##
**Group theoretical methods in approximation theory, elementary number theory, and computational signal geometry.**
*(English)*
Zbl 0643.22005

Approximation theory V, Proc. 5th Int. Symp., College Station/Tex. 1986, 129-171 (1986).

[For the entire collection see Zbl 0606.00013.]

This is an interesting expository paper which asks “What groups govern the symmetry properties of the finite Fourier transform and cardinal spline interpolation?” It answers the question by: “finite and real Heisenberg groups, respectively.” In the process some applications to the quadratic reciprocity law of elementary number theory, analog radar signal design and laser optics are found. Many concepts from harmonic analysis are sketched quickly and many references are given.

The paper begins with a sketch of some examples of harmonic analysis. It reviews Fourier analysis on the circle or torus, including Gelfand’s proof of the theorem of Wiener which says that if f is a non-vanishing continuous function on the circle with an absolutely convergent Fourier series, then 1/f also has an absolutely convergent Fourier series. This result is used later in the paper. Next one finds a summary of results on Fourier analysis on n-dimensional Euclidean space followed by a definition of the reduced Heisenberg group of a locally compact abelian topological group G. Explicit results from Fourier analysis on the sphere are listed.

The 1st key principle of harmonic analysis on Lie groups is stated as that of replacing the 1-dimensional characters from commutative harmonic analysis with the (matrix entries of) higher dimensional continuous irreducible unitary representations of Lie groups.

The 1st part of the paper ends with applications to periodic multilayers from the theory of optical filters and to a formula for the Jacobi symbol from number theory. Both results involve Chebyshev polynomials of the 2nd kind coming from characters of SU(2,\({\mathbb{C}}).\)

The second part of the paper emphasizes the importance of the Heisenberg group over \({\mathbb{Z}}/n{\mathbb{Z}}\) and over \({\mathbb{R}}\). The Heisenberg group over a ring R is the group of \(3\times 3\) upper triangular matrices over R with ones on the diagonal. The finite Fourier transform is obtained as an intertwining operator between two n-dimensional representations of the finite Heisenberg group. A study is made of the spectrum and trace of the finite Fourier cotransform. The latter is a Gauss sum and thus its evaluation implies the quadratic reciprocity law.

Then a 2nd key principle of harmonic analysis is enunciated as: express classical linear transformations L as intertwining operators and use this interpretation to derive properties of L.

The 3rd part of the paper concerns the real Heisenberg group. The study of this group has been applied to the quadratic reciprocity law of number theory, theta functions, and quantum mechanics. Here the author is concerned with applications to cardinal spline interpolation and computational signal geometry (radar theory and laser optics). In particular it is noted that the Fourier transform on the real line can be realized as an operator intertwining two representations of the real Heisenberg group. The Heisenberg nilmanifold is obtained from the real Heisenberg group by considering the quotient space modulo the integer lattice. The Weil-Brezin map which appears in the Poisson-Weil factorization of the Fourier cotransform is used in various ways in the rest of the paper. For example, the author derives a theorem of Subbotin- Schoenberg concerning the existence and uniqueness of solutions to the problem of cardinal spline interpolation. The same technique leads to a theorem about cardinal spline expansions of functions in the Paley-Wiener space of entire holomorphic functions of exponential type at most \(\pi\) that are square integrable over \({\mathbb{R}}\). This is a sampling theorem which is foundational for digital signal processing, particularly in compact and video disc technology. The paper closes with a review of the theory of the Leray-Maslov-Souriau index and its application to the quadratic reciprocity law. And the Weil-Brezin isomorphism is used to study theta functions.

This is an interesting expository paper which asks “What groups govern the symmetry properties of the finite Fourier transform and cardinal spline interpolation?” It answers the question by: “finite and real Heisenberg groups, respectively.” In the process some applications to the quadratic reciprocity law of elementary number theory, analog radar signal design and laser optics are found. Many concepts from harmonic analysis are sketched quickly and many references are given.

The paper begins with a sketch of some examples of harmonic analysis. It reviews Fourier analysis on the circle or torus, including Gelfand’s proof of the theorem of Wiener which says that if f is a non-vanishing continuous function on the circle with an absolutely convergent Fourier series, then 1/f also has an absolutely convergent Fourier series. This result is used later in the paper. Next one finds a summary of results on Fourier analysis on n-dimensional Euclidean space followed by a definition of the reduced Heisenberg group of a locally compact abelian topological group G. Explicit results from Fourier analysis on the sphere are listed.

The 1st key principle of harmonic analysis on Lie groups is stated as that of replacing the 1-dimensional characters from commutative harmonic analysis with the (matrix entries of) higher dimensional continuous irreducible unitary representations of Lie groups.

The 1st part of the paper ends with applications to periodic multilayers from the theory of optical filters and to a formula for the Jacobi symbol from number theory. Both results involve Chebyshev polynomials of the 2nd kind coming from characters of SU(2,\({\mathbb{C}}).\)

The second part of the paper emphasizes the importance of the Heisenberg group over \({\mathbb{Z}}/n{\mathbb{Z}}\) and over \({\mathbb{R}}\). The Heisenberg group over a ring R is the group of \(3\times 3\) upper triangular matrices over R with ones on the diagonal. The finite Fourier transform is obtained as an intertwining operator between two n-dimensional representations of the finite Heisenberg group. A study is made of the spectrum and trace of the finite Fourier cotransform. The latter is a Gauss sum and thus its evaluation implies the quadratic reciprocity law.

Then a 2nd key principle of harmonic analysis is enunciated as: express classical linear transformations L as intertwining operators and use this interpretation to derive properties of L.

The 3rd part of the paper concerns the real Heisenberg group. The study of this group has been applied to the quadratic reciprocity law of number theory, theta functions, and quantum mechanics. Here the author is concerned with applications to cardinal spline interpolation and computational signal geometry (radar theory and laser optics). In particular it is noted that the Fourier transform on the real line can be realized as an operator intertwining two representations of the real Heisenberg group. The Heisenberg nilmanifold is obtained from the real Heisenberg group by considering the quotient space modulo the integer lattice. The Weil-Brezin map which appears in the Poisson-Weil factorization of the Fourier cotransform is used in various ways in the rest of the paper. For example, the author derives a theorem of Subbotin- Schoenberg concerning the existence and uniqueness of solutions to the problem of cardinal spline interpolation. The same technique leads to a theorem about cardinal spline expansions of functions in the Paley-Wiener space of entire holomorphic functions of exponential type at most \(\pi\) that are square integrable over \({\mathbb{R}}\). This is a sampling theorem which is foundational for digital signal processing, particularly in compact and video disc technology. The paper closes with a review of the theory of the Leray-Maslov-Souriau index and its application to the quadratic reciprocity law. And the Weil-Brezin isomorphism is used to study theta functions.

Reviewer: A.A.Terras

### MSC:

22E30 | Analysis on real and complex Lie groups |

43A80 | Analysis on other specific Lie groups |

78A60 | Lasers, masers, optical bistability, nonlinear optics |

11A15 | Power residues, reciprocity |

41A15 | Spline approximation |

22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |