A sampling theorem for transforms with discontinuous kernels

References

• Annaby , MH and Hassan , HA . 1998 . A sampling theorem associatedwith boundary-value problems with not necessarily simple eigenvalues . Internat. J. Math. & Math. Sci. , 21 : 571 – 580 .
   Google Scholar
• Annaby , MH and Zayed , AI . 1998 . On the use of Green's function insampling theory . J. Integral Equations and Applications , 10 : 117 – 139 .
   Google Scholar
• Annaby , MH . 1999 . Interpolation expansions associated with nonselfadjoint boundary value problems . Comm. Appl. Anal. , 3 : 545 – 561 .
   Google Scholar
• Annaby , MH and Freiling , G . 2000 . Sampling integrodifferential transforms arising from second order differential operators . Math. Nach. , 216 : 25 – 43 .
   Google Scholar
• Annaby , MH and Freiling , G . 2000 . Sampling expansions associated with Kamke problems . Math. Zeit. , 234 : 163 – 189 .
   Web of Science ®Google Scholar
• Benzinger , HE . 1977 . Pointwise and norm convergence of class of biorthonormal expansions . Trans. Amer. Math. Soc. , 231 : 259 – 271 .
   Web of Science ®Google Scholar
• Birkhoff , GD . 1908 . Boundary-value problems and expansion problems of ordinary linear differential equations . Trans. Amer. Math. , 9 : 373 – 395 .
   Google Scholar
• Butzer , PL . 1983 . A survey of the Whittaker-Shannon sampling theorem and some of its extensions . J. Math. Res. Exposition , 3 : 185 – 212 .
   Web of Science ®Google Scholar
• Butzer , PL and Schöttler , G . 1994 . Sampling theorems associated with fourth and higher order self-adjoint eigenvalue problems . J. Comput. Appl. Math. , 51 : 159 – 177 .
   Web of Science ®Google Scholar
• Campbell , LL . 1964 . A comparison of the sampling theorems of Kramer and Whittaker . SIAM J. Appl. Math. , 12 : 117 – 130 .
   Google Scholar
• Coddington EA Levinson N 1955 Theory of Ordinary Differential Equations, McGraw-Hill New York
   Google Scholar
• Everitt , WN , Schöttler , G and Butzer , PL . 1994 . Sturm-Liouville boundary value problems and Lagrange interpolation series . Rendiconti di Matematica, Roma (7) , 14 : 87 – 126 .
   Google Scholar
• Everitt WN Nasri-Roudsari G 1999 Interpolation and sampling theories, and linear ordinary boundary value problems In: J.R. Higgins and R.L. Stens (Eds.) Sampling Theory in Fourier and Signal Analysis; Advancedtopics pp. 96–129 Oxford University Press Oxford
   Google Scholar
• Everitt , WN and Nasri-Roudsari , G . 1999 . Sturm-Liouville problems with coupled boundary conditions and Lagrange interpolation series . Computational Appl. Math. , 1 : 319 – 347 .
   Google Scholar
• Everitt , WN and Nasri-Roudsari , G . 2000 . Sturm-Liouville problems with coupled boundary conditions and Lagrange interpolation series: II . Rendiconti di Matematica, Roma (7) , 20 : 199 – 238 .
   Google Scholar
• Everitt , WN and Poulkou , A . 2002 . Kramer analytic kernels and first-order boundary value problems . J. Computational Appl. Math. , 148 : 29 – 47 .
   Google Scholar
• Everitt , WN , Hayman , WK and Nasri-Roudsari , G . 1997 . On the representation of holomorphic functions by integrals . Applicable Analysis , 65 : 95 – 102 .
   Google Scholar
• Everitt , WN , Nasri-Roudsari , G and Rehberg , J . 1998 . A note on the analytic form of the Kramer sampling theorem . Results in Math. , 34 : 310 – 319 .
   Google Scholar
• Haddad , A , Yau , K and Thomas , J . 1967 . General methods for the derivation of the sampling theorems . IEEE Trans. Inf. Theory , IT-13 : 227 – 230 .
   Google Scholar
• Higgins JR 1996 Sampling Theory in Fourier and Signal Analysis: Foundations, Oxford University Press Oxford
   Google Scholar
• Jank G Volkmann L 1985 Meromorphe Funktionen und differentialgleichungen., Birkhäuser Basel
   Google Scholar
• Kramer , HP . 1959 . A generalized sampling theorem . J. Math. Phys. , 38 : 68 – 72 .
   Google Scholar
• Kobayashi , M . 1990 . Eigenfunction expansions: a discontinuous version . SIAM J. Appl. Math. , 50 : 910 – 917 .
   Google Scholar
• Mihailov , VP . 1962 . Riesz bases in L 2(0, 1) . Sov. Math. , 3 : 851 – 855 .
   Google Scholar
• Muravei , LA . 1967 . Riesz bases in L 2(−1, 1) . Proc. Steklov Inst. Mat. , 91 : 117 – 136 .
   Google Scholar
• Naimark MA 1967 Linear Differential Operators I., George Harrap London
   Google Scholar
• Rasulov ML 1967 Methods of Contour Integration, North-Holland Publ. Co Amsterdam
   Google Scholar
• Shannon , CE . 1949 . Communications in the presence of noise . Proc. IRE , 37 : 10 – 21 .
   Google Scholar
• Tamarkin , JD . 1927 . Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions . Math. Zeit. , 27 : 1 – 54 .
   Google Scholar
• Weiss , P . 1957 . Sampling theorems associated with Sturm-Liouville systems . Bull. Amer. Math. Soc. , 163 : 242
   Google Scholar
• Whittaker , E . 1915 . On the functions which are represented by the expansion of the interpolation theory . Proc. Roy. Soc. Edin. Sect. A , 35 : 181 – 194 .
   Google Scholar
• Young RM 1980 Introduction to Nonharmonic Fourier Series, Academic Press New York
   Google Scholar
• Zayed , AI . 1991 . On Kramer's sampling theorem associated with general Sturm-Liouville problems and Lagrange interpolation . SIAM J. Appl. Math. , 51 : 575 – 604 .
   Web of Science ®Google Scholar
• Zayed AI 1993 Advances in Shannon's Sampling Theory, CRC Press Boca Raton
   Google Scholar
• Zayed , AI , Hinsen , G and Butzer , PL . 1990 . On Lagrange interpolationsand Kramer-type sampling theorems associated with Sturm-Liouville problems . SIAM J. Appl. Math. , 50 : 893 – 909 .
   Web of Science ®Google Scholar
• Zayed , AI , EL-Sayed , MA and Annaby , MH . 1991 . On Lagrange interpolations and Kramer's sampling theorem associated with self-adjoint boundary value problems . J. Math. Anal. Appl. , 158 : 269 – 284 .
   Web of Science ®Google Scholar
• Zayed , AI and García , AG . 1998 . Kramer's sampling theorem with discontinuous kernels . Results Math. , 34 : 197 – 206 .
   Google Scholar