By Domenico Marinucci
Random Fields at the Sphere offers a accomplished research of isotropic round random fields. the most emphasis is on instruments from harmonic research, starting with the illustration thought for the gang of rotations SO(3). Many contemporary advancements at the approach to moments and cumulants for the research of Gaussian subordinated fields are reviewed. This historical past fabric is used to examine spectral representations of isotropic round random fields after which to enquire intensive the houses of linked harmonic coefficients. homes and statistical estimation of angular strength spectra and polyspectra are addressed in complete. The authors are strongly inspired by way of cosmological purposes, specifically the research of cosmic microwave history (CMB) radiation information, which has initiated a demanding new box of mathematical and statistical learn. excellent for mathematicians and statisticians attracted to purposes to cosmology, it's going to additionally curiosity cosmologists and mathematicians operating in crew representations, stochastic calculus and round wavelets
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Extra info for Random fields on the sphere : representation, limit theorems and cosmological applications
Sample text
Vdπ be an orthonormal basis of V. , vdπ , that is, dπ π (g) v j = πi j (g) vi , πi j (g) = π (g) v j , vi V . 20) where δab is the Kronecker symbol, that is: δab = 1 if a = b and δab = 0 if a b. More explicitly, the elements of the matrix representations πi j are mutually orthogonal. e. ) Let [π] ∈ G, representations of G. We denote by Mπ the space of matrix coefficients associated with [π], that is, Mπ is the closed subspace of L2 (G) generated by the mappings of the type g → π (g) v, w V , where (π, V) is a representative element of [π] and v, w ∈ V.
We have also the Plancherel identity f (g) h(g)dg = f, G d πi j V d πi j h, V . e. 2 f 2 (g)dg = G d bi j , ij a fact will have a great importance in the spectral theory of spherical random fields developed in the following chapters. 48 (1) Let G be a compact commutative group. Then, in this case the Peter-Weyl Theorem reduces to the statement that the characters, that is, the continuous homomorphisms from G to S 1 = {z ∈ C : |z| = 1} are an orthonormal basis of L2 (G). 1). (2) Suppose that G is a finite commutative group.
3 Actions Let X be any set. A (left) action A of a group G on X is a mapping A : G × X → X : (g, x) → A(g, x), verifying the two properties: (i) A(g, A(h, x)) = A(gh, x), for every g, h ∈ G, and (ii) A(e, x) = x, where e is the identity element of G. When there is no risk of confusion, we write g · x instead of A(g, x). A group action is said to be transitive if, for every x, y ∈ X, there exists g ∈ G such that y = g · x. A set X admitting an action (resp. a transitive action) of a group G is called a G-set (resp.