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Construction of a Fourier transform based on Haar measure in structure (Rn, +, T)

Żukowicz M

Zeszyty Naukowe / Akademia Morska w Szczecinie

2014

nr 39 (111)

181--184

An important aspects of learning are the theoretical elements of mathematics and the methods of obtaining the function or designbecause it allows for further development of the issues and the ability to apply this problem in practice. The Fourier transform is very useful in science and technology. The article shows how and in what structures is possible to use the Fourier transform in the set Rn. The article describes the project of compact-open topology, which is essential to the construction of the Haar measure and integral. It was also described the concept of the nature of a locally compact topological group, because a group of characters is also essential to the construction of the Fourier transform and inverse Fourier transform. It has been proven that a group of characters is a locally compact topological group, which allows the introduction of the Haar measure. It has given an example application of the theory of Haar integral when calculating the Fourier transform for some of the electronics.

On ∞-entropy points in real analysis

Korczak-Kubiak E., Loranty A., Pawlak R. J.

Opuscula Mathematica

2014

Vol. 34, no. 4

799--812

We will consider ∞-entropy points in the context of the possibilities of approximation mappings by the functions having ∞-entropy points and belonging to essential (from the point of view of real analysis theory) classes of functions: almost continuous, Darboux Baire one and approximately continuous functions.

Čech-completeness and related properties of the generalized compact-open topology

Holá L`., Zsilinszky L.

Journal of Applied Analysis

2010

Vol. 16, nr 1

151-169

The generalized compact-open topology τc on partial continuous functions with closed domains in X and values in Y is studied. If Y is a non-countably compact Čech-complete space with a Gδ-diagonal, then τc is Čech-complete, sieve complete and satisfies the p-space property of Arhangel'skii, respectively, if and only if X is Lindelof and locally compact. Lindelofness, paracompactness and normality of τc is also investigated. New results are obtained on Čech-completeness, sieve completeness and the p-space property for the compact-open topology on the space of continuous functions with a general range Y.

Homeomorphism groups and the topologist's sine curve

Dijkstra J. J., Tahri R.

Bulletin of the Polish Academy of Sciences. Mathematics

2010

Vol. 58, no 3

269-272

It is shown that deleting a point from the topologist's sine curve results in a locally compact connected space whose autohomeomorphism group is not a topological group when equipped with the compact-open topology.