Uniform and non-uniform sampling of bandlimited functions at minimal density with a few additional samples
Abstract
Abstract A family of sampling theorems for the reconstruction of bandlimited functions from their samples is presented. Taking one or more additional samples is shown to yield more rapidly convergent series with lower truncation errors, as well as to facilitate the reconstruction of bandlimited functions with polynomial growth on the real line. The theorems apply to both uniform sampling and a large class of non-uniform sampling sets, i.e., complete interpolating sequences for the Paley–Wiener space. A number of examples and numerical illustrations accompany the general theory. These include uniform sampling, uniform sampling with finitely many points moved, periodic non-uniform sampling, and sampling at the zeros of Bessel functions $$J_\nu (\pi x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>J</mml:mi><mml:mi>ν</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>π</mml:mi><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> for non-integer $$v > -1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>v</mml:mi><mml:mo>></mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> .