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.
Highlights
The theory of generalized functions [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25], an overview is given in Figure 1
To overcome this we use a theorem of Laurent Schwartz (Lemma 1) which states that convolution products exist if one of the two factors is summable (∈ OC 0 and F (OC 0) ), and equivalently, multiplication products exist if one of the two factors is differentiable (∈ O M and F (O M) )
In 2015, we used it to prove that Poisson’s summation formula (PSF) holds within the generalized functions setting under exactly the same conditions [57] (Lemma 2 below), and in 2018 we used it to prove conditions such that the PSF nested into itself is true [59]
Summary
The theory of generalized functions [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25], an overview is given in Figure 1 (cf. Section 2.3), is known for its generality and for its strictness not to allow arbitrary multiplication and arbitrary convolution among generalized functions [26,27,28,29,30,31,32,33,34] and, as a result of this, other theories have already been proposed to evade these difficulties. A major barrier is the fact that tempered distributions cannot be multiplied or convolved with arbitrary other tempered distributions To overcome this we use a theorem of Laurent Schwartz (Lemma 1) which states that convolution products exist if one of the two factors is (finitely) summable (∈ OC 0 ), and equivalently, multiplication products exist if one of the two factors is (infinitely) differentiable (∈ O M ). It has, to the best of our knowledge, never been shown so far that discretizations and periodizations can be neutralized within the setting of generalized functions.
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