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  • Publication
    Establishing cut-points for physical activity classification using triaxial accelerometer in middle-aged recreational marathoners
    (2018-08-29) Hernando, Carlos; Hernando Fuster, Carla; Collado, Eladio Joaquin; Panizo, Nayara; Martinez Navarro, Ignacio; Hernando, Barbara
    The purpose of this study was to establish GENEA (Gravity Estimator of Normal Everyday Activity) cut-points for discriminating between six relative-intensity activity levels in middle-aged recreational marathoners. Nighty-eight (83 males and 15 females) recreational marathoners, aged 30-45 years, completed a cardiopulmonary exercise test running on a treadmill while wearing a GENEA accelerometer on their non-dominant wrist. The breath-by-breath- O-2 data was also collected for criterion measure of physical activity categories (sedentary, light, moderate, vigorous, very vigorous and extremely vigorous). GENEA cut-points for physical activity classification was performed via Receiver Operating Characteristic (ROC) analysis. Spearman's correlation test was applied to determine the relationship between estimated and measured intensity classifications. Statistical analysis were done for all individuals, and separating samples by sex. The GENEA cut-points established were able to distinguish between all six-relative intensity levels with an excellent classification accuracy (area under the ROC curve (AUC) values between 0.886 and 0.973) for all samples. When samples were separated by sex, AUC values were 0.881-0.973 and 0.924-0.968 for males and females, respectively. The total variance in energy expenditure explained by GENEA accelerometer data was 78.50% for all samples, 78.14% for males, and 83.17% for females. In conclusion, the wrist-worn GENEA accelerometer presents a high capacity of classifying the intensity of physical activity in middle-aged recreational marathoners when examining all samples together, as well as when sample set was separated by sex. This study suggests that the triaxial GENEA accelerometers (worn on the non-dominant wrist) can be used to predict energy expenditure for running activities.
  • Publication
    Modeling Sampling in Tensor Products of Unitary Invariant Subspaces
    (Hindawi, 2016-10-01) García García, Antonio; Ibort Latre, Luis Alberto; Muñoz-Bouzo, María José; Ministerio de Economía y Competitividad (España)
    The use of unitary invariant subspaces of a Hilbert space H is nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces of L-2( R) and also periodic extensions of finite signals are remarkable examples where this occurs. As a consequence, the availability of an abstract unitary sampling theory becomes a useful tool to handle these problems. In this paper we derive a sampling theory for tensor products of unitary invariant subspaces. This allows merging the cases of finitely/infinitely generated unitary invariant subspaces formerly studied in the mathematical literature; it also allows introducing the several variables case. As the involved samples are identified as frame coefficients in suitable tensor product spaces, the relevant mathematical technique is that of frame theory, involving both finite/infinite dimensional cases.
  • Publication
    A geometric construction of isospectral magnetic graphs
    (Springer, 2023-07-01) Fabila Carrasco, John Stewart; Lledó Macau, Fernando; Post, Olaf
    We present a geometrical construction of families of finite isospectral graphs labelled by different partitions of a natural number r of given length s (the number of summands). Isospectrality here refers to the discrete magnetic Laplacian with normalised weights (including standard weights). The construction begins with an arbitrary finite graph GG with normalised weight and magnetic potential as a building block from which we construct, in a first step, a family of so-called frame graphs (FFa)a∈N . A frame graph FFa is constructed contracting a copies of G along a subset of vertices V0 . In a second step, for any partition A=(a1,…,as) of length s of a natural number r (i.e., r=a1+⋯+as ) we construct a new graph FFA contracting now the frames FFa1,…,FFas selected by A along a proper subset of vertices V1⊂V0 . All the graphs obtained by different s-partitions of r≥4 (for any choice of V0 and V1 ) are isospectral and non-isomorphic. In particular, we obtain increasing finite families of graphs which are isospectral for given r and s for different types of magnetic Laplacians including the standard Laplacian, the signless standard Laplacian, certain kinds of signed Laplacians and, also, for the (unbounded) Kirchhoff Laplacian of the underlying equilateral metric graph. The spectrum of the isospectral graphs is determined by the spectrum of the Laplacian of the building block G and the spectrum for the Laplacian with Dirichlet conditions on the set of vertices V0 and V1 with multiplicities determined by the numbers r and s of the partition.
