I hold an engineering undergraduate degree in EC. When I went to a good school for my post-grad, I found that research in many EC/CS departments required a lot of core math expertise which my UG lacked. It took me a whole lot of time to accustom myself and complete the courses. At the end of the coursework, I was able to thoroughly appreciate the development of the theory (in my case, real analysis, measure and probability theory).
When working on an engineering problem, I was able to understand fundamental ideas in many papers, thanks to the courses. Yet when it came to using the math for solving my own problem, I found things arduously difficult. In other words, non-intuitive math was well beyond me.
I regularly find top academic researchers in engineering fluently pulling off sophisticated mathematical results in their work. What tips could you suggest for someone with an engineering (or non-math) background requiring mathematical sophistication for their research?
My background is in Mechanical and Aerospace Engineering, so I cannot comment on EC/CS directly. However, much of my work is mathematical in nature, so generally speaking I would offer the following:
Do not be intimidated those fluent mathematical results. In my experience, for every "hit" that has an elegant result there were probably multiple "misses" that did not lead anywhere.
Balance perseverance with distance. Even techniques that seem ideally suited to your problem will inevitably have challenges, otherwise someone would likely have published the result already. It is important to stay positive and keep working at it. At the same time, sometimes it helps to take a step back and work on something else for awhile and give your mind a break. When you get some distance from your original problem, you may find an insight that helps you to overcome your roadblock.
Read widely. I am sure you have been reading papers related to whatever problems you are working on, but oftentimes there are techniques used in tangentially related or even unrelated fields that may be adapted to work on a problem in your area. Even if such studies do not lead to a usable result for your current work, they may lead to inspiration down the road.
Just like in your classes, learning to apply mathematical techniques to research problems takes time and practice. If it is possible, try to scale down your problem as much as you can to something very basic. Finding a solution to the more basic problem will increase your confidence in the method. Then you can slowly start to scale your problem back up. By attacking the problem this way, you get a number of small successes that help build your background with whatever techniques you use. Overcoming the smaller problems can give insight into the issues you may be having with your larger problem as well.