Sampling Theory in Shift-Invariant Spaces

Sampling theory in shift-invariant spaces extends classical bandlimited reconstruction to a broad class of function spaces generated by the integer shifts of one or more prototype functions. In this framework, a shift-invariant space is defined as the closure of all finite linear combinations of translates of a fixed generator or a finite set of generators. The objective is to recover any element of such a space from discrete measurements taken at suitably chosen sampling points. Unlike the Shannon–Whittaker theorem, which applies to globally bandlimited signals, modern formulations accommodate irregular sampling, localised reconstruction and mixed-norm settings. Stability and convergence are governed by frame and Riesz basis conditions on the translated generators, ensuring that perturbations in samples lead to controlled errors in the reconstruction. Recent advances have explored function spaces beyond L2, notably mixed-norm Lebesgue spaces and Wiener amalgam spaces, to capture both local and global behaviour. Such generalisations underpin practical algorithms in signal processing, imaging and communications, where generators may be splines, Gaussians or compactly supported functions. Iterative schemes and convolution-based methods enable finite‐interval reconstruction, while probabilistic sampling strategies allow for recovery from randomly selected data points. The interplay between abstract functional‐analytic conditions and concrete numerical algorithms has driven a deeper understanding of aliasing, truncation errors and pointwise estimates, with implications for phase retrieval, compressed sensing and large‐scale data acquisition.

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Recent studies have established generalised sampling and stability theorems in shift-invariant subspaces of Lebesgue and Wiener amalgam spaces with mixed‐norms. By identifying appropriate reproducing kernels and exploiting mixed integrability conditions, these works derive new inequalities that guarantee the convergence of iterative reconstruction algorithms. The resulting framework unifies irregular and average sampling, offering explicit error bounds in Lp and amalgam norms and paving the way for robust numerical schemes.

Investigations into random convolution sampling have shown that signals in multiply generated shift-invariant subspaces of weighted mixed Lebesgue spaces can be approximated by finite-dimensional models with high probability. Under mild conditions on the generators and convolution kernels, large random samples suffice to ensure sampling stability. This probabilistic approach provides a practical pathway to design sampling systems that are resilient to noise and incomplete data, with applications in distributed sensor networks and high‐dimensional signal recovery.

Current work on phase retrieval in Gaussian shift-invariant spaces demonstrates stable reconstruction of complex-valued signals from spectrogram measurements. By formulating a Riesz basis expansion along parallel lines in the time-frequency plane, researchers have derived explicit reconstruction formulas and stability estimates on compact intervals. The corresponding algorithm offers provably convergent and efficient approximation from finitely many spectrogram samples and extends to other function spaces, including Paley–Wiener spaces, highlighting the versatility of shift-invariant sampling methods in time-frequency analysis.

Sampling Theory in Shift-Invariant Spaces publication trend

The graph below shows the total number of articles in sampling theory in shift-invariant spaces across all publications each year (not limited to Nature Index journals).

Technical terms

Shift-invariant space: A function space closed under integer shifts of one or more generating functions, forming the basis for sampling and reconstruction.

Generator: A prototype function whose integer translates span the shift-invariant space and determine sampling properties.

Riesz basis: A stable and complete sequence in a Hilbert space that allows unique, well-conditioned expansions and reconstructions.

Frame: An overcomplete system of functions that provides redundancy and robustness for signal representation and reconstruction.

Wiener amalgam space: A function space combining local and global norms to measure both fine‐scale and large‐scale behaviour of signals.

Gabor transform: A time-frequency representation obtained by applying a windowed Fourier transform, yielding spectrogram measurements.

References

1. On Generalizations of Sampling Theorem and Stability Theorem in Shift-Invariant Subspaces of Lebesgue and Wiener Amalgam Spaces with Mixed-Norms. Symmetry (2021).
2. The random convolution sampling stability in multiply generated shift invariant subspace of weighted mixed Lebesgue space. AIMS Mathematics (2022).
3. Stable Gabor Phase Retrieval in Gaussian Shift-Invariant Spaces via Biorthogonality. Constructive Approximation (2023).