand section 3.5 of Ram P. Kanwal.
There are three properties listed here:
- is absolutely and uniformly convergent for all values of s and t in the circle
- It satisfies integrodifferential equations
- Direct Comparison Test (Wikipedia): If and then is also absolutely convergent. (Make note of variants in the Wikipedia Page in NET Notebook.) Here,
- Two Assumptions: 1. , 2. is a constant.
- Schwartz Inequality (Wikipedia): Fot two square integrable functions,
- Convergence of Geometric Series (Wikipedia): If
In step 2, they are changing limits from m=2 to m=1. If it is confusing, you can try substituting p = m-1.
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)