  • Publication
    Descriptions of Relativistic Dynamics with World Line Condition
    (MDPI, 2019-10-19) Ciaglia, Florio Maria; Di Cosmo, Fabio; Ibort Latre, Luis Alberto; Marmo, Giuseppe; Comunidad de Madrid; Ministerio de Economía y Competitividad (España)
    In this paper, a generalized form of relativistic dynamics is presented. A realization of the Poincaré algebra is provided in terms of vector fields on the tangent bundle of a simultaneity surface in R4 . The construction of this realization is explicitly shown to clarify the role of the commutation relations of the Poincaré algebra versus their description in terms of Poisson brackets in the no-interaction theorem. Moreover, a geometrical analysis of the "eleventh generator" formalism introduced by Sudarshan and Mukunda is outlined, this formalism being at the basis of many proposals which evaded the no-interaction theorem.
  • Publication
    On the Structure of Finite Groupoids and Their Representations
    (MDPI, 2019-03-20) Ibort Latre, Luis Alberto; Rodriguez, Miguel A.; Comunidad de Madrid; Ministerio de Economía y Competitividad (España)
    In this paper, both the structure and the theory of representations of finite groupoids are discussed. A finite connected groupoid turns out to be an extension of the groupoids of pairs of its set of units by its canonical totally disconnected isotropy subgroupoid. An extension of Maschke's theorem for groups is proved showing that the algebra of a finite groupoid is semisimple and all finite-dimensional linear representations of finite groupoids are completely reducible. The theory of characters for finite-dimensional representations of finite groupoids is developed and it is shown that irreducible representations of the groupoid are in one-to-one correspondence with irreducible representation of its isotropy groups, with an extension of Burnside's theorem describing the decomposition of the regular representation of a finite groupoid. Some simple examples illustrating these results are exhibited with emphasis on the groupoids interpretation of Schwinger's description of quantum mechanical systems.
  • Publication
    Using accelerometry for evaluating energy consumption and running intensity distribution throughout a marathon according to sex
    (MDPI, 2020-09-01) Hernando, Carlos; Hernando Fuster, Carla; Martinez Navarro, Ignacio; Collado Boira, Eladio; Panizo, Nayara; Hernando, Barbara
    The proportion of females participating in long-distance races has been increasing in the last years. Although it is well-known that there are differences in how females and males face a marathon, higher research may be done to fully understand the intrinsic and extrinsic factors affecting sex differences in endurance performance. In this work, we used triaxial accelerometer devices to monitor 74 males and 14 females, aged 30 to 45 years, who finished the Valencia Marathon in 2016. Moreover, marathon split times were provided by organizers. Several physiological traits and training habits were collected from each participant. Then, we evaluated several accelerometry- and pace-estimated parameters (pacing, average change of speed, energy consumption, oxygen uptake, running intensity distribution and running economy) in female and male amateur runners. In general, our results showed that females maintained a more stable pacing and ran at less demanding intensity throughout the marathon, limiting the decay of running pace in the last part of the race. In fact, females ran at 4.5% faster pace than males in the last kilometers. Besides, their running economy was higher than males (consumed nearly 19% less relative energy per distance) in the last section of the marathon. Our results may reflect well-known sex differences in physiology (i.e., muscle strength, fat metabolism, VO2max), and in running strategy approach (i.e., females run at a more conservative intensity level in the first part of the marathon compared to males). The use of accelerometer devices allows coaches and scientific community to constantly monitor a runner throughout the marathon, as well as during training sessions.
  • Publication
    Manifolds of classical probability distributions and quantum density operators in infinite dimensions
    (Springer, 2019-12-01) Ciaglia, F. M.; Ibort Latre, Luis Alberto; Jost, J.; Marmo, Giuseppe; Agencia Estatal de Investigación (España); Ministerio de Economía y Competitividad (España)
    The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of C∗-algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given C∗-algebraA which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states S of a possibly infinitedimensional, unital C∗-algebra A is partitioned into the disjoint union of the orbits of an action of the group G of invertible elements of A . Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space H are smooth, homogeneous Banach manifolds of G = GL(H), and, when A admits a faithful tracial state τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through τ is a smooth, homogeneous Banach manifold for G .
