Research Interests

Sampling Theory Wavelets Medical Imaging
Fractional Fourier Transform Sinc Approximations Boundary-Value Problems
Special Functions Integral Transforms Generalized Functions
Chromatic Derivatives Prolate Wave Functions Miscelaneous

 

Below are samples of my publications in these areas, for a full list of publications, click here.

Sampling Theory (Shannon)

1) . “New Perspectives on Approximation and Sampling Theory” (A. Zayed and G. Schmeisser, Editors), Bikhauser, 2015.

2) Shift-invariant and Sampling Spaces Associated with the Fractional Fourier Transform Domain, IEEE Transactions on Signal Processing, Vol. 60, No. 3, 2012, pp. 1627-1637 (jointly with A. Bhandari.).

3) Sampling Theorem for Bandlimited Hardy Space Functions Generated by Regge Problem, Applied and Computational Harmonic Analysis, Vol. 31 (2011), pp. 125-142 (jointly with M. Shubov).

4) On the Notion of Bandlimitedness and its Generalizations, Journal of the Argentinean Mathematical Society, Revista De La Union Matematica Argentina, Vol. 49 (2008) pp. 99-109.

5) “Sampling expansions of functions having values in a Banach space,” Proceedings of the American Mathematical Societ, Vol. 133, # 12 (2005), pp. 3597- 3607 (jointly with D. Han).

6) “A q-analogue of the Whittaker-Shannon-Kotel’nikov Sampling theorem, Proceedings of the American Mathematical Societ, Vol. 131, pp. 3711-3719 (2003), (jointly with M. Ismail).

7) “ Lagrange Interpolation and Sampling Theorems,” jointly with P. Butzer, Chapter 3 in Theory and Applications of Non-Uniform Sampling,” F. Marvasti, Editor, Kluwer/Plenum Publishing Corporation (2001), pp. 123-168.

8) “ Sampling on a string, " The Journal of Fourier Analysis and Applications , Vol. 8 (2002), pp. 211-231 (jointly with A. Boumenir), .

9) "Advances in Shannon's Sampling Theory, " CRC Press, Boca Raton, Florida, 1993.

10) “ Kramer's sampling theorem for multidimensional signals and its relationship with Lagrange-type interpolation,"Journal of multidimensional systems and signal processing, Vol. 3 (1992), pp. 323-340.

Wavelets:

1) “The continuous wavelet transform,” Standard Mathematical Tables and Formulae, 33rd Edition, CRC Press (2017).

2) “Wavelets: Continuous Transform and Series,” Encyclopedia of Optical and Photonic Engineering, Second Edition, August 30, 2015. http://www.taylorandfrancis.com/books/details/9781439850978

3) Preface to Multi-scale Analysis,” Multi-scale Analysis and Modeling, Research Monograph:, Springer-Verlag, 2012

4) “Wavelets, Multiscale Analysis, and Their Applications: An Introduction, Chapter 1,” in Wavelets and Multiscale Analysis, J. Cohen and Ahmed Zayed, Editors; Birkhauser Publishing February 2011

5) Shift-invariant and Sampling Spaces Associated with the Fractional Fourier Transform Domain, IEEE Transactions on Signal Processing, Vol. 60, No. 3, 2012, pp. 1627-1637 (jointly with A. Bhandari.),

6) “Texture Identification of Tissues Using Directional Wavelet, Ridgelet, and Curvelet Transforms, in “Frames and Operator Theory in Analysis and Signal Processing,” Contemporary Mathematics Series, American Mathematical Society, Vol. 451, pp. 89-118 (2008), Jointly with L. Dettori,.

7) “The Wavelet, Directional Wavelet, and Ridgelet Transforms With Applications in Texture Identification,” jointly with L. Dettori,, in “Modern Mathematical Models, Methods and Algorithms for Real World Systems” edited by A.H. Siddiqi, I. Duff and O.Christensen, Anamaya  Publisher, New Delhi-London (2007).

