The rst part of this diploma thesis deals with the topic of nding equivalent norms and characterizations for vector-valued Besov and Triebel-Lizorkin spaces Bs p,q(E) and F s p,q(E). We will deduce general criteria by transferring and extending a theorem of Bui, Paluszy«ski and Taibleson from the scalar to the vector-valued case. By using special norms and characterizations we will derive neces...

In this article, we formalized the notion of the integral of a complex-valued function considered as a sum of its real and imaginary parts. Then we defined the measurability and integrability in this context, and proved the linearity and several other basic properties of complex-valued measurable functions. The set of properties showed in this paper is based on [15], where the case of real-valu...

As a byproduct of Theorem 3.1 we can give an estimate of the truncation error which arises if one ignores all the samples outside a finite interval. More precisely, we have the following corollary. Theorem 4.1: Let us define the truncation error E N (t) as follows: E N (t) = F(t) 0 jnjN F (t n)S n (t): Then jE N (t)j 2 2p0q bA p C ^ (1 0 p D)(p 0 q 0 1)N p0q01 ; t in the compact set K (14) wher...

This study extends the framework of adaptive dynamics to function-valued traits. Such adaptive traits naturally arise in a great variety of settings: variable or heterogeneous environments, age-structured populations, phenotypic plasticity, patterns of growth and form, resource gradients, and in many other areas of evolutionary ecology. Adaptive dynamics theory allows analysing the long-term ev...

The following propositions are true: (1) Let n, m be natural numbers, a be a function from [: Segn, Segm :] into R, and p, q be finite sequences of elements of R. Suppose that (i) dom p = Seg n, (ii) for every natural number i such that i ∈ dom p there exists a finite sequence r of elements of R such that dom r = Segm and p(i) = ∑ r and for every natural number j such that j ∈ dom r holds r(j) ...

The main purpose of this paper is to establish the Milloux inequality of E-valued meromorphic function from the complex plane ℂ to an infinite dimensional complex Banach space E with a Schauder basis. As an application, we study the Borel exceptional values of an E-valued meromorphic function and those of its derivatives; results are obtained to extend some related results for meromorphic scala...