In this work we take interest in frames and modulation spaces. On the basis of their properties, we show how frame expansions can be used to prove the boundedness of a particular class of Hilbert operators on modulation spaces taking advantage of the special category of piece-wise polynomial functions known as B-splines.

This thesis concerns the Wiener algebra of periodic functions with absolutely convergent Fourier series, and the "Wiener-Lévy functions", meaning functions that preserve absolute convergence of Fourier series under composition. Results regarding the properties of functions in the Wiener algebra are established, as well as properties of Wiener-Lévy functions. By the Wiener-Lévy Theorem, analyticity is a sufficient condition for a function to be a Wiener-Lévy function, and in this paper, we find the necessary condition of being real analytic in the real and imaginary variable, separately.

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This paper examines the Riemann zeta-function and its relation to the prime distribution. In this work, I present the journey from the Dirichlet series to the Riemann hypothesis. Furthermore, I discuss the prime counting function, the Riemann prime counting function and the Riemann explicit function for distribution of primes. This paper explains that the non-trivial zeros of the zeta-function are the key to understand the prime distribution.

In this essay, we present a proof of the great Picard theorem by showing that a holomorphic function with an essential singularity attains infinitely many complex values in the vicinity of the singularity. / Vi kommer att presentera ett bevis på Picards stora sats genom att visa att en holomorf funktion med en väsentlig singularitet antar oändligt många komplexa värden i ett område av singulariteten.

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