Interpolation of functions from generalized Paley-Wiener spaces

We consider the problem of reconstruction of functions f from generalized Paley-Wiener spaces in terms of their values on complete interpolating sequence {Zn}. We characterize the set of data sequences {f(zn)} and exhibit an explicit solution to the problem. Our development involves the solution of a particular ∂- problem.

[1]  L. Hörmander,et al.  An introduction to complex analysis in several variables , 1973 .

[2]  P. Koosis Introduction to H[p] spaces , 1999 .

[3]  R. Young,et al.  An introduction to non-harmonic Fourier series , 2001 .

[4]  Timothy S. Murphy,et al.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , 1993 .

[5]  W. Madych Summability of Lagrange type interpolation series , 2001 .

[6]  I. J. Schoenberg Cardinal interpolation and spline functions VII. The behavior of cardinal spline interpolants as their degree tends to infinity , 1974 .

[7]  J. Garnett,et al.  Bounded Analytic Functions , 2006 .

[8]  Kristian Seip,et al.  Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt's ($A_p$) condition , 1995 .

[9]  Paul Koosis,et al.  Introduction to Hp Spaces , 1999 .

[10]  B. S. Pavlov,et al.  Unconditional bases of exponentials and of reproducing kernels , 1981 .

[11]  I. J. Schoenberg,et al.  Cardinal interpolation and spline functions , 1969 .

[12]  W. Madych,et al.  The Recovery of Irregularly Sampled Band Limited Functions via Tempered Splines , 1994 .

[13]  W. K. Hayman,et al.  LECTURES ON ENTIRE FUNCTIONS (Translations of Mathematical Monographs 150) , 1998 .

[14]  G. Goluzin Geometric theory of functions of a complex variable , 1969 .

[15]  S. Warschawski On conformal mapping of infinite strips , 1942 .

[16]  R. A. Hunt,et al.  Weighted norm inequalities for the conjugate function and Hilbert transform , 1973 .

[17]  A. Gol’dberg,et al.  Asymptotic behavior of the logarithmic derivative of an entire function of completely regular growth , 1978 .

[18]  Yurii Lyubarskii,et al.  Lectures on entire functions , 1996 .

[19]  N. Wiener,et al.  Fourier Transforms in the Complex Domain , 1934 .

[20]  J. R. Higgins,et al.  Five short stories about the cardinal series , 1985 .