Integrabilty of fourier transforms, general monotonicity, and related problems
- Autores: Askhat Mukanov
- Directores de la Tesis: Sergey Tikhonov (dir. tes.)
- Lectura: En la Universitat Autònoma de Barcelona ( España ) en 2018
- Idioma: español
- Tribunal Calificador de la Tesis: Joan Orobitg i Huguet (presid.), Santiago Boza Rocho (secret.), Elijah Liflyand (voc.)
- Programa de doctorado: Programa de Doctorado en Matemáticas por la Universidad Autónoma de Barcelona
- Materias:
- Enlaces
- Resumen
This thesis is devoted to the study of integrability and convergence properties of Fourier series and transforms.
The main results are the following.
1. We investigate the integrability properties of trigonometric series with general monotone coefficients and prove the Hardy-Littlewood-type results, i.e., equivalences of the norms of sums of trigonometric series and weighted norms of their Fourier coefficients. We prove such equivalences for the Lorentz and weighted Lebesgue spaces. Here we deal with the trigonometric series with general monotone coefficients.
2. We study the smoothness properties of functions that can be represented by trigonometric series with general monotone coefficients. The equivalence of the Lp-modulus of smoothness of such functions and weighted sums of their Fourier coefficients is proved.
3. We obtain the multidimensional versions of Boas-type theorem on integrability properties of the Fourier transforms of monotone in each variable functions.
4. Finally, we study criteria for the uniform convergence of trigonometric series with general monotone coefficients. In particular, we generalize the well-known Chaundy-Jolliffe criterion for the uniform convergence of sine series and obtain the corresponding result for cosine series. Moreover, we prove necessary and sufficient conditions for partial Fourier sums of such series to have certain convergence rate.
