Introduction of İnner Distributions and Their Approximation with Blaschke Distributions. Characterization of Blaschke Distributions
M. Saliji, V. M. Erakovikj

Abstract. We define inner distributions on the unite disk as boundary values of classical inner functions. Also, we introduce Blaschke distributions on the unite disk. We give properties of the introduced inner and Blaschke distributions. We prove that an inner distribution on the unite disk can be uniformly approximated by a sequence of Blaschke distributions. We characterize the Blaschke distributions in the spirit of Fatou’s theorem
KeyWords and Phrases: Boundary values of distributions, inner distributions, Blaschke product.
2000 Mathematics Subject Classiffcations: 46T30, 46F20.

References.

[1] H. Bremermann, Distributions, Complex Variables and Fourier Transforms, Addison-Wesley, Reading, Massachusetts, 1965.
[2] R. D. Carmichael, D. Mitrovi´c, Distributions and Analytic functions, Longman Scientific & Technical, London, 1989.
[3] J. B. Conway, Functions of One Complex Variable, second edition, SpringerVerlag, New York, 1978.
[4] P. L. Duren, Theory of Hp Spaces, Acad. Press, New York, 1970.
[5] G. Grubb, Distributions and Operators, Springer, New York, 2009.
[6] J. Mashreghi, T. Ransford, Approximation in the closed unit ball, D´epartement de math´ematiques et de statistique, Universit´e Laval, 1045 avenue de la Medecine, Quebec Canada, G1V 0A6, 2017.
[7] V. Manova-Erakovic, V. Reckoski, A note on the analytic representations of convergent sequences in S’, Filomat 29:6 , University of Nis, Serbia, 2015, pp.1419-1424.
[8] V. Manova-Erakovic, V. Reckoski, A Note on the analytic representations of sequences in space, Proceedings of the Sixth International Scientific Conference-FMNS 2015, South-West University “Neofit Rilski”, Blagoevgrad, Bulgaria, 2015, pp.57-61.
[9] V. Manova-Erakovic, Bounded subsets of distributions in D′ generated with boundary values of functions of the space Hp, 1 < p < ∞, Godisen zbornik na Institutot za matematika, Annuaire, ISSN 0351-7241, 2001, pp.31-40.
[10] V. Manova-Erakovic, Introduction of Blaschke distributions and approximation of distribution in D′ by a sequence of finite Blaschke distributions, Matematicki Bilten, Kniga 24 (L) Tome, Skopje, ISSN 0351-336X, 2000, pp.69-76.
[11] N. Pandeski , V. Manova-Erakovic, Distributional boundary values of the Blaschke product, Matematicki Bilten, Kniga 29 (LV) Tome, Skopje, ISSN 0351-336X, 2005, pp.17-22 .
[12] S. Pilipovi´c , B. Stankovi´c, Prostori Distribucija, Novi Sad, 2000.
[13] W. Rudin , Functional Analysis, Mc Graw-Hill, Inc., 1970.
[14] L. Schwartz , Th˙eorie des distributions, Hermann, Paris, 1966.
[15] S. L. Sobolev: Some applications of functional analysis in mathematical physics (3rd ed.), American Mathematical Society, 1991.
[16] F. Treves, Topological vector spaces, Distributions and Kernels, Academic Press, New York, 1967.
[17] V. S. Vladimirov, Methods of the Theory of Generalized Functions, Taylor & Francis, London, 2002.