Bases and isomorphisms in spaces of analytic functions

We will discuss the construction of bases in a space of analytic functions for a given domain and isomorphic classification of spaces of analytic functions. We will focus on results in one dimensional case. In one dimensional case, we consider the construction of bases in two different ways. Using one of them, we construct interpolational bases for the space of analytic functions on a compactum K and in that part, results of Leja, Walsh, and Zahariuta are used. Then, isomorphic classification follows by the use of Potential Theory. Using the second way, we construct a common basis for the spaces of analytic functions of a regular pair “compact set domain” by the Hilbert methods that was proposed by Zahariuta. GKS-duality is used for both of the cases. In multidimensional case, some results about bases and isomorphisms of spaces of analytic functions in several variables that were proved by Zahariuta are represented (see also Aytuna). Since a multidimensional analogue of GKS-duality does not exist, interpolational bases cannot be constructed as in one dimensional case. But the bases constructed by Hilbert methods proves to be applicable for studying the isomorphism of the space of analytic functions on D to the space of analytic functions on the unit circle of n-dimensional complex plane. Keywords: Hilbert scales, spaces of analytic functions, Green potential, regularity, GKS-duality

Turkish-German heritage speakers' predictive use of case: webcam-based vs. in-lab eye-tracking

Recently, Özge et al. have argued that Turkish and German monolingual 4-year-old children can interpret case-marking predictively disregarding word order. Heritage speakers (HSs) acquire a heritage language at home and a majority societal language which usually becomes dominant after school enrollment. Our study directly compares two elicitation modes: in-lab and (remote) webcam-based eye-tracking data collection. We test the extent to which in-lab effects can be replicated in webcam-based eye-tracking using the exact same design. Previous research indicates that Turkish HSs vary more in the comprehension and production of case-marking compared to monolinguals. Data from 49 participants–22 Turkish monolinguals and 27 HSs–were analyzed using a binomial generalized linear mixed-effects regression model. In the Accusative condition, participants looked for the suitable Agent before it is appeared in speech. In the Nominative condition, participants looked for the suitable Patient before it is appeared in speech. HSs were able to use morphosyntactic cues on NP1 to predict the thematic role of NP2. This study supports views in which core grammatical features of languages, such as case, remain robust in HSs, in line with the Interface Hypothesis. We were able to replicate the effect of the predictive use of case in monolinguals using webcam-based eye-tracking, but the replication with heritage speakers was not successful due to variability in data collection contexts. A by-participant analysis of the results revealed individual variation in that there were some speakers who do not use case-marking predictively in the same way as most monolinguals and most HSs do. These findings suggest that the predictive use of case in heritage speakers is influenced by different factors, which may differ across individuals and affect their language abilities. We argue that HSs should be placed on a native-speaker continuum to explain variability in language outcomes.Peer Reviewe

Turkish-German heritage speakers’ predictive use of case: webcam-based vs. in-lab eye-tracking

Recently, Özge et al. have argued that Turkish and German monolingual 4-year- old children can interpret case-marking predictively disregarding word order. Heritage speakers (HSs) acquire a heritage language at home and a majority societal language which usually becomes dominant after school enrollment. Our study directly compares two elicitation modes: in-lab and (remote) webcam- based eye-tracking data collection. We test the extent to which in-lab eects can be replicated in webcam-based eye-tracking using the exact same design. Previous research indicates that Turkish HSs vary more in the comprehension and production of case-marking compared to monolinguals. Data from 49 participants–22 Turkishmonolinguals and 27 HSs–were analyzed using a binomial generalized linear mixed-eects regression model. In the Accusative condition, participants looked for the suitable Agent before it is appeared in speech. In the Nominative condition, participants looked for the suitable Patient before it is appeared in speech. HSs were able to usemorphosyntactic cues on NP1 to predict the thematic role of NP2. This study supports views in which core grammatical features of languages, such as case, remain robust in HSs, in line with the Interface Hypothesis. We were able to replicate the eect of the predictive use of case in monolinguals using webcam-based eye-tracking, but the replication with heritage speakers was not successful due to variability in data collection contexts. A by- participant analysis of the results revealed individual variation in that there were some speakers who do not use case-marking predictively in the same way asmost monolinguals and most HSs do. These findings suggest that the predictive use of case in heritage speakers is influenced by dierent factors, whichmay dier across individuals and aect their language abilities. We argue that HSs should be placed on a native-speaker continuum to explain variability in language outcomes