8) “Construction of orthonormal wavelets using Kampe de Feriet functions," Proceedings of the American Mathematical Society, Vol. 130 (2002), pp.2893-2904.

9) Wavelets in Closed Forms,” jointly with G. Walter, Chapter 5 in “Wavelet Transforms and Time-frequency Signal Analysis,” L. Debnath, Editor, Birkhäuser Publishing Company (2001) , pp. 121-143.

10) “Shannon-type Wavelets and the Convergence of their Associated Wavelet Series,” in “Modern Sampling Theory: Mathematics and Applications,” J. Benedetto and P. Ferreira, Editors, Birkhause Publishing Company (2001), pp. 135-152..

11) "Pointwise convergence of a class of non-orthogonal wavelet expansions," Proceedings of the American Mathematical Society, Vol. 128 (2000), pp. 3629-3637.

Medical Imaging:

1) “Texture Identification of Tissues Using Directional Wavelet, Ridgelet, and Curvelet Transforms, in “ Frames and Operator Theory in Analysis and Signal Processing, ” Contemporary Mathematics Series, American Mathematical Society, Vol. 451, pp. 89-118 (2008).

2) The Wavelet, Directional Wavelet, and Ridgelet Transforms With Applications in Texture Identification,” jointly with L. Dettori,, in “ Modern Mathematical Models, Methods and Algorithms for Real World Systems ” edited by A.H. Siddiqi, I. Duff and O.Christensen, Anamaya Publisher, New Delhi-London (2007).

Fractional Fourier Transform:

1) "On Sampling Theorems for Fractional Fourier Transforms and Series,” Proceedings of the SampTa 17, the 12th International Conference on Sampling Theory and Applications, Tallinn, Estonia (2017).

2) “On the Invalidity of Fourier Series Expansions of Fractional Order” J. Fractional Calculus and Applied Analysis, Vol. 18, No. 6 (2015) pp. 1507-1517 (jointly with P. Massupost).

3) Shift-Invariant and Sampling Spaces Associated with the Fractional Fourier Transform Domain, (jointly with A. Bhandari.), IEEE Transactions on Signal Processing, Vol. 60, No. 3, (2012), pp. 1627-1637.

4) “ Fractional Wigner distribution and ambiguity functions,” (jointly with V. B. Shakhmurov) J. Fractional Calculus and Applied Analysis, Vol. 6. No. 4 (2003), pp. 473-490

5) “A class of fractional integral transforms: A generalization of the fractional Fourier transform,” IEEE Transactions on Signal Processing, Vol. 50 (2002), pp. 619-627.

6) “New Sampling Formulae for the Fractional Fourier Transform,” (jointly with A. Garcia) the Journal of Signal Processing, Vol. 77 (1999), pp. 111-114.

7) “Hilbert transform associated with the fractional Fourier transform,” IEEE Signal Proc. Letters, Vol 5, No 8 (1998), pp. 206-208.

8) "A convolution and product theorem for the fractional Fourier transform," IEEE Signal Processing Letters, Vol. 5, No. 4 (1998), pp. 101-103.

9) "Fractional Fourier transform of generalized functions," Journal of Integral Transforms and Special Functions, Vol. 7 No. 4(1998), pp. 299-312.

10) "On the relationship between the Fourier and fractional Fourier transforms," IEEE Signal Processing Letters, Vol. 3 (1996), pp. 310-311.

11) " On the relationship between the fractional Fourier transform and the Riemann-Liouville fractional integral, Proceedings of the International Association for Mathematics and Computers in Simulation (IMACS), May 2000.

Sinc Approximations:

1) “A comparison between the Adomian decomposition and the sinc-Galerkin methods for solving nonlinear boundary-value problems,” Journal of Computational Analysis and Applications,Vol. 7, No. 1(2005)), pp. 5-20 (jointly with E. Deeba and J. Yoon)

2) “Sinc-Galerkin method for solving linear sixth order boundary-value problems,”Mathematics of Computation,” American Mathematical Society, Vol. 73, No. 247 (2004), pp. 1325-1343 (jointly with M. El-Gamel and J. Cannon).

