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Does it mean the application of the theorems of Harmonic Analysis to real-world problems, say, engineering or data science problems, and that Applied Harmonic Analysis is not about theory or proofs? And since it's "applied", does that mean there's no measure theory involved, except for the very minimal: monotone convergence and dominated convergence theorems?

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Just because a subject has an applied moniker attached to it does not mean that the subject does not require proofs and theorems. Applied mathematicians are mathematicians foremost, and as such they work to provide theoretical guarantees concerning their objects of study.

Applied Harmonic Analysis follows the tradition of classical Harmonic Analysis in that aims to represent acquired data by some decomposition of basis functions. Classical Harmonic Analysis deconstructs a signal into sinusoids of varying frequencies, whereas modern applied Harmonic analysis aims to represent signals in a variety of different bases. These bases are usually attached to some Hilbert space framework.

These bases include wavelets, orthogonal polynomials, reproducing kernels, and others. To provide a rigorous analysis of these representations, often convergence results require that the estimated functions reside in some Hilbert space (most frequently) or Banach space (such as $C[0,1]$). For example, Representer Theorems lean on projections available in a Hilbert space to yield guarantees as to the structure of the solution to certain regularized optimization problems.

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