We develop numerical implementations of several criteria to asses
the regularity of functions. The criteria are based on finite
difference method and harmonic analysis: Littlewood-Paley theory
and wavelet analysis.
As a first application of the methods,
we study the regularity of conjugacies between critical circle maps
(i.e., differentiable homeomorphisms with a critical point)
with golden mean rotation number.
These maps have a very well developed mathematical theory
as well as a wealth of numerical studies.
We compare the results produced by our methods among themselves
and with theorems in the mathematical literature.
We confirm that several of the features that are predicted
by the mathematical results are observable by numerical computation,
and compute reliably some universal numbers.
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