The Ubiquitous Appearance of BUPUs

While partitions of unity are a standard tool in differential geometry or the theory of PDEs, they seem to play a minor role in the context of Harmonic Analysis. In the present note we give a survey of the use of the so-called BUPUs, i.e., Bounded Uniform Partitions of Unity, demonstrating that they form a universal tool for many branches of Fourier Analysis, including engineering applications. Although all these tools can be established in the context of LCA (locally compact Abelian) groups, we present the results here in the context of the Euclidean spaces \({{{\mathbb R}^d}}\) . BUPUs are also in many ways a crucial tool for an alternative approach to Fourier Analysis developed by the author, avoiding the classical pathway via Lebesgue integration and Abstract Harmonic Analysis, usually starting with the classical theory of Fourier series and then going on to discuss the Fourier transform which appears to be naturally defined on the Banach convolution algebra \(\big ( {{{\boldsymbol L}^1} \negthinspace ({\mathbb R}^d)}, \, \|{\,\cdot \,}\|_1 \big )\) . The approach given here is based on the Segal algebra \(\big ( {{{\boldsymbol S}_{\negthinspace 0}}({\mathbb R}^d)}, \|{\,\cdot \,}\|_{{\boldsymbol S}_{\negthinspace 0}} \big )\) (known as Feichtinger’s algebra), and the dual space \(({{{\boldsymbol S}_{\negthinspace 0}^{\prime }}({\mathbb R}^d)} , \|{\,\cdot \,} \|_{{{\boldsymbol S}_{\negthinspace 0}^{\prime }}} ) \) , whose elements are called mild distributions. In this way it is possible to develop most of the applications of Fourier Analysis relevant for engineering applications, but also for abstract Harmonic Analysis from scratch, using only elementary functional analytic tools. The goal of this chapter is to point out the crucial role played by BUPUs in this context. Details are given in a series of papers (many of which are cited in the current text).