Fourier analysis
Representing functions as sums of sines and cosines.
Founded: 1822 — Fourier's Théorie analytique de la chaleur
The claim
No credible dismissal found. The 1807 Paris Academy committee (Lagrange, Laplace, Monge, Lacroix) objected to representing arbitrary functions as trigonometric series on rigor grounds, not utility — the underlying problem (heat conduction) was never in question.
What happened
Tide-predicting machine (1872) — Kelvin's harmonic analysis of tidal constituents — an early, concrete engineering application.
50 years — founding (1822) to Kelvin's tide predictor (1872)
Verdict: Never dismissed, applied anyway.
Confidence: medium-high.