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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

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Jasmina Veta Buralieva (University Goce Delcev, Stip)

Abelian results for the directional short-time Fourier transform

(joint work with K. Hadzi-Velkova Saneva and S. Atanasova)

We study the directional short-time Fourier transform (DSTFT) of Lizorkin distributions. DSTFT on the space $L^{1}(\mathbb R^{n}) $ was introduced and investigated by Giv in [4]. Saneva and Atanasova extended this transform on the space of tempered distributions [5]. Here, we analyze the continuity of the DSTFT on the closed subspace of $ \mathcal S(\mathbb R^{n}) $, i.e. on the space $ \mathcal S_{0}(\mathbb R^{n}) $ of highly time-frequency localized functions over $ \mathbb R^{n} $. We also prove the countinuity of the directional synthesis operator on the space $ \mathcal S(\mathbb Y^{2n}) $. Using the obtained continuity results, we will define the DSTFT on space $ \mathcal S'_{0}(\mathbb R^{n}) $ of Lizorkin distributions, and prove an Abelian type result for this transform.

Keywords: (Directional short-time Fourier transform, distributions, quasiasymptotic behavior, Abelian theorems.)

References:

[1] F. Treves: Topological vector spaces, distributions and kernels, Academic press, New York-London, 1967.
[2] L. Schwartz: Theorie des distributions a valeurs vectorielles. I, Ann.Inst. Fourier Grenpble 7 (1957), 1-141.
[3] J. Sebastiao e Silva: Sur la definition et la structure des distributions vectorielles, Portugal. Math 19 (1960), 1-80.
[4] H.H. Giv: Directional short-time Fourier transform, J. Math. Anal. Appl. 399 (2013), 100-107.
[5] K. Hadzi-Velkova Saneva, S. Atanasova: Directional short-time Fourier transform of distributions, J. of Inequaility and Appl. Vol. 124, no.1, 2016, 1-10.
[6] S. Kostadinova, S. Pilipović, K. Hadzi-Velkova Saneva and J. Vindas: The ridglet transform of distribution., Integral transforms Spec. Func. Vol. 25, No. 5, 2014, 344-358.
[7] S. Kostadinova, S. Pilipović, K. Hadzi-Velkova Saneva and J. Vindas: The ridglet transform of distribution and quasiasymptotic behaviour of distributions., Operator Theory: Advances and Applications, Vol. 245, 2015, 183-195.
[8] S. Pilipović, J. Vindas: Multideimensional Tauberian theorems for vector-valued distributions, Publ. Inst. Math. (Beograd), Publications de l'institut mathematique, Nouvelle serie, tome 95 (109) (2014), 1?28.
[9] S. Pilipović, B. Stanković and J. Vindas: Asymptotic behavior of generalized functions, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.
[10] K. Saneva, R. Aceska, S. Kostadinova: Some Abelian and Tauberian results for the short-time Fourier transform, Novi Sad Journal of Math., Vol. 43, No. 2, 2013, 81-89.

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