### Browsing by Author "Vuojamo, Vesa"

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Item Harnack's Inequality(2012) Vuojamo, Vesa; Matematiikan ja systeemianalyysin laitos; Perustieteiden korkeakoulu; School of Science; Kinnunen, JuhaIn the theory of partial differential equations there are only few simple equations that can be given an explicit formula for solutions. In general equations are studied by proving certain existence and regularity results. Smoothness and boundedness are examples of regularity results for solutions. Harnack's inequality is a result which states that the essential supremum and infimum of a solution are comparable. This means that an upper bound for the essential supremum can be obtained from the infimum by multiplying it by a constant. This constant may only depend on the geometry of the sets considered and on the structure of the equation. The proof of Harnack's inequality is based on Caccioppoli-type estimates and Moser's iteration. This iteration leads to separate estimates for both supremum and infimum of solutions. These estimates can be combined to give the Harnack's inequality by means of using parabolic John-Nirenberg lemma or a measure theoretical Bombieri's lemma. Bombieri's lemma is considerably less technical than the parabolic version of John-Nirenberg lemma. In this thesis a proof is given for Harnack's inequality for a certain type of parabolic partial differential equations. A similar result was previously proven in an article by N. Trudinger in 1968. Trudinger's proof is simplified by replacing the parabolic John-Nirenberg lemma with Bombieri's lemma. Calculations are also carried out in more detail.Item On Hypercomplex and Time-Frequency Analysis(Aalto University, 2020) Vuojamo, Vesa; Eriksson, Sirkka-Liisa, Dr., Helsinki University; Finland; Turunen, Ville, Dr., Aalto University, Finland; Matematiikan ja systeemianalyysin laitos; Department of Mathematics and Systems Analysis; Perustieteiden korkeakoulu; School of Science; Kinnunen, Juha, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandIn this thesis we study two topics in harmonic analysis. In the first half we concentrate on Clifford analysis and, in particular, derive Cauchy-type formulas for certain regular functions. In the second half of this thesis we focus on time-frequency analysis and prove characterizations of properties quadratic time-frequency transforms. Clifford analysis is a branch of mathematical analysis applying Clifford algebras to study generalizations of complex analysis. These algebras are used to construct higher dimensional analogues to complex numbers. In this context the complex analytic functions are generalized to monogenic functions which are null-solutions of certain Cauchy-Riemann or Dirac operators. Many of the results in complex analysis may be translated into higher dimensions. However, these results depend on the choice of the operator defining the family of monogenic functions. We study the theory known as modified Clifford analysis. This theory is based on the modified Cauchy-Riemann operator which is closely connected to the hyperbolic space. Working in the Poincaré upper half-space model of hyperbolic geometry, we find the k-hyperbolic harmonic fundamental solutions. Using these solutions we also prove a Cauchy-type integral formula for k-hypermonogenic functions. In the second part of this thesis we focus on time-frequency analysis. The goal of this field of study is to find representations which combine the features of both the signal and its Fourier transform. Using time-frequency representations such as time-frequency transforms signals can be described and manipulated jointly in time and in frequency. If the signal is music, a time-frequency transforms acts as its mathematical musical score. We study quadratic time-frequency transforms which may be interpreted as time-frequency energy densities of a given signal. By the Heisenberg uncertainty relation, a signal cannot be perfectly localized in time and simultaneously have a definite frequency. This precludes the existence of a perfect time-frequency energy density. Nevertheless, such an energy density may be approximated in some sense using quadratic time-frequency transforms. We consider the Cohen class of covariant time-frequency transforms and prove characterizations of several properties linked to energy densities and transformations of signals. Most of these properties are characterized in terms of the quantization, the integral kernel and the evaluation at the time-frequency origin of the given transform.Item Time–Frequency Analysis in Rn(SPRINGER BIRKHAUSER, 2021-02) Vuojamo, Vesa; Turunen, Ville; Orelma, Heikki; Department of Mathematics and Systems Analysis; Analysis; Tampere UniversityA time–frequency transform is a sesquilinear mapping from a suitable family of test functions to functions on the time–frequency plane. The goal is to quantify the energy present in the signal at any given time and frequency. The transform is further specified by imposing conditions such as those stemming from basic transformations of signals and those which an energy density should satisfy. We present a systematic study on how properties of a time–frequency transform are reflected in the associated evaluation at time–frequency origin, integral kernel and quantization and discuss some examples of time–frequency transforms.Item Vasenivariantit Riemannin metriikat Heisenbergin ryhmässä(2010) Vuojamo, Vesa; Peltonen, Kirsi; Informaatio- ja luonnontieteiden tiedekunta; Peltonen, Kirsi