On sequence spaces for Fr\'echet frames

We analyze the construction of a sequence space $\widetilde{\Theta}$, resp. a sequence of sequence spaces, in order to have $\{g_i\}$ as a $\widetilde{\Theta}$-frame or Banach frame for a Banach space $X$, resp. pre-$F$-frame or $F$-frame for a Fr\'echet space $X_F=\cap_{s\in {\mathbb N}_0} X_s$, where $\{X_s\}_{s\in {\mathbb N}_0}$ is a sequence of Banach spaces

Fr\'echet frames, general definition and expansions

We define an {\it $(X_1,\Theta, X_2)$-frame} with Banach spaces $X_2\subseteq X_1$, $|\cdot|_1 \leq |\cdot|_2$, and a $BK$-space (\Theta, \snorm[\cdot]). Then by the use of decreasing sequences of Banach spaces ${X_s}_{s=0}^\infty$ and of sequence spaces ${\Theta_s}_{s=0}^\infty$, we define a general Fr\' echet frame on the Fr\' echet space $X_F=\bigcap_{s=0}^\infty X_s$. We give frame expansions of elements of $X_F$ and its dual $X_F^*$, as well of some of the generating spaces of $X_F$ with convergence in appropriate norms. Moreover, we give necessary and sufficient conditions for a general pre-Fr\' echet frame to be a general Fr\' echet frame, as well as for the complementedness of the range of the analysis operator $U:X_F\to\Theta_F$.Comment: A new section is added and a minor revision is don

Semigroups of operators on the space of generalized functions exp A′

AbstractThe inductive limit of spaces exppA′, p ϵ N (Pilipović, SIAM J. Math. Anal., 17 1986, 477–484) whose elements have unique orthonormal series expansions with exponential growth rate of the corresponding coefficients is to be studied in the first part of the paper. Then, we determine some semigroups of operators on this space. This enables us to solve some classes of infinite order partial differential equations

Regularity properties of distributions through sequences of functions

We give necessary and sufficient criteria for a distribution to be smooth or uniformly H\"{o}lder continuous in terms of approximation sequences by smooth functions; in particular, in terms of those arising as regularizations $(T\ast\phi_{n})$.Comment: 10 page

Classes of generalized functions with finite type regularities

We introduce and analyze spaces and algebras of generalized functions which correspond to Hölder, Zygmund, and Sobolev spaces of functions. The main scope of the paper is the characterization of the regularity of distributions that are embedded into the corresponding space or algebra of generalized functions with finite type regularities

Distributional versions of Littlewood's Tauberian theorem

We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We apply these Tauberian results to deduce a number of Tauberian theorems for power series where Ces\`{a}ro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.Comment: 15 page

Topological properties of regular generalized function algebras

We investigate density of various subalgebras of regular generalized functions in the special Colombeau algebra of generalized functions.Comment: 6 page

Sex differences of human corpus callosum revealed by polar coordinate system: magnetic resonance imaging study

Background: Evaluation of morphological and size changes related to various pathological conditions of the corpus callosum (CC) requires the data about sex dimorphism of the CC. The purpose of our study is to define potential morphological sex differences of the CC by the use of polar coordinate system as a system of measurements. Materials and methods: After division of the CC into three equal segments by the use of polar coordinate system, we investigated the length of the hemisphere (A-A’), the CC size as its midsagittal section area (CCA), the size of its segments (C1, C2, C3), thickness of the thinnest part of the CC (TCC) and the angular coordinate (a angle) of dorsal point of the TCC in a sample of 30 human brains magnetic resonance images (15 males and 15 females, age 20–50 years). Results: We found significantly larger CCA, C3 segment and the TCC in males. Statistically significant correlation in both, males and females, was found between parameters of the CCA and of all of its segments (C1, C2, C3), the C1 and C2, the C2 and C3 segments, as well as like as between the C2 and TCC. Sex differences were also in findings of significant correlation between the C1 and C3 segments, between CCA and TCC, and of significant negative correlation between the a angle and A-A’ only in females. Conclusions: We concluded that the use of polar coordinate system appropriately reflects the anatomical and encephalometric characteristics of human CC