By Ricardo Abreu-Blaya, Juan Bory-Reyes (auth.), Daniel Alpay, Annemarie Luger, Harald Woracek (eds.)

From a mathematical viewpoint it really is interesting to gain that almost all, if now not all, of the notions bobbing up from the speculation of analytic services within the open unit disk have opposite numbers whilst one replaces the integers through the nodes of a homogeneous tree. it's also attention-grabbing to achieve entire functionality idea, diverse from the classical concept of a number of complicated variables, will be developped whilst one considers hypercomplex (Clifford) variables, Fueter polynomials and the Cauchy-Kovalevskaya product, instead of the classical polynomials in 3 self sustaining variables.

This quantity includes a collection of papers at the issues of Clifford research and wavelets and multiscale research, the latter being understood in a really huge experience. the idea of wavelets is mathematically wealthy and has many sensible applications.

Contributors: R. Abreu-Blaya, J. Bory-Reyes, F. Brackx, Sh. Chandrasekaran, N. de Schepper, P. Dewilde, D.E. Dutkay, okay. Gustafson, H. Heyer, P.E.T. Jorgensen, T. Moreno-García, L. Peng, F. Sommen, M.W. Wong, J. Zhao, H. Zhu

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**Sample text**

M ﬁxed, we then call Ckm = λ ∈ Cm : λ = λA ei1 ∧ ei2 ∧ · · · ∧ eik , A = (i1 , i2 , . . , the space spanned by the outer products of k different basis vectors. Note that the 0-vectors and 1-vectors are simply the scalars and Cliﬀord-vectors; the 2-vectors are also called bivectors. 30 F. Brackx, N. De Schepper and F. 3. Embeddings of Rm By identifying the point (x1 , . . , xm ) ∈ Rm with the 1-vector x given by x = xj ej , the space Rm is embedded in the Cliﬀord algebra Cm as the subspace of 1vectors R1m of the real Cliﬀord algebra Rm .

Brackx, N. De Schepper and F. 3. 21. With respect to the orthonormal frame η j (j = 1, . . , m), the Dirac operator takes the form ∂x = ∂x j η j . j Proof. For the sake of clarity, we do not use the Einstein summation convention. We have consecutively ∂xj ej = ∂x = j j,k ⎛ ⎝ = t ∂xj g jk ek = ∂xj g jk Et ATtk j,k,t ⎞ ATtk g jk ∂xj ⎠ Et . 3) this becomes ∂x = t 1 ∂X t Et = λt ∂x t η t . 22. (i) A Cm -valued function F (x1 , . . , xm ) is called left monogenic in an open region of Rm if in that region ∂x [F ] = 0.

Then we consider the Cliﬀord algebra-valued inner product f, g h(x) f † (x) g(x) dxM = Rm where dxM stands for the Lebesgue measure on Rm ; the associated norm then reads f 2 = [ f, f ]0 . The unitary right Cliﬀord-module of Cliﬀord algebra-valued measurable functions on Rm for which f 2 < ∞ is a right Hilbert-Cliﬀord-module which we denote by L2 (Rm , h(x) dxM ). 46 F. Brackx, N. De Schepper and F. Sommen In particular, taking h(x) ≡ 1, we obtain the right Hilbert-module of square integrable functions: L2 Rm , dxM = f : Lebesgue measurable in Rm for which 1/2 f 2 = Rm |f (x)|20 dxM <∞ .