3) “Sinc-Galerkin method for solving non-linear boundary-value problems,”Journal of Computers and Mathematics with Applications, Vol. 48, No. 9 (2004), pp. 1285-1298 (jointly withM. El-Gamel).

4) “A Comparison between the wavelet-Galerkin and the Sinc-Galerkin methods in solving non-homogeneous heat equations,”Contemporary Mathematics, American Mathematical Society, Vol. 313 (2002), pp. 97-116. (Jointly with M. El Gamel),

Boundary-Value Problems:

1) “Discontinuous boundary-value problems: Expansion and sampling theorems,” (jointly with M. Annaby, G. Freiling), Journal of Integral Equations and Applications, Vol. 16 (2004), pp. 1-24.

2) “An inversion theorem for integral transforms related to singular Sturm-Liouville problems on a half Line,” (jointly with C. Shin), Acta Mathematica Hungaria, Vol. 97 (2002), pp. 273-286.

3) “Polynomial growth solutions of Sturm-Liouvill equations on a half-line and their zero distribution,” Mathematische Nachrichten, (jointly with C. Shin and A. Tovbis), Vol. 263-264, pp. 204-217, January, 2004 .

40 "A new role of Green's function in interpolation and sampling theory, "the Journal of Mathematical Analysis and Applications, Vol. 175 (1993), pp. 222-238.

Special Functions and Orthogonal Polynomials:

1) “A q-analogue of the Whittaker-Shannon-Kotel’nikov Sampling theorem, Proceedings of the American Mathematical Society, Vol. 131, pp. 3711-3719 (2003), (jointly with M. Ismail).

2) "A proof of new summation formulae by using sampling theorems," Proceedings of the American Mathematical Society, Vol. 117, No. 3 (1993), pp. 699-710.

3) "A new role of sampling theory in the theory of special functions," Proceedings of the 13th IMACS World congress on Computation and Applied Mathematics, Dublin, Ireland 1991, C. Brezinski, editor, Elsevier Publ., Amsterdam (1993).

40 "Generalized Jacobi transforms," jointly with E. Deeba, the Journal of Applicable Analysis, Vol. 48 (1993), pp. 63-79.

5) "Jacobi polynomials as generalized Faber polynomials," Transactions of the American Mathematical Society, Vol. 321, No. 1 (1990), pp. 363-378.

Integral Transforms:

 

1) “Linear Transformations in Signal and Optical Systems,” Handbook of Operator Theory, D. Alpay, Editor, Springer-Verlag, (2015), P. 833-874.

2) Paley-Wiener Subspace of Vectors in a Hilbert Space with Applications to Integral Transforms, Journal of Mathematical Analysis and Applications, Vol. 353 (2009), pp. 566-582. (jointly with I. Pesenson)

3) "On the extension of the Zak transform," (jointly with P. Mikusinski), Journal of Methods and Applications of Analysis, Vol. 2 (1995),pp. 160-172.

4) "Radon transform of Boehmians," (jointly with P. Mikusinski), Proceedings of the American Mathematical Society, Vol. 118 (1993), pp. 561-570.

5) "On the inversion of integral transforms associated with Sturm-Liouville problems," jointly with G. Walter, Journal of Mathematical Analysis and Applications, Vol. 164, No. 1(1992), pp. 285-306.

6) "Inversion of integral transforms associated with a class of perturbed heat equations," jointly with D. Haimo, Journal of Mathematical Analysis and Applications, Vol. 163, No. 1(1992), pp. 113-135.

Generalized and Hyperfunctions:

1) "On the theta semi-group," journal of Complex Analysis and Operator Theory, (2012) 6: 565-583 (jointly with W. Urbina),.

