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A general theory of reproducing kernels and reproducing kernel Hilbert spaceson a right quaternionic Hilbert space is presented. Positive operator valuedmeasures and their connection to a class of generalized quaternionic coherentstates are examined. A Naimark type extension theorem associated with thepositive operator valued measures is proved in a right quaternionic Hilbertspace. As illustrative examples, real, complex and quaternionic reproducingkernels and reproducing kernel Hilbert spaces arising from Hermite and Laguerrepolynomials are presented. In particular, in the Laguerre case, the Naimarktype extension theorem on the associated quaternionic Hilbert space isindicated.

General construction of Reproducing Kernels on a quaternionic Hilbert space