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
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.