  • Publication
    Estimation of energy consumed by middle-aged recreational marathoners during a marathon using accelerometry-based devices
    (Springer Nature, 2020-01-30) Hernando, Carlos; Hernando Fuster, Carla; Martinez-Navarro, Ignacio; Collado-Boira, Eladio; Panizo, Nayara; Hernando, Barbara
    As long-distance races have substantially increased in popularity over the last few years, the improvement of training programs has become a matter of concern to runners, coaches and health professionals. Triaxial accelerometers have been proposed as a one of the most accurate tools to evaluate physical activity during free-living conditions. In this study, eighty-eight recreational marathon runners, aged 30-45 years, completed a marathon wearing a GENEActiv accelerometer on their non-dominant wrist. Energy consumed by each runner during the marathon was estimated based on both running speed and accelerometer output data, by applying the previously established GENEActiv cut-points for discriminating the six relative-intensity activity levels. Since accelerometry allowed to perform an individualized estimation of energy consumption, higher interpersonal differences in the number of calories consumed by a runner were observed after applying the accelerometry-based approach as compared to the speed-based method. Therefore, pacing analyses should include information of effort intensity distribution in order to adjust race pacing appropriately to achieve the marathon goal time. Several biomechanical and physiological parameters (maximum oxygen uptake, energy cost of running and running economy) were also inferred from accelerometer output data, which is of great value for coaches and doctors.
  • Publication
    The sky invariant: A new conformal invariant for Schwarzschild spacetime
    (World Scientific Publishing, 2022-09-30) Lafuente, A.; Ibort Latre, Luis Alberto; Lafuente, J.; Comunidad de Madrid; Ministerio de Economía y Competitividad (España)
    A new class of conformal invariants for a given spacetime M is introduced exploiting the conformal geometry of any light ray . Each congruence of light rays passing through a given point p defines the sky S(p) of such point. The new conformal invariants are defined on the bundle of skies of the spacetime M, being called sky invariants accordingly. The natural conformal covariant derivative defined on a light ray and its associated covariant calculus allows us to show the existence of a natural conformal invariant differential of arc that, together with the restriction of the curvature of the conformal covariant derivative, can be used to construct a sky invariant that will be called the sky curvature. An algorithm, that can be implemented on any symbolic manipulation software system, to compute the sky curvature will be discussed and the main ideas and the explicit computation of the sky curvature are illustrated in Schwarzschild spacetime.
  • Publication
    Structural backward stability in rational eigenvalue problems solved via block Kronecker linearizations
    (Springer, 2023-03) Martínez Dopico, Froilán César; Quintana Ponce, María del Carmen; Van Dooren, Paul; Comunidad de Madrid; Ministerio de Economía y Competitividad (España); Ministerio de Ciencia e Innovación (España); Universidad Carlos III de Madrid
    In this paper we study the backward stability of running a backward stable eigenstructure solver on a pencil S(λ) that is a strong linearization of a rational matrix R(λ) expressed in the form R(λ)=D(λ)+C(λIℓ−A)−1B, where D(λ) is a polynomial matrix and C(λIℓ−A)−1B is a minimal state-space realization. We consider the family of block Kronecker linearizations of R(λ), which have the following structure [...] where the blocks have some specific structures. Backward stable eigenstructure solvers, such as the QZ or the staircase algorithms, applied to S(λ) will compute the exact eigenstructure of a perturbed pencil Sˆ(λ):=S(λ)+ΔS(λ) and the special structure of S(λ) will be lost, including the zero blocks below the anti-diagonal. In order to link this perturbed pencil with a nearby rational matrix, we construct in this paper a strictly equivalent pencil S˜(λ)=(I−X)Sˆ(λ)(I−Y) that restores the original structure, and hence is a block Kronecker linearization of a perturbed rational matrix R˜(λ)=D˜(λ)+C˜(λIℓ−A˜)−1B˜, where D˜(λ) is a polynomial matrix with the same degree as D(λ). Moreover, we bound appropriate norms of D˜(λ)−D(λ), C˜−C, A˜−A and B˜−B in terms of an appropriate norm of ΔS(λ). These bounds may be, in general, inadmissibly large, but we also introduce a scaling that allows us to make them satisfactorily tiny, by making the matrices appearing in both S(λ) and R(λ) have norms bounded by 1. Thus, for this scaled representation, we prove that the staircase and the QZ algorithms compute the exact eigenstructure of a rational matrix R˜(λ) that can be expressed in exactly the same form as R(λ) with the parameters defining the representation very near to those of R(λ). This shows that this approach is backward stable in a structured sense. Several numerical experiments confirm the obtained backward stability results.
