Landau's necessary density conditions for the Hankel transform

We will prove an analogue of Landau's necessary conditions [Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967).] for spaces of functions whose Hankel transform is supported in a measurable subset S of the positive semi-axis. As a special case, necessary density conditions for the existence of Fourier-Bessel frames are obtained.

[1]  I. Daubechies.  Ten Lectures on Wavelets , 1992 .

[2]  S. Bergman.  The kernel function and conformal mapping , 1950 .

[3]  Holger Rauhut,et al.  Radial Multiresolution in Dimension Three , 2003 .

[4]  Marcinkiewicz–Zygmund inequalities and interpolation by spherical harmonics , 2006, math/0611503.

[5]  Michael D. Rawn,et al.  On nonuniform sampling expansions using entire interpolating functions, and on the stability of Bessel-type sampling expansions , 1989, IEEE Trans. Inf. Theory.

[6]  F. Riesz.  FUNCTIONAL ANALYSIS , 2020, Systems Engineering Principles and Practice.

[7]  Karlheinz Gröchenig,et al.  On Landau's Necessary Density Conditions for Sampling and Interpolation of Band-Limited Functions , 1996 .

[8]  Craig A. Tracy,et al.  Mathematical Physics © Springer-Verlag 1994 Level Spacing Distributions and the Bessel Kernel , 1993 .

[9]  H. Pollard,et al.  Complete Sets of Bessel and Legendre Functions , 1947 .

[10]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[11]  H. Landau.  Sampling, data transmission, and the Nyquist rate , 1967 .

[12]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

[13]  J. Ramanathan,et al.  Incompleteness of Sparse Coherent States , 1995 .

[14]  Holger Rauhut.  Time-Frequency and Wavelet Analysis of Functions with Symmetry Properties , 2005 .

[15]  J. R. Higgins.  An Interpolation Series Associated with the Bessel–Hankel Transform , 1972 .

[16]  H. Landau.  Necessary density conditions for sampling and interpolation of certain entire functions , 1967 .

[17]  K. Stempak,et al.  Transplantation and multiplier theorems for Fourier-Bessel expansions , 2006 .

[18]  K. Seip.  Reproducing formulas and double orthogonality in Bargmann and Bergman spaces , 1991 .

[19]  L. Roncal,et al.  Littlewood–Paley–Stein gk-functions for Fourier–Bessel expansions , 2010 .

[20]  Kristian Seip,et al.  On Fourier frames , 2000 .

[21]  Óscar Ciaurri,et al.  A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform , 2000 .

[22]  J. L. Varona,et al.  Fourier series of functions whose Hankel transform is supported on [0, 1] , 1994 .

[23]  K. Grōchenig,et al.  Landau's necessary density conditions for LCA groups , 2008, 0803.3529.

[24]  J. L. Varona,et al.  Mean and Weak Convergence of Fourier-Bessel Series , 1993 .