Let us start with the basic definition of a signal. What is a signal? A signal is a function that conveys information about the behaviour or attributes of a phenomenon, which is a function of space, time, or both.
Signals have many uses, such as in medical diagnosis, to analyse electrocardiogram (ECG) signals in order to assess the heart; in communications, to transmit and decode radio signals; and also in speech recognition and image compression.
Signal processing is a field of science and engineering that deals with the analysis, modification, and synthesis of signals.
Now that we know about signals and signal processing, let us come to an important concept in mathematics and engineering, called the sampling theory. Sampling theory is the study of how to accurately reconstruct a continuous signal or function from a set of discrete samples. This theory lays the foundation for converting analog signals, like sound or images, into digital form and vice-versa.
Sampling theory finds use in digital signal processing (DSP), data compression, audio and image processing, and in communications.
In sampling theory, a sample is a small portion of data, taken from a larger population, used to analyse and make inferences about that entire population. It is a data point represented as a vector (a vector is a quantity that has both magnitude and direction) of values corresponding to various attributes or variables.
One of the main phenomena studied in sampling theory, is reconstruction, which is the process of recreating the original signal from its samples. In the reconstruction process, we need information about data at given points. However, in practical applications, we may end up obtaining information only in the neighbourhood points which should be used in the reconstruction algorithm.
The question of whether and how this reconstruction can be done is called the sampling problem.
Sampling and reconstruction methods often assume that the signal lies in what is known as a shift-invariant space (SIS). SIS is a function space that remains unchanged under shifts in its domain.
In a previous work, the sampled data were defined as the inner product of samples of the vector-valued functions with respect to different vectors or directions. This type of sample data can be regarded as linear functional data of vector-valued functions. Such an approach of dealing with functional data is highly relevant in machine learning.
Functional data analysis is attracting a lot of research interest in recent years. Here the data consist of multiple realisations of what is known as Hilbert space-valued random variables. A Hilbert space is a convenient finite dimensional or infinite dimensional vector space used in mathematics
Vast amounts of functional data have been collected from cardiac signals, earthquake measurements, traffic data, healthcare records, etc. The main problem with functional data is functional data classification.
In this study, the authors Mr. Md Hasan Ali Biswas, from the Department of Mathematics, Indian Institute of Science (IISc) Bangalore, Bengaluru, India, Mr. Rohan Joy and Prof. Ramakrishnan Radha from the Department of Mathematics, Indian Institute of Technology (IIT) Madras, Chennai, India, and Prof. Felix Krahmer from the Department of Mathematics, Technical University of Munich, Munich, Germany, have investigated the problem of sampling and reconstruction in principal shift-invariant spaces generated by Hilbert space-valued functions.
The aim of this study was to reconstruct a signal from various samples taken along different directions as perfectly as possible. First, a stable set of sampling was done, and it was proven that a given set was a stable set of sampling. Secondly, a reconstruction formula for a signal was presented from its integer samples. Lastly, the reconstruction of a signal was done in the case of a perturbed and irregular sampling set, and their impact on the reconstruction process was examined.
The findings of this study were not straightforward generalisations, but involved significant theoretical contributions. These results could be used for future research.
Prof. Xiaoping Shen, from Ohio University, Ohio, United States, appreciated the work done by the authors of this paper with the following comments: “The paper titled “Sampling in Shift-Invariant Spaces Generated by Hilbert Space-Valued Functions” (accepted in Mathematical Methods in the Applied Sciences, John Wiley & Sons, 2025) presents a rigorous and well-structured contribution to sampling theory in shift-invariant spaces (SIS) generated by Hilbert space-valued functions. The authors successfully generalize classical SIS frameworks from scalar and vector-valued settings to a more general Hilbert space-valued context. The work is mathematically sound, building on solid theoretical tools such as block Laurent operators, reproducing kernel Hilbert spaces, and Riesz basis theory. The paper provides clear stability criteria, equivalent formulations, and reconstruction results that are both novel and relevant to applications in signal processing and functional data analysis.
This publication, appearing in a reputable international journal published by Wiley and indexed in major databases such as MathSciNet and Scopus, represents a Ph.D.-level contribution to applied mathematics.”
Article by Akshay Anantharaman
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