  • Publication
    The space of light rays: Causality and L-boundary
    (Springer, 2022-06) Bautista, A.; Ibort Latre, Luis Alberto; Lafuente, J.
    The space of light rays N of a conformal Lorentz manifold (M,C) is, under some topological conditions, a manifold whose basic elements are unparametrized null geodesics. This manifold N, strongly inspired on R. Penrose’s twistor theory, keeps all information of M and it could be used as a space complementing the spacetime model. In the present review, the geometry and related structures of N, such as the space of skies Σ and the contact structure H, are introduced. The causal structure of M is characterized as part of the geometry of N. A new causal boundary for spacetimes M prompted by R. Low, the L-boundary, is constructed in the case of 3–dimensional manifolds M and proposed as a model of its construction for general dimension. Its definition only depends on the geometry of N and not on the geometry of the spacetime M. The properties satisfied by the L–boundary ∂M permit to characterize the obtained extension M = M ∪ ∂ M and this characterization is also proposed for general dimension.
  • Publication
    Quantum tomography and Schwinger's picture of quantum mechanics
    (IOP Science, 2022-07-08) Ciaglia, Florio Maria; Di Cosmo, Fabio; Ibort Latre, Luis Alberto; Marmo, G.; Comunidad de Madrid; European Commission; Ministerio de Economía y Competitividad (España); Universidad Carlos III de Madrid
    In this paper the problem of tomographic reconstruction of states is investigated within the so-called Schwingers picture of quantum mechanics in which a groupoid is associated with every quantum system. The attention is focussed on spin tomography: in this context the groupoid of interest is the groupoid of pairs over a finite set. In a nutshell, this groupoid is made up of transitions between all possible pairs of outcomes belonging to a finite set. In addition, these transitions possess a partial composition rule, generalizing the notion of groups. The main goal of the paper consists in providing a reconstruction formula for states on the groupoid-algebra associated with the observables of the system. Using the group of bisections of this groupoid, which are special subsets in one-to-one correspondence with the outcomes, a frame is defined and it is used to prove the validity of the tomographic reconstruction. The special case of the set of outcomes being the set of integers modulo n, with n odd prime, is considered in detail. In this case the subgroup of discrete affine linear transformations, whose graphs are linear subspaces of the groupoid, provides a quorum in close analogy with the continuous case.
  • Publication
    Quasi-triangularization of matrix polynomials over arbitrary fields
    (Elsevier, 2023-05-15) Anguas Marquez, Luis Miguel; Martínez Dopico, Froilán César; Hollister, R.; Mackey, D. S.; Ministerio de Economía y Competitividad (España); Ministerio de Ciencia e Innovación (España)
    In [19], Taslaman, Tisseur, and Zaballa show that any regular matrix polynomial P (λ)over an algebraically closed field is spectrally equivalent to a triangular matrix polynomial of the same degree. When P (λ)is real and regular, they also show that there is a real quasi-triangular matrix polynomial of the same degree that is spectrally equivalent to P (λ), in which the diagonal blocks are of size at most 2 × 2. This paper generalizes these results to regular matrix polynomials P (λ)over arbitrary fields F, showing that any such P (λ) can be quasi-triangularized to a spectrally equivalent matrix polynomial over F of the same degree, in which the largest diagonal block size is bounded by the highest degree appearing among all of the F-irreducible factors in the Smith form for P (λ).
