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Adapting the recently developed randomized dyadic structures, we introducethe notion of spline function in geometrically doubling quasi-metric spaces.Such functions have interpolation and reproducing properties as the linearsplines in Euclidean spaces. They also have H\"older regularity. This is usedto build an orthonormal basis of H\"older-continuous wavelets with exponentialdecay in any space of homogeneous type. As in the classical theory, waveletbases provide a universal Calder\'on reproducing formula to study and developfunction 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 extracondition {(like 'reverse doubling', 'small boundary' of balls, etc.)} on thespace of homogeneous type is required, our results extend a long line of workson the subject.

Orthonormal bases of regular wavelets in spaces of homogeneous type