## České vysoké učení technické v Praze

## Fakulta jaderná a fyzikálně inženýrská

vás zve na přednášku pořádanou v rámci Fakultního kolokvia:

## Dr. Lyonell Boulton

(Heriot-Watt University and FJFI ČVUT)

**Deformed Trigonometric Functions and Applications**

** **

Přednáška se bude konat ve středu dne **24 10. 2018 **od **15.30** hodin **v místnosti 115** v Břehové ulici

(FJFI ČVUT, Břehová 7, Praha 1)

**Abstrakt:**

People with elementary mathematical training will be familiar with the standard trigonometric functions sine and cosine. The shape and periodicity of these functions is determined by the complete symmetry of the circle. According to the celebrated Fourier's Theorem, one-dimensional signals of any reasonable shape can be decomposed into fundamental frequencies of dilated sine/cosine functions. One of the central mathematical objects to be discussed in this colloquium are the generalised trigonometric functions p-sine and p-cosine which were introduced by Eilbert, Otani and Lindqvist over forty years ago. In order to define them, the symmetries of the circle are broken in a certain controlled manner in which the usual geometry is replace with a different geometry depending on a parameter p greater than one. The usual trigonometric functions correspond to the case p equal 2.

As it turns, one can also establish versions of Fourier's Theorem for these generalised trigonometric functions. This is related to deep results about how spaces of regular functions (Sobolev spaces) are embedded into spaces of functions which after taking a power p have finite amplitude graph area (Lebesgue spaces). Many interesting questions about how signals decompose into these deformed modes remain open. Answers to some of them could have a crucial impact in the field of signal processing of non-smooth data. After showing how to construct the p-trigonometric functions and how to generalise Fourier's Theorem, I will be describing the mathematical context of some of these questions and their solution.