  • Publication
    On bundles of matrix pencils under strict equivalence
    (Elsevier, 2023-02-01) Terán Vergara, Fernando de; Martínez Dopico, Froilán César; Comunidad de Madrid; Ministerio de Ciencia e Innovación (España); Universidad Carlos III de Madrid
    Bundles of matrix pencils (under strict equivalence) are sets of pencils having the same Kronecker canonical form, up to the eigenvalues (namely, they are an infinite union of orbits under strict equivalence). The notion of bundle for matrix pencils was introduced in the 1990's, following the same notion for matrices under similarity, introduced by Arnold in 1971, and it has been extensively used since then. Despite the amount of literature devoted to describing the topology of bundles of matrix pencils, some relevant questions remain still open in this context. For example, the following two: (a) provide a characterization for the inclusion relation between the closures (in the standard topology) of bundles; and (b) are the bundles open in their closure? The main goal of this paper is providing an explicit answer to these two questions. In order to get this answer, we also review and/or formalize some notions and results already existing in the literature. We also prove that bundles of matrices under similarity, as well as bundles of matrix polynomials (defined as the set of m x n matrix polynomials of the same grade having the same spectral information, up to the eigenvalues) are open in their closure.
  • Publication
    Strongly minimal self-conjugate linearizations for polynomial and rational matrices
    (SIAM, 2022-09) Martínez Dopico, Froilán César; Quintana Ponce, María del Carmen; Van Dooren, Paul; Comunidad de Madrid; Ministerio de Economía y Competitividad (España); Ministerio de Ciencia e Innovación (España); Universidad Carlos III de Madrid
    We prove that we can always construct strongly minimal linearizations of an arbitrary rational matrix from its Laurent expansion around the point at infinity, which happens to be the case for polynomial matrices expressed in the monomial basis. If the rational matrix has a particular self-conjugate structure, we show how to construct strongly minimal linearizations that preserve it. The structures that are considered are the Hermitian and skew-Hermitian rational matrices with respect to the real line, and the para-Hermitian and para-skew-Hermitian matrices with respect to the imaginary axis. We pay special attention to the construction of strongly minimal linearizations for the particular case of structured polynomial matrices. The proposed constructions lead to efficient numerical algorithms for constructing strongly minimal linearizations. The fact that they are valid for any rational matrix is an improvement on any other previous approach for constructing other classes of structure preserving linearizations, which are not valid for any structured rational or polynomial matrix. The use of the recent concept of strongly minimal linearization is the key for getting such generality. Strongly minimal linearizations are Rosenbrock's polynomial system matrices of the given rational matrix, but with a quadruple of linear polynomial matrices (i.e., pencils): L(λ):=[A(λ)C(λ)−B(λ)D(λ)], where A(λ) is regular, and the pencils [A(λ)−B(λ)] and [A(λ)C(λ)] have no finite or infinite eigenvalues. Strongly minimal linearizations contain the complete information about the zeros, poles, and minimal indices of the rational matrix and allow one to very easily recover its eigenvectors and minimal bases. Thus, they can be combined with algorithms for the generalized eigenvalue problem for computing the complete spectral information of the rational matrix.
  • Publication
    Diagonal scalings for the eigenstructure of arbitrary pencils
    (SIAM, 2022-09) Martínez Dopico, Froilán César; Quintana Ponce, María del Carmen; Van Dooren, Paul; Comunidad de Madrid; Ministerio de Economía y Competitividad (España); Ministerio de Ciencia e Innovación (España); Universidad Carlos III de Madrid
    In this paper we show how to construct diagonal scalings for arbitrary matrix pencils λB−A , in which both A and B are complex matrices (square or nonsquare). The goal of such diagonal scalings is to “balance” in some sense the row and column norms of the pencil. We see that the problem of scaling a matrix pencil is equivalent to the problem of scaling the row and column sums of a particular nonnegative matrix. However, it is known that there exist square and nonsquare nonnegative matrices that cannot be scaled arbitrarily. To address this issue, we consider an approximate embedded problem, in which the corresponding nonnegative matrix is square and can always be scaled. The new scaling methods are then based on the Sinkhorn--Knopp algorithm for scaling a square nonnegative matrix with total support to be doubly stochastic or on a variant of it. In addition, using results of U. G. Rothblum and H. Schneider [Linear Algebra Appl., 114--115 (1989), pp. 737--764], we give simple sufficient conditions on the zero pattern for the existence of diagonal scalings of square nonnegative matrices to have any prescribed common vector for the row and column sums. We illustrate numerically that the new scaling techniques for pencils improve the accuracy of the computation of their eigenvalues.
