Return to Answer

When asking this question, in principle one should make a distinction between two cases:

1. When $n=1$, the space $PW_{S}$ is a space of entire functions of exponential type, with all of their unique properties.
2. The case $n\geq2$ is much more involved and as far as I know is not fully understood, some results were obtained by Olevskii and Ulanovskii ("On multi-dimensional sampling and interpolation").

A proof to the fact that a uniformly discrete set forms a Bessel sequence can be found in thm 17, on chapter 2 in Young's book "An Introduction to Non-Harmonic Fourier Series" in the case $n=1$. In the general case, one can extend a result by Ingham (thm 2, "Some trigonometrical inequalities with applications to the theory of series") together with the fact that the inequality $$\sum_{\lambda\in\Lambda}|f(\lambda)|^2 \leq C||f||^2_{L^2(\mathbb R^n)}$$ holds for all $f\in PW_S$ is equivalent to having the inequality $$||\sum_{\lambda\in\Lambda}c_\lambda e^{i\lambda t}||^2_{L^2(S)}\leq C\sum_{\lambda\in\Lambda}|c_\lambda|^2$$ hold for all finite sequence of coefficients $\{c_\lambda\}$.
The fact that a sampling set is always relatively dense follows from Landau's necessary condition (appears in "Necessary density conditions for sampling and interpolation of certain entire functions"), and applies to both cases.

When asking this question, in principle one should make a distinction between two cases:

1. When $n=1$, the space $PW_{S}$ is a space of entire functions of exponential type, with all of their unique properties.

2. The case $n\geq2$ is much more involved and as far as I know is not fully understood, some results were obtained by Olevskii and Ulanovskii ("On multi-dimensional sampling and interpolation").

   A proof to the fact that a uniformly discrete set forms a Bessel sequence can be found in thm 17, on chapter 2 in Young's book "An Introduction to Non-Harmonic Fourier Series" in the case $n=1$. In the general case, one can extend a result by Ingham (thm 2, "Some trigonometrical inequalities with applications to the theory of series") together with the fact that the inequality $$\sum_{\lambda\in\Lambda}|f(\lambda)|^2 \leq C||f||^2_{L^2(\mathbb R^n)}$$ holds for all $f\in PW_S$ is equivalent to having the inequality $$||\sum_{\lambda\in\Lambda}c_\lambda e^{i\lambda t}||^2_{L^2(S)}\leq C\sum_{\lambda\in\Lambda}|c_\lambda|^2$$ hold for all finite sequence of coefficients $\{c_\lambda\}$.
   The fact that a sampling set is always relatively dense follows from Landau's necessary condition (appears in "Necessary density conditions for sampling and interpolation of certain entire functions"), and applies to both cases.

When asking this question, in principle one should make a distinction between two cases:

1. When $n=1$, the space $PW_{S}$ is a space of entire functions of exponential type, with all of their unique properties.
2. The case $n\geq2$ is much more involved and as far as I know is not fully understood, some results were obtained by Olevskii and Ulanovskii ("On multi-dimensional sampling and interpolation").

A proof to the fact that a uniformly discrete set forms a Bessel sequence can be found in thm 17, on chapter 2 in Young's book "An Introduction to Non-Harmonic Fourier Series" in the case $n=1$. In the general case, one can extend a result by Ingham (thm 2, "Some trigonometrical inequalities with applications to the theory of series") together with the fact that the inequality $$\sum_{\lambda\in\Lambda}|f(\lambda)|^2 \leq C||f||^2_{L^2(S)}$$ holds for all $f\in PW_S$ is equivalent to having the inequality $$||\sum_{\lambda\in\Lambda}c_\lambda e^{i\lambda t}||^2_{L^2(S)}\leq C\sum_{\lambda\in\Lambda}|c_\lambda|^2$$ hold for all finite sequence of coefficients $\{c_\lambda\}$.
The fact that a sampling set is always relatively dense follows from Landau's necessary condition (appears in "Necessary density conditions for sampling and interpolation of certain entire functions"), and applies to both cases.

When asking this question, in principle one should make a distinction between two cases:

1. When $n=1$, the space $PW_{S}$ is a space of entire functions of exponential type, with all of their unique properties.
2. The case $n\geq2$ is much more involved and as far as I know is not fully understood, some results were obtained by Olevskii and Ulanovskii ("On multi-dimensional sampling and interpolation").

A proof to the fact that a uniformly discrete set forms a Bessel sequence can be found in thm 17, on chapter 2 in Young's book "An Introduction to Non-Harmonic Fourier Series" in the case $n=1$. In the general case, one can extend a result by Ingham (thm 2, "Some trigonometrical inequalities with applications to the theory of series") together with the fact that the inequality $$\sum_{\lambda\in\Lambda}|f(\lambda)|^2 \leq C||f||^2_{L^2(\mathbb R^n)}$$ holds for all $f\in PW_S$ is equivalent to having the inequality $$||\sum_{\lambda\in\Lambda}c_\lambda e^{i\lambda t}||^2_{L^2(S)}\leq C\sum_{\lambda\in\Lambda}|c_\lambda|^2$$ hold for all finite sequence of coefficients $\{c_\lambda\}$.
The fact that a sampling set is always relatively dense follows from Landau's necessary condition (appears in "Necessary density conditions for sampling and interpolation of certain entire functions"), and applies to both cases.

When asking this question, in principle one should make a distinction between two cases:

1. When $n=1$, the space $PW_{S}$ is a space of entire functions of exponential type, with all of their unique properties.
2. The case $n\geq2$ is much more involved and as far as I know is not fully understood, some results were obtained by Olevskii and Ulanovskii ("On multi-dimensional sampling and interpolation").

A proof to the fact that a uniformly discrete set forms a Bessel sequence can be found in thm 17, on chapter 2 in Young's book "An Introduction to Non-Harmonic Fourier Series" (the case $n=1$).
The fact that a sampling set is always relatively dense follows from Landau's necessary condition (appears in "Necessary density conditions for sampling and interpolation of certain entire functions"), and applies to both cases.

When asking this question, in principle one should make a distinction between two cases:

1. When $n=1$, the space $PW_{S}$ is a space of entire functions of exponential type, with all of their unique properties.
2. The case $n\geq2$ is much more involved and as far as I know is not fully understood, some results were obtained by Olevskii and Ulanovskii ("On multi-dimensional sampling and interpolation").

A proof to the fact that a uniformly discrete set forms a Bessel sequence can be found in thm 17, on chapter 2 in Young's book "An Introduction to Non-Harmonic Fourier Series" in the case $n=1$. In the general case, one can extend a result by Ingham (thm 2, "Some trigonometrical inequalities with applications to the theory of series") together with the fact that the inequality $$\sum_{\lambda\in\Lambda}|f(\lambda)|^2 \leq C||f||^2_{L^2(S)}$$ holds for all $f\in PW_S$ is equivalent to having the inequality $$||\sum_{\lambda\in\Lambda}c_\lambda e^{i\lambda t}||^2_{L^2(S)}\leq C\sum_{\lambda\in\Lambda}|c_\lambda|^2$$ hold for all finite sequence of coefficients $\{c_\lambda\}$.
The fact that a sampling set is always relatively dense follows from Landau's necessary condition (appears in "Necessary density conditions for sampling and interpolation of certain entire functions"), and applies to both cases.