We survey various mathematical aspects of the uncertainty principle, including Heisenberg’s inequality and its variants, local uncertainty inequalities, logarithmic uncertainty inequalities, results relating to Wigner distributions, qualitative uncertainty principles, theorems on approximate concentration, and decompositions of phase space.

The Uncertainty Principle: A Mathematical Survey

We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere or, alternatively, of strictly spacelimited functions that are optimally concentrated in the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology, and numerical analysis. The spherical Slepian functions can be found by solving either an algebraic eigenvalue problem in the spectral domain or a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap, the spatiospectral projection operator commutes with a Sturm--Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small spatial region and a large spherical harmonic bandwidth, the spherical concentration problem reduces to its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.

https://www.jstor.org/stable/pdfplus/20453835.pdf

Spatiospectral Concentration on a Sphere

Polynomials are one of the principal tools of classical numerical analysis. When a function needs to be interpolated, integrated, differentiated, etc, it is assumed to be approximated by a polynomial of a certain fixed order (though the polynomial is almost never constructed explicitly), and a treatment appropriate to such a polynomial is applied. We introduce analogous techniques based on the assumption that the function to be dealt with is band-limited, and use the well developed apparatus of prolate spheroidal wavefunctions to construct quadratures, interpolation and differentiation formulae, etc, for band-limited functions. Since band-limited functions are often encountered in physics, engineering, statistics, etc, the apparatus we introduce appears to be natural in many environments. Our results are illustrated with several numerical examples.

https://iopscience.iop.org/article/10.1088/0266-5611/17/4/315/pdf

Prolate Spheroidal Wave Functions, Quadrature, and Interpolation

It is often advantageous to investigate the relationship between two geophysical data sets in the spectral domain by calculating admittance and coherence functions. While there exist powerful Cartesian windowing techniques to estimate spatially localized (cross-)spectral properties, the inherent sphericity of planetary bodies sometimes necessitates an approach based in spherical coordinates. Direct localized spectral estimates on the sphere can be obtained by tapering, or multiplying the data by a suitable windowing function, and expanding the resultant field in spherical harmonics. The localization of a window in space and its spectral bandlimitation jointly determine the quality of the spatiospectral estimation. Two kinds of axisymmetric windows are here constructed that are ideally suited to this purpose: bandlimited functions that maximize their spatial energy within a cap of angular radius ?0, and spacelimited functions that maximize their spectral power within a spherical harmonic bandwidth L. Both concentration criteria yield an eigenvalue problem that is solved by an orthogonal family of data tapers, and the properties of these windows depend almost entirely upon the spacebandwidth product N0= (L+ 1) ?0/?. The first N0_1 windows are near perfectly concentrated, and the best-concentrated window approaches a lower bound imposed by a spherical uncertainty principle. In order to make robust localized estimates of the admittance and coherence spectra between two fields on the sphere, we propose a method analogous to Cartesian multitaper spectral analysis that uses our optimally concentrated data tapers. We show that the expectation of localized (cross-)power spectra calculated using our data tapers is nearly unbiased for stochastic processes when the input spectrum is white and when averages are made over all possible realizations of the random variables. In physical situations, only one realization of such a process will be available, but in this case, a weighted average of the spectra obtained using multiple data tapers well approximates the expected spectrum. While developed primarily to solve problems in planetary science, our method has applications in all areas of science that investigate spatiospectral relationships between data fields defined on a sphere.

Localized spectral analysis on the sphere

The estimation of potential fields such as the gravitational or magnetic potential at the surface of a spherical planet from noisy observations taken at an altitude over an incomplete portion of the globe is a classic example of an ill-posed inverse problem. We show that this potential-field estimation problem has deep-seated connections to Slepian's spatiospectral localization problem which seeks bandlimited spherical functions whose energy is optimally concentrated in some closed portion of the unit sphere. This allows us to formulate an alternative solution to the traditional damped least-squares spherical harmonic approach in geodesy, whereby the source field is now expanded in a truncated Slepian function basis set. We discuss the relative performance of both methods with regard to standard statistical measures such as bias, variance and mean squared error, and pay special attention to the algorithmic efficiency of computing the Slepian functions on the region complementary to the axisymmetric polar gap characteristic of satellite surveys. The ease, speed, and accuracy of our method make the use of spherical Slepian functions in earth and planetary geodesy practical.

/pdf/spherical-slepian-functions-and-the-polar-gap-in-geodesy-3pf1jsgkm9.pdf

Spherical Slepian functions and the polar gap in geodesy

Slepian, Landau and Pollak found that a certain finite convolution integral operator on the real line commutes with a much simpler second order differential operator. This opens the way to a detailed analysis of the space of “time and band limited functions” which has found applications in several fields. Here we exhibit some exact analogues of these commutation results with the real line replaced by either a non-Abelian group or a symmetric space. The sphere may be the most natural example for the applications.

F. A. Grünbaum

Papers

Differential Operators Commuting with Finite Convolution Integral Operators: Some Nonabelian Examples