On Inverses of the Dirac Comb

Abstract

We determine tempered distributions which convolved with a Dirac comb yield unity and tempered distributions, which multiplied with a Dirac comb, yield a Dirac delta. Solutions of these equations have numerous applications. They allow the reversal of discretizations and periodizations applied to tempered distributions. One of the difficulties is the fact that Dirac combs cannot be multiplied or convolved with arbitrary functions or distributions. We use a theorem of Laurent Schwartz to overcome this difficulty and variants of Lighthill’s unitary functions to solve these equations. The theorem we prove states that double-sided (time/frequency) smooth partitions of unity are required to neutralize discretizations and periodizations on tempered distributions.

On the Reversibility of Discretization

“…Here, we continue a series of studies [82][83][84][85] towards a sampling theorem on tempered distributions. The present study is, more precisely, the second half of [85] which is a foundation for this one.…”

“…Here, we continue a series of studies [82][83][84][85] towards a sampling theorem on tempered distributions. The present study is, more precisely, the second half of [85] which is a foundation for this one. Our primary goal is the embedding of the Whittaker-Kotel'nikov-Shannon sampling theorem [95][96][97][98][99][100][101][102][103][104][105] in generalized functions theory, more precisely, its embedding within the space of tempered distributions.…”

“…and the answer is infinite, unless the "scale" T > 0, the smallest measuring unit, is specified [110]. Equivalently, the multiplication product between two Dirac delta functions remains undetermined (Figure 1), unless their relative "position uncertainty" T > 0 is known [85]. In any measurement, some uncertainty T > 0 is present (Figure 2, left) because measuring devices with infinite bandwidth do not exist.…”

“…between discretization ⊥⊥⊥ and two kinds of regularization,ˆ and ∩ , at a respective scale T (see an example in Figure A6), • is the concatenation of operations and id is the identity operation. The reason whyˆ T can be replaced by ∩ T and vice versa is Lemma 5 in [85]. Regularization [11,[13][14][15]17,30,35,49] is well-established today, used for example in [47,69,91,111,[125][126][127][128][129][130].…”

“…Our primary goal is the embedding of the Whittaker-Kotel'nikov-Shannon sampling theorem [95][96][97][98][99][100][101][102][103][104][105] in generalized functions theory, more precisely, its embedding within the space of tempered distributions. Tempered distributions, in turn, lie in the intersection of distributions and ultradistributions, see Figure 1 in [85], and distributions and ultradistributions are particular hyperfunctions [106][107][108][109]. Hyperfunctions arise as a difference of pairs of holomorphic functions [26,40] and "generalized functions" are nothing else than "Randwerte", boundary values, of these pairwise arising holomorphic functions [3,4,8].…”