Reference request: why integral transforms? I understand there have been previous questions on why/how exactly integral transforms arise, but I am here asking specifically for reference requests to sources other than full-length books.
My question: what is the motivation for integral transforms and how do/have they arisen? What are some appealing applications (not necesarily applicable applications, just applications of integral transforms to approach notable problems across certain fields, ie. number theory) of integral transforms?
References I am looking for: Short to medium length expository works. Concise books, notes, articles, etc.
 A: 
Question: What is the motivation for integral transforms and how do/have they arisen?

Answer: For the answer of the above questions please find the answer (given by me) in the link entitled "How do Integral Transforms work". 
Here the first paragraph gives the motivation for integral transforms and the latter part gives the answer of  how do/have they arisen ?.

Question: What are some appealing applications (not necesarily applicable applications, just applications of integral transforms to approach notable problems across certain fields, ie. number theory) of integral transforms?

Answer: Integral transformations are used for problems in linear differential equations, to boundary and initial value problems in partial differential equations and continuum mechanics. They provide effective ways to solve a variety of problems arising in the engineering, optical, and physical sciences. There are many applications of probability that rely on integral transforms, such as "pricing kernel" or stochastic discount factor, or the smoothing of data recovered from robust statistics.
For more about it, I am providing some references (as per your demand)
"Lecture Notes in Integral Transforms" by Eugenia Ferrari
"Applications of integral transforms in fractional diffusion processes" by Francesco Mainardi
"Integral Transforms And Their Applications" by  Prof.Sarthok Sircar
"Some Schemata for Applications of the Integral Transforms of Mathematical Physics" by Yuri Luchko
along with the references given here How do Integral Transforms work. 
