BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Sanja Atanasova (Faculty of Electrical Engineering and Information Technologies\, “Ss. Cyril and Methodius” University in Skopje\, North Macedonia) DTSTART:20201211T132000Z DTEND:20201211T141000Z DTSTAMP:20260423T004658Z UID:WMSEE/8 DESCRIPTION:Title: In tegral transforms through the prism of distribution theory\nby Sanja A tanasova (Faculty of Electrical Engineering and Information Technologies\, “Ss. Cyril and Methodius” University in Skopje\, North Macedonia) as part of Women in Mathematics in South-Eastern Europe\n\n\nAbstract\nThe Fo urier transform is probably the most widely applied signal processing tool in science and engineering. It reveals the frequency composition of a tim e series by transforming it from the time domain into the frequency domain . However\, it does not reveal how the signals frequency contents vary wit h time. A straightforward solution to overcoming the limitations of the Fo urier transform the concept of the short-time Fourier transform (STFT). Th e short-time Fourier transform is a very effective device in the study of function spaces. However\, significant barrier in application of the STFT is the fact that the fixed window function has to be predefined\, which le ads to a poor time-frequency resolution and\, in general\, the absence of a sufficiently good reconstruction algorithm. The Wavelet transform (WT) i s used to overcome some of the shortcomings of the STFT. With the dilatati on and translation of the window function\, the WT has better phase modula tion in the spectral domain. However\, the self-similarity caused by the t ranslation and the overlap in the frequency domain becomes non-avoidable s ince they do not permit straightforwardly the transfer of scale informatio n into proper frequency information. The Stockwell transform (ST) also dec omposes a signal into temporal and frequency components. In contrast to th e WT\, the ST exhibits a frequency-invariant amplitude response and covers the whole temporal axis creating full resolutions for each designated fre quency. It is invertible\, and recovers the exact phase and the frequency information without reconstructing the signal. The problem with the ST is its redundancy. But\, there have been different strategies in order to imp rove the performance and the application of the ST.\n\nOn the other hand\, the STFT\, as a tool of the time-frequency analysis\, contains localized time and frequency information of a function. Another idea is to localize information in time\, frequency\, and direction\, which leads to direction ally sensitive variant of STFT\, which gives the Directional short time Fo urier transform (DSTFT).\n\nIn mathematics\, distributions extend the noti on of functions. Distribution theory is a power tool in applied mathematic s and the extension of integral transforms to generalized function spaces is an important subject with a long tradition. The theory is developed by proving that these transforms are well defined on the appropriate spaces o f distribution. These is done by proving continuity results for these tran sforms on so called test function spaces\, and then extending the definiti ons on distributions. In this talk\, i consider several integrals transfor ms (STFT\, WT\, ST\, DSTFT) and try to make short survey on their behaviou r on distributions.\n\nThere are several approaches to the theory of distributions\, but in all of them one quickly learn that distri butions do not have point values\, as functions do\, despite the fact that they are called generalized functions. Natural generalization of this notion is the quasiasymptotic behavior of distributions. It is an old su bject that has found applications in various fields of pure and app lied mathematics\, physics\, and engineering. In the second part of my talk\, I use Abelian and Tauberian ideas for asymptotic analysis of the mentioned integral transforms to characterize the asymptotic properties of a distribution.\n LOCATION:https://researchseminars.org/talk/WMSEE/8/ END:VEVENT END:VCALENDAR