2) "On the Lame series representation of analytic hyperfunctions on a two-dimensional complex
manifold,” in “Micro-local Analysis and Complex Fourier Analysis, Editors, K. Fujita and M. Morimoto, World Scientific Publisher (2002), pp. 317-328.

3) "Fractional Fourier transform of generalized functions," Journal of Integral Transforms and Special Functions, Vol. 7 No. 4(1998), pp. 299-312.

4) "Wavelet expansions of analytic hyperfunctions," Journal of Integral Transforms and Special Functions, Vol. 3 (1995), pp. 305-320.

5) " Wavelet transforms of periodic generalized functions, "the Journal of Mathematical Analysis and Applications., Vol. 183, 2(1994), pp. 391-412.

6) "Generalized Faber expansions of hyperfunctions on analytic curves," the Journal of the Mathematical Society of Japan, Vol. 42, No. 1 (1990), pp. 155-170.

 

Chromatic Derivatives:

1) Chromatic Expansions in Function Spaces, Transactions of the American Mathematical Society, Vol. 366, No. 8, (2014), pp. 4097-4125.

2) “Chromatic Expansions and the Bargman Transform,” Chapter 6 in Multi-scale Analysis and Modeling, Research Monograph,  (Xiaoping Shen and Ahmed Zayed, Editors), Springer-Verlag, 2012.

3) Chromatic Expansions in Reproducing-kernel Hilbert Spaces, Progress in Analysis, Proceedings of the 8th ISAAC Congress, Moscow, Russia, 2011, People’s Friendship University of Russia, Publisher (2012), pp. 346-355.

4) "Multidimensional chromatic derivatives and series expansions," (jointly with A. Ignjatovic) Proceedings of the American Mathematical Society, Vol. 139, No. 10 (2011), pp. 3513-3525.

5) "Chromatic derivatives of generalized functions," Journal of Integral Transforms and Special Functions, Vol. 22, No. 4-5 (2011), pp. 383-390.

6) "Generalizations of Chromatic Derivatives and Series Expansions," (Institute of Electrical and Electronics Engineers) IEEE Transactions on Signal Processing, Vol. 58, No. 3, (2010), pp. 1638- 1647.

 

Prolate Spheroidal Wave Functions:

1) “Numerical Solution to an Energy Concentration Problem Associated with the Special Affine Fourier Transformation,” (jointly with Amara Ammari and Tahar Moumni), Frames and other Bases in Abstract and Function Spaces, I. Pesenson, et al Editors, Birkhauser (2017),pp. 161-184

2) The Prolate Spheroidal Wave Function,” Standard Mathematical Tables and Formulae, 33rd Edition, CRC Press (2017).

3) Maximally concentrated signals in the special affine Fourier transformation domain, Proceedings of 2015 International Conference on Sampling Theory and Applications (SampTA) July 2015, http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7148841.

4) Solution of the Energy Concentration Problem in Reproducing-Kernel Hilbert Space, Journal of the Society for Industrial and Applied Mathematics (SIAM) on Applied Mathematics, Vol. 75, No. 1 (2015), pp. 21-37.

5) A Generalization of the Prolate Spheroidal Wave Functions with Applications to Sampling (jointly with Taher Moumni), Journal of Integral Transforms and Special Functions, Vol. 25, No. 6, (2014), pp. 433-447.

6) A Generalization of the Prolate Spheroidal Wave Functions,” Proceedings of the American Mathematical Society, Vol. 135, (2007), pp. 2193-2203.

 

 

Miscelaneous:

1) Topics in Harmonic Analysis and Ergodic Theory,” Contemporary Mathematics Series, American Mathematical Society, Volume 444, (2007), J. Rosenblatt, A. Stokolos, and A. Zayed, Editors.

2) The Zak transform," Encyclopedia of Mathematics, Supplement III, Kluwer Publications, 2001.

3) Density deconvolution of different conditional distributions,” (jointly with M. Pensky), Annals of the Institute of Statistical Mathematics, Vol. 54 (2002), pp. 701-712.

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