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On the Duality of Regular and Local Functions Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2017 doi:10.20944/preprints201705.0175.v1 mathematics ISSN 2227-7390 www.mdpi.com/journal/mathematics Article On the Duality of Regular and Local Functions Jens V. Fischer Institute of Mathematics, Ludwig Maximilians University Munich, 80333 Munich, Germany; E-Mail: jens.fi[email protected]; Tel.: +49-8196-934918 1 Abstract: In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty 2 principle. They both express Fourier dualities within the space of tempered distributions and 3 these dualities are furthermore the inverses of one another. While Poisson’s summation 4 formula expresses a duality between discretization and periodization, Heisenberg’s 5 uncertainty principle expresses a duality between regularization and localization. We define 6 regularization and localization on generalized functions and show that the Fourier transform 7 of regular functions are local functions and, vice versa, the Fourier transform of local 8 functions are regular functions. 9 Keywords: generalized functions; tempered distributions; regular functions; local functions; 10 regularization-localization duality; regularity; Heisenberg’s uncertainty principle 11 Classification: MSC 42B05, 46F10, 46F12 1 1.2 Introduction 13 Regularization is a popular trick in applied mathematics, see [1] for example. It is the technique 14 ”to approximate functions by more differentiable ones” [2]. Its terminology coincides moreover with 15 the terminology used in generalized function spaces. They contain two kinds of functions, ”regular 16 functions” and ”generalized functions”. While regular functions are being functions in the ordinary 17 functions sense which are infinitely differentiable in the ordinary functions sense, all other functions 18 become ”infinitely differentiable” in the ”generalized functions sense” [3]. In this way, all functions 19 are being infinitely differentiable. Localization, in contrast to that, is another popular technique. It 20 allows for example to integrate functions which could not be integrated otherwise, if we think of ”locally 21 integrable” functions or if we think of the Short-Time Fourier Transform (STFT), capable to analyze 22 infinitely extended signals. Although, regularization and localization appear to be quite different, a © 2017 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2017 doi:10.20944/preprints201705.0175.v1 2 of 12 23 connection between these operations is no surprise. It is ubiquitous in the literature. The theorem 24 below, however, appears in wider sense. It holds within the space of tempered distributions and is 25 directly related to Heisenberg’s uncertainty principle. It is moreover the inverse of an already known 26 discretization-periodization duality. 27 Section 2 provides an introduction to the notations used and previous results. Section 3 presents a 28 justification for Section 4 where regularization and localization within the space of tempered distributions 29 are defined. Section 5 provides symbolic calculation rules based on these definitions, needed to prove 30 the theorem in Section 6. Section 7 connects these results to results in a previous study and Section 8, 31 finally, concludes this study and provides an outlook. 32 2. Preliminaries n n n Let δkT be the Dirac impulse shifted by k ∈ Z units of T ∈ R+ = {t ∈ R : 0 < tν < ∞, 1 6 ν 6 n}, kT being componentwise multiplication, within the space S′ ≡ S′(Rn) of tempered distributions (generalized functions that do not grow faster than polynomials) and let IIIT := X δkT k∈Zn ′ ′ n 33 be the Dirac comb. Then δkT ∈ S and IIIT ∈ S for any T ∈ R+ are tempered distributions [4–6]. ′ 34 We shortly write δ instead of δkT if k = 0. The Fourier transform F in S is defined as usual and such 35 that F1 = δ and Fδ = 1 where 1 is the function being constantly one [3,4,7–12]. The Dirac comb 36 is moreover known for its excellent discretization (sampling) and periodization properties [7,11–13]. ′ ′ 37 While multiplication IIIT · f in S samples a function f ∈ S , the corresponding convolution product ′ 38 IIIT ∗ f periodizes f in S . ′ 39 The following two lemmas summarize the demands that must be put on f ∈ S such that f can ′ 40 be sampled or periodized in S . Recall that smoothness, i.e., infinite differentiability, is not a demand. 