Saying that Turing and Church "invented computing" is too vague of a justification. I want numbers and details. Folks like Babbage were already thinking procedurally in practical enough terms that once the technological capability (not the theoretical capabilities) caught up, algorithms like the FFT could be discovered and put to use.
Turing's work was so important that he, not Babbage, is generally recognized as the founder of computer science.
One example of the importance of Church's work is Lisp. Church's lambda calculus is the foundation of Lisp, which pioneered nearly all the features of modern programming languages [1]. The designer of Smalltalk, Alan Kay, called it "the greatest single programming language ever designed" [2] and talked about its influence on Smalltalk quite a bit [3].
And your example, the Fourier transform, was itself a mathematical tool long before the first computer was built, and the first published FFT algorithm also dates from the 1930s. [4]
1: http://www.paulgraham.com/diff.html
2: https://www.quora.com/What-did-Alan-Kay-mean-by-Lisp-is-the-...
3: http://worrydream.com/EarlyHistoryOfSmalltalk/
4: https://en.wikipedia.org/wiki/Fast_Fourier_transform#History
The entire field of computing was invented by pure mathematicians like Turing and Church working in the 1930s. At the time, no one had any idea what the applications, if any, would be.
It's like any science. Some work has immediate applications. Other work is pure exploration of the unknown.
And it is pure exploration of the unknown that leads to the truly revolutionary discoveries. You can't set out to discover penicillin when you don't even know it exists.