We establish spectral expansions of homogeneous and isotropic random fieldstaking values in the $3$-dimensional Euclidean space $E^3$ and in the space$\mathsf{S}^2(E^3)$ of symmetric rank $2$ tensors over $E^3$. The former is amodel of turbulent fluid velocity, while the latter is a model for the randomstress tensor or the random conductivity tensor. We found a link between thetheory of random fields and the theory of finite-dimensional convex compacta.
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