Abstract
We use apreviously obtained characterization of test functions of w-Tempered Ultradistributions to charcterize the space w-Tempered Ultradistributions using Riesz representation Theorem.
Highlights
In this paper, We use the characterization of the space Sw of test functions of w−tempered ultradistribution in terms of their short-time Fourier transform to characterize w−tempered ultradistribution using Riesz representation theorem
It is well known that Fourier series are a good tool to represent periodic functions
The short-time Fourier transform works by first cutting off the function by multiplying it by another function called window apply the Fourier transform
Summary
We use the characterization of the space Sw of test functions of w−tempered ultradistribution in terms of their short-time Fourier transform to characterize w−tempered ultradistribution using Riesz representation theorem. ( [6], [7])The short-time Fourier transform (STFT) of a function or distribution f on Rn with respect to a non-zero window function g is formally defined as νgf (x, ξ) = f (t)g(t − x)e−2πit.ξdt = (f Txg)(ξ) =< f, MξTxg > . The composition of Tx and Mξ is the time-frequency shift (MξTxg)(t) = e2πix.ξg(t − x), and its Fourier transform is given by
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