Basics of a generalized Wiman-Valiron theory for monogenic Taylor series of finite convergence radius

In this paper, we develop the basic concepts for a generalized Wiman-Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system . These functions are called monogenic or Clifford holomorphic inside the ball. We introduce growth orders, the maximum term and a generalization of the central index for monogenic Taylor series of finite convergence radius. Our goal is to establish explicit relations between these entities in order to estimate the asymptotic growth behavior of a monogenic function in a ball in terms of its Taylor coefficients. Furthermore, we exhibit a relation between the growth order of such a function f and the growth order of its partial derivatives

On the approximation of analytic functions by infinite series of fractional Ruscheweyh derivatives bases

This paper presented a new Ruscheweyh fractional derivative of fractional order in the complex conformable calculus sense. We applied the constructed complex conformable Ruscheweyh derivative (CCRD) on a certain base of polynomials (BPs) in different regions of convergence in Fréchet spaces (F-spaces). Accordingly, we investigated the relation between the approximation properties of the resulting base and the original one. Moreover, we deduced the mode of increase (the order and type) and the $\mathbb{T}_{\rho}$-property of the polynomial bases defined by the CCRD. Some bases of special polynomials, such as Bessel, Chebyshev, Bernoulli, and Euler polynomials, have been discussed to ensure the validity of the obtained results

Expansions of generalized bases constructed via Hasse derivative operator in Clifford analysis

The present paper investigates the approximation of special monogenic functions (SMFs) in infinite series of hypercomplex Hasse derivative bases (HHDBs) in Fréchet modules (F-modules). The obtained results ensure the existence of such representation in closed hyperballs, open hyperballs, closed regions surrounding closed hyperballs, at the origin, and for all entire SMFs (ESMFs). Furthermore, we discuss the mode of increase (order and type) and the $T_{\rho}$-property. This study enlightens several implications for some associated HHDBs, such as hypercomplex Bernoulli polynomials, hypercomplex Euler polynomials, and hypercomplex Bessel polynomials. Based on considering a more general class of bases in F-modules, our results enhance and generalize several known results concerning approximating functions in terms of bases in the complex and Clifford settings

Famílias normais e crescimento de funções polimonogénicas

Doutoramento em MatemáticaEste trabalho tem como objectivo contribuir para um estudo de famílias normais de funções meromórficas especiais assim como para o estudo do comportamento assimptótico de funções polimonogénicas no domínio da Análise Hipercomplexa. Neste contexto, obtemos condições necessárias e/ou suficientes de normalidade para famílias de funções meromórficas especiais, nomeadamente a generalização do Teorema de Marty e a Lema de Zalcman. Para a classe de funções polimonogénicas são demonstradas desigualdades do tipo de Cauchy e algumas generalizações de resultados da teoria de Wiman e Valiron. Consequentemente, são obtidas relações entre o máximo módulo da função, o termo máximo e índice central da sua respectiva série de Taylor-Almansi. Aplicam-se estes resultados ao crescimento assimptótico desta classe de funções. Como aplicação, são obtidos teoremas sobre soluções assimptóticas de determinadas equações diferenciais de derivadas parciais e a classificação de algumas soluções das mesmas.The aim of this work is to provide some contributions to the study of normal family of special meromorphic functions as well as to the study of the asymptotic behaviour of polymonogenic functions in the framework of Hypercomplex Analysis. In this context we have obtained necessary and/or sufficient normality conditions for families of special meromorphic functions, in particular, a generalization of Marty’s criterion and also of Zalcman’s lemma. We prove inequalities of Cauchy-type estimates for a class of polymonogenic functions and also some generalizations of results of the Wiman-Valiron theory. Consequently, relations of the maximum modulus, the maximum term and the norm of the central index with respect to their Taylor-Almansi series expansion are obtained. These results are applied to the asymptotic growth behaviour of those functions classes. As applications we establish theorems on the asymptotic of solutions of certain partial differential equations which allow us to provide a classification of some of such solutions