On certain results of Fourier analysis on non-abelian discrete groups.

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The topics of this thesis lie in the intersection of harmonic analysis, functional analysis and operator algebras. Recently, advancement made in the area of quantum information has ignited a wave of interest in the non-commutative aspects of analysis. This, in turn, helped to connect the tools in operator algebras with the techniques used in harmonic analysis to produce various deep results. The first part of the thesis will discuss about the characterization of positive definite, radial functions on some non-commutative groups. The subject of positive definite functions is widely studied due to its link with the area of Fourier Multipliers. So, a characterization of positive, definite functions will help in identifying the class of Fourier Multipliers which is completely positive. There is also some surprising consequences which slightly improves the classical Schoenberg-Bochner theorem. The second part of the thesis focus on the non-commutative Khintchine inequality and its relation with the Z2 property. The classical Khintchine inequality has been studied extensively and greatly improved by many mathematicians. In our setting, we show that the existence of the Z2 property in an orthonormal system will result in its matrix-valued coefficient function satisfying a Khintchine-type inequality. For the case of an abelian group, we also obtain some partial converse result on that, where the existence of a Khtinchine-type inequality will imply that the group has finite Z2 constants. In some examples, the optimal constants can be computed.

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