Some compact and non-compact embedding theorems for the function spaces defined by fractional Fourier transform

Yıl 2021, Cilt: 50 Sayı: 6, 1620 - 1635, 14.12.2021

Erdem Toksoy Ayşe Sandıkçı

https://doi.org/10.15672/hujms.795924

Öz

The fractional Fourier transform is a generalization of the classical Fourier transform through an angular parameter $\alpha $. This transform uses in quantum optics and quantum wave field reconstruction, also its application provides solving some differrential equations which arise in quantum mechanics. The aim of this work is to discuss compact and non-compact embeddings between the spaces $A_{\alpha ,p}^{w,\omega }\left(\mathbb{R}^{d}\right) $ which are the set of functions in ${L_{w}^{1}\left(\mathbb{R}^{d}\right) }$ whose fractional Fourier transform are in ${L_{\omega}^{p}\left(\mathbb{R}^{d}\right) }$. Moreover, some relevant counterexamples are indicated.

Anahtar Kelimeler

Fractional Fourirer transform, weighted Lebesgue spaces, compact embedding

Kaynakça

• [1] T. Alieva, V. Lopez, F. Agullo-Lopez and L.B. Almeida, The fractional Fourier trans- form in optical propagation problems, J. Modern Opt. 41 (5), 1037–1044, 1994.
• [2] L.B. Almeida, The fractional Fourier transform and time-frequency representations, IEEE Trans. Signal Process. 42 (11), 3084–3091, 1994.
• [3] L.B. Almeida, Product and convolution theorems for the fractional Fourier transform, IEEE Signal Process. Lett. 4 (1), 15–17, 1997.
• [4] A. Bultheel and H. Martinez, A shattered survey of the fractional Fourier transform, K.U.Leuven: Department of Computer Science, Report TW337, 2002.
• [5] P. Dimovski , S. Pilipovic, B. Prangoski and J. Vindas, Translation-modulation in- variant Banach spaces of ultradistributions, J. Fourier Anal. Appl. 25 (3), 819–841, 2019.
• [6] R.G. Dorsch, A.W. Lohmann, Y. Bitran, D. Mendlovic and H.M. Ozaktas, Chirp filtering in the fractional Fourier domain, Appl. Optics, 33 (32), 7599–7602, 1994.
• [7] H.G. Feichtinger, A compactness criterion for translation invariant Banach spaces of functions, Anal. Math. 8, 165–172, 1982.
• [8] H.G. Feichtinger, Compactness in translation invariant Banach spaces of distributions and compact multipliers, J. Math. Anal. Appl. 102, 289–327, 1984.
• [9] H.G. Feichtinger and A.T. Gürkanlı, On a family of weighted convolution algebras, Int. J. Math. Sci. 13 (3), 517–526, 1990.
• [10] R.H. Fischer, A.T. Gürkanlı and T.S. Liu, On a family of weighted spaces, Math. Slovaca, 46 (1), 71–82, 1996.
• [11] K. Gröchenig, Weight functions in time-frequency analysis, in: Pseudodifferential Operators: Partial Differential Equations and Time-Frequency Analysis, Fields Inst. Commun. 52, 343–366, Amer. Math. Soc., Providence, RI, 2007.
• [12] A.T. Gürkanlı, Compact embeddings of spaces $A_{w,\omega }^{p}\left(\mathbb{R}^{d}\right) $, Taiwanese J. Math. 12 (7), 1757–1767, 2008.
• [13] A.C. McBride and F.H. Kerr, On Namias’s fractional Fourier transforms, IMA J. Appl. Math. 39, 159–175, 1987.
• [14] V. Namias, The fractional order of Fourier transform and its application in quantum mechanics, J. Inst. Math. Appl. 25, 241–265, 1980.
• [15] H.M. Ozaktas, M.A. Kutay and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing, Chichester, John Wiley and Sons, 2001.
• [16] H. Reiter, Classical Harmonic Analysis and Locally Compact Group, Oxford Univer- sity Press, 1968.
• [17] W. Rudin, Real and Complex Analysis, New York, McGraw-Hill, 1966.
• [18] W. Rudin, Functional Analysis, McGraw-Hill, 1973.
• [19] E. Toksoy and A. Sandıkçı, On function spaces with fractional Fourier transform in weighted Lebesgue spaces, J. Inequal. Appl. 2015 (1), Article Id:87, 1–10, 2015.
• [20] N. Wiener, Hermitian polynomials and Fourier analysis, J. Math. Phys. 8, 70–73, 1929.
• [21] A.I. Zayed, On the relationship between the Fourier and fractional Fourier transforms, IEEE Signal Proc. Let. 3, 310–311, 1996.