Lieb-Thirringova nerovnost – Joona NIEMINEN

Abstract:

This paper examines the natural construction and motivation for the widely used and studied space L2(R2). The general motivation for constructing the Lebesgue spaces is explored as well as the topic of why exactly the family of spaces L2 is advantageous over other said other Lebesgue spaces. Turns out that because L2 are not only complete but also Hilbert spaces that they are preferred. Motivated by this, it is pointed out that Hilbert spaces are especially convenient because orthogonality makes it possible to readily compute Fourier coefficients for any member of the Hilbert space. Lastly, the paper examines some of the modern literature inspired by work of Lieb and Thirring where a number of estimates for upper bounds of orthonormal series are proposed in the context of L2(R2). …víceméně

Abstract:

This paper examines the natural construction and motivation for the widely used and studied space L2(R2). The general motivation for constructing the Lebesgue spaces is explored as well as the topic of why exactly the family of spaces L2 is advantageous over other said other Lebesgue spaces. Turns out that because L2 are not only complete but also Hilbert spaces that they are preferred. Motivated by this, it is pointed out that Hilbert spaces are especially convenient because orthogonality makes it possible to readily compute Fourier coefficients for any member of the Hilbert space. Lastly, the paper examines some of the modern literature inspired by work of Lieb and Thirring where a number of estimates for upper bounds of orthonormal series are proposed in the context of L2(R2). …víceméně