41 It is a given fact for all functions in generalized function spaces. Also recall that OM is the space of ′ ′ ′ 42 multiplication operators in S and OC is the space of convolution operators in S according to Laurent 43 Schwartz’ theory of distributions [4,5,14–20]. ′ ′ 44 Lemma 1 (Discretization). Generalized functions f ∈ S can be sampled in S if and only if f ∈ OM . Proof. Any uniform discretization (sampling) in S′ corresponds to forming the product ′ IIIT · f in S ′ ′ 45 where IIIT ∈ S is the Dirac comb. Furthermore, IIIT is no regular function, i.e., IIIT ∈ S \OM . On ′ 46 the other hand, for any multiplication product in S , it is required that at least one of the two factors is in ′ 47 OM . Hence, f ∈ OM ⊂ S . Otherwise the product does not exist. Vice versa, if f ∈ OM then IIIT · f ′ ′ 48 exists due to S ·OM ⊂ S . 49 An equivalent statement is the following lemma. ′ ′ ′ 50 Lemma 2 (Periodization). Generalized functions f ∈ S can be periodized in S if and only if f ∈ OC . Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2017 doi:10.20944/preprints201705.0175.v1 3 of 12 Proof. Any periodization in S′ corresponds to forming the convolution product ′ IIIT ∗ f in S ′ ′ ′ 51 where IIIT ∈ S is the Dirac comb. Furthermore, IIIT is of no rapid descent, i.e., IIIT ∈ S \OC . On ′ 52 the other hand, for any convolution product in S , it is required that at least one of the two factors is in ′ ′ ′ ′ 53 OC . Hence, f ∈ OC ⊂ S . Otherwise the convolution product does not exist. Vice versa, if f ∈ OC ′ ′ ′ 54 then IIIT ∗ f exists due to S ∗ OC ⊂ S . In a previous study [19], we used these insights in order to define operations of discretization ⊥⊥⊥T and ′ ′ periodization △△△T in S . While discretization is an operation ⊥⊥⊥T : OM → S , f 7→ ⊥⊥⊥T f := IIIT · f, ′ ′ periodization is an operation △△△T : OC → S , g 7→ △△△T g := IIIT ∗ g, respectively. Starting from these two definitions we proved that F( ⊥⊥⊥f) = △△△(Ff) and (1) F( △△△g) = ⊥⊥⊥(Fg) (2) ′ 55 hold in S , both being expressions of Poisson’s Summation Formula. We shortly write ⊥⊥⊥ and △△△ 56 instead of ⊥⊥⊥T and △△△T if Tν = 1 for all 1 6 ν 6 n. Recall moreover that these rules are a consequence of the Fourier duality F(α · f) = Fα ∗ Ff and (3) F(g ∗ f) = Fg ·Ff in S′ (4) ′ ′ 57 for any α ∈ OM , g ∈ OC and f ∈ S which is, according to Laurent Schwartz’ theory of generalized 58 functions, the widest possible comprehension of both, multiplication and convolution within the space of ′ ′ 59 tempered distributions [4,14,18]. It lies at the very heart of S . Many calculation rules in S , including 60 Equations (1), (2), (7), (8), (11), (12) and Lemmas 1, 2, 3, 4 rely on it. 61 3. Feasibilities 62 The following two lemmas provide justifications for the way we will define regularizations and ′ ′ 63 localizations in S below. They will allow us to invert discretizations and periodizations in S . ′ 64 Lemma 3 (Regularization). Let ϕ ∈ S. Then for any f ∈ S , ϕ ∗ f can be sampled. ′ 65 Proof. This is a consequence of the fact that S ∗ S ⊂ OM [4,10,14,18] and Lemma 1. 66 An equivalent statement is the following lemma. ′ 67 Lemma 4 (Localization). Let ϕ ∈ S. Then for any f ∈ S , ϕ · f can be periodized. ′ ′ ′ 68 Proof. It follows from the fact that S·S ⊂ OC [4,18], which is the Fourier dual F(S ∗ S ) = F(OM ) ′ 69 of S ∗ S ⊂ OM , and Lemma 2. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2017 doi:10.20944/preprints201705.0175.v1 4 of 12 ′ 70 It is interesting to observe that ϕ ∗ and ϕ · stretch and compress f ∈ S , respectively. This property 71 is moreover independent of the actual choice of ϕ ∈ S. It can therefore be attributed to the operations of 72 convolution and multiplication themselves. 73 4. Definitions 74 ”One of the main applications of convolution is the regularization of a distribution” [14] or the 75 regularization of ordinary functions which are not being infinitely differentiable in the conventional 76 functions sense. Its actual importance lies furthermore in the fact that it is the reversal of discretization. 77 Regularization is usually understood as the approximation of generalized functions via approximate 78 identities [2,8–10,14,21–23]. In this paper, however, we extend this idea by allowing any ϕ ∈ S ′ ′ 79 and by allowing even ordinary functions f ∈ S to be used for regularizations in S . This approach 80 naturally includes the special case of choosing approximate identities without unnecessarily restricting 81 our theorem below. Lemmas 3 and 4 above justify the following two definitions. Definition 5 (Regularization). Let ϕ ∈ S. Then for any tempered distribution f ∈ S′ we define another tempered distribution by ∩ ϕf := ϕ ∗ f (5) ′ 82 which is a regular, slowly growing function in OM ⊂ S . The operation ∩ ϕ is called regularization, 83 approximation, interpolation or smoothing of f by means of ϕ. It is a linear continuous operation ′ ′ 84 ∩ ϕ : S → OM , f 7→ ∩ ϕf. The result of ∩ ϕ is called regular function of f in S . f ∩ ϕf ∩ ✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻ −−→ϕ ... ... ... ... 85 Figure 1. The regularization of generalized function f yields regular function ∩ ϕf.
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