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We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D_4=SO(4,2)/(SO(4)times SO(2)) of the conformal group SO(4,2) (locally isomorphic to SU(2,2)) in 1+3 dimensions. The manifold D_4 can be mapped one-to-one onto the future tube domain C^4_+ of the complex Minkowski space through a Cayley transformation, where other kind of (electromagnetic) wavelets have already been proposed in the literature. We study the unitary irreducible representations of the conformal group on the Hilbert spaces L^2_h(D_4,d u_lambda) and L^2_h(C^4_+,dtilde u_lambda) of square integrable holomorphic functions with scale dimension lambda and continuous mass spectrum, prove the isomorphism (equivariance) between both Hilbert spaces, admissibility and tight-frame conditions, provide reconstruction formulas and orthonormal basis of homogeneous polynomials and discuss symmetry properties and the Euclidean limit of the proposed conformal wavelets. For that purpose, we firstly state and prove a lambda-extension of Schwingers Master Theorem (SMT), which turns out to be a useful mathematical tool for us, particularly as a generating function for the unitary-representation functions of the conformal group and for the derivation of the reproducing (Bergman) kernel of L^2_h(D_4,d u_lambda). SMT is related to MacMahons Master Theorem (MMT) and an extension of both in terms of Loucks SU(N) solid harmonics is also provided for completeness. Convergence conditions are also studied.
We study a number of possible extensions of the Ramanujan master theorem, which is formulated here by using methods of Umbral nature. We discuss the implications of the procedure for the theory of special functions, like the derivation of formulae co
We revise the use of 8-dimensional conformal, complex (Cartan) domains as a base for the construction of conformally invariant quantum (field) theory, either as phase or configuration spaces. We follow a gauge-invariant Lagrangian approach (of nonlin
A (special case of a) fundamental result of Horn and Loewner [Trans. Amer. Math. Soc. 1969] says that given an integer $n geq 1$, if the entrywise application of a smooth function $f : (0,infty) to mathbb{R}$ preserves the set of $n times n$ positive
The adiabatic theorem refers to a setup where an evolution equation contains a time-dependent parameter whose change is very slow, measured by a vanishing parameter $epsilon$. Under suitable assumptions the solution of the time-inhomogenous equation
We prove that on an arbitrary metric measure space a countable collection of test plans is sufficient to recover all $rm BV$ functions and their total variation measures. In the setting of non-branching ${sf CD}(K,N)$ spaces (with finite reference me