  • Publication
    Renal function recovery strategies following marathon in amateur runners
    (Frontiers Media, 2022-02-28) Hernando, Carlos; Hernando Fuster, Carla; Panizo, Nayara; Collado Boira, Eladio; Folch Ayora, Ana; Martinez Navarro, Ignacio; Hernando, Barbara
    Long distance races have a physiological impact on runners. Up to now, studies analyzing these physiological repercussions have been mainly focused on muscle and cardiac damage, as well as on its recovery. Therefore, a limited number of studies have been done to explore acute kidney failure and recovery after performing extreme exercises. Here, we monitored renal function in 76 marathon finishers (14 females) from the day before participating in a marathon until 192 h after crossing the finish line (FL). Renal function was evaluated by measuring serum creatinine (sCr) and the glomerular filtration rate (GFR). We randomly grouped our cohort into three intervention groups to compare three different strategies for marathon recovery: total rest (REST), continuous running at their ventilatory threshold 1 (VT1) intensity (RUN), and elliptical workout at their VT1 intensity (ELLIPTICAL). Interventions in the RUN and ELLIPTICAL groups were performed at 48, 96, and 144 h after marathon running. Seven blood samples (at the day before the marathon, at the FL, and at 24, 48, 96, 144, and 192 h post-marathon) and three urine samples (at the day before the marathon, at the finish line, and at 48 h post-marathon) were collected per participant. Both heart rate monitors and triaxial accelerometers were used to control the intensity effort during both the marathon race and the recovery period. Contrary to our expectations, the use of elliptical machines for marathon recovery delays renal function recovery. Specifically, the ELLIPTICAL group showed a significantly lower ∆GFR compared to both the RUN group (p = 4.5 × 10−4) and the REST group (p = 0.003). Hence, we encourage runners to carry out an active recovery based on light-intensity continuous running from 48 h after finishing the marathon. In addition, full resting seems to be a better strategy than performing elliptical workouts.
  • Publication
    Generalized sampling: from shift-invariant to U-invariant spaces
    (World Scientific Publishing, 2015-05-01) Fernández Morales, Héctor Raúl; García García, Antonio; Hernández Medina, Miguel Ángel; Muñoz-Bouzo, María José; Ministerio de Economía y Competitividad (España); Ministerio de Ciencia e Innovación (España)
    The aim of this article is to derive a sampling theory in U-invariant subspaces of a separable Hilbert space ℋ where U denotes a unitary operator defined on ℋ. To this end, we use some special dual frames for L2(0, 1), and the fact that any U-invariant subspace with stable generator is the image of L2(0, 1) by means of a bounded invertible operator. The used mathematical technique mimics some previous sampling work for shift-invariant subspaces of L2(ℝ). Thus, sampling frame expansions in U-invariant spaces are obtained. In order to generalize convolution systems and deal with the time-jitter error in this new setting we consider a continuous group of unitary operators which includes the operator U.
  • Publication
    The Kramer sampling theorem revisited
    (Springer, 2014-10-01) García García, Antonio; Hernández-Medina, M. A.; Muñoz-Bouzo, María José; Ministerio de Ciencia e Innovación (España)
    The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. Besides, it has been the cornerstone for a significant mathematical literature on the topic of sampling theorems associated with differential and difference problems. In this work we provide, in an unified way, new and old generalizations of this result corresponding to various different settings; all these generalizations are illustrated with examples. All the different situations along the paper share a basic approach: the functions to be sampled are obtaining by duality in a separable Hilbert space through an -valued kernel K defined on an appropriate domain.
  • Publication
    Riesz bases associated with regular representations of semidirect product groups
    (Springer Science and Business Media LLC, 2019-12) García García, Antonio; Pérez Villalón, Gerardo; Ministerio de Economía y Competitividad (España)
    This work is devoted to the study of Bessel and Riesz systems of the type Lγ f γ ∈ obtained from the action of the left regular representation Lγ of a discrete non abelian group which is a semidirect product, on a function f ∈ 2(). The main features about these systems can be conveniently studied by means of a simple matrix-valued function F(ξ ). These systems allow to derive sampling results in principal -invariant spaces, i.e., spaces obtained from the action of the group on a element of a Hilbert space. Since the systems Lγ f γ ∈ are closely related to convolution operators, a connection with C∗-algebras is also established.