  Adapting the recently developed randomized dyadic structures, we introduce
the notion of spline function in geometrically doubling quasi-metric spaces.
Such functions have interpolation and reproducing properties as the linear
splines in Euclidean spaces. They also have H\"older regularity. This is used
to build an orthonormal basis of H\"older-continuous wavelets with exponential
decay in any space of homogeneous type. As in the classical theory, wavelet
bases provide a universal Calder\'on reproducing formula to study and develop
function space theory and singular integrals. We discuss the examples of $L^p$
spaces, BMO and apply this to a proof of the T(1) theorem. As no extra
condition {(like 'reverse doubling', 'small boundary' of balls, etc.)} on the
space of homogeneous type is required, our results extend a long line of works
on the subject.
