The Poisson summation formula (PSF), which relates the sampling of an analog signal with the periodization of its Fourier transform, plays a key role in the classical sampling theory. In its current forms, the formula is only applicable to a limited class of signals in . However, this assumption on the signals is too strict for many applications in signal processing that require sampling of non-decaying signals. In this paper we generalize the PSF for functions living in weighted Sobolev spaces that do not impose any decay on the functions. The only requirement is that the signal to be sampled and its weak derivatives up to order 1∕2 + ε for arbitrarily small ε > 0, grow slower than a polynomial in the sense. The generalized PSF will be interpreted in the language of distributions.
Rakesh Chawla, Andrea Rizzi, Matthias Finger, Federica Legger, Matteo Galli, Sun Hee Kim, Jian Zhao, João Miguel das Neves Duarte, Tagir Aushev, Hua Zhang, Alexis Kalogeropoulos, Yixing Chen, Tian Cheng, Ioannis Papadopoulos, Gabriele Grosso, Valérie Scheurer, Meng Xiao, Qian Wang, Michele Bianco, Varun Sharma, Joao Varela, Sourav Sen, Ashish Sharma, Seungkyu Ha, David Vannerom, Csaba Hajdu, Sanjeev Kumar, Sebastiana Gianì, Kun Shi, Abhisek Datta, Siyuan Wang, Anton Petrov, Jian Wang, Yi Zhang, Muhammad Ansar Iqbal, Yong Yang, Xin Sun, Muhammad Ahmad, Donghyun Kim, Matthias Wolf, Anna Mascellani, Paolo Ronchese, , , , , , , , , , , , , , , , , , , , , , , ,
Giovanni Piccioli, Lenka Zdeborová, Emanuele Troiani
François Maréchal, Jonas Schnidrig, Cédric Terrier