On the Duality of Regular and Local Functions

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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 21. 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

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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

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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

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′ 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 ......

Figure 1. The regularization of generalized function f yields regular function ∩ ϕf.

86 Regular functions are functions in the ordinary functions sense which are inﬁnitely differentiable

87 in the ordinary functions sense, a function property that is of immense value in many branches of ′ 88 mathematics. Regular functions belong to OM because S ∗ S ⊂ OM , see e.g. [4,10,18]. They maintain

89 the being ’tempered’ property, i.e., they do not grow faster than polynomials, which is common to all

90 tempered distributions but add the regularity of ϕ to f. It follows that regularized functions can always

91 be sampled according to Lemma 1.

92 Regularizations are treated in many mathematical textbooks [2,5,12,14,15,21] and scientiﬁc papers

93 [1,24–27]. They are also known in terms of ”regularizers” [1,25–27], ”smooth cutoff functions” [2,28]

94 and ”molliﬁers” [16,29–33], a term that goes back (see [16], p.63) to K.O. Friedrichs [29]. Regularized

95 rect functions (characteristic functions of an interval) are known as ”mesa function”, ”tapered box” [12] ∞ 96 or ”tapered characteristic function” and ”taper function” [34] or as ”C bell” or ”smoothed top hat”

97 function in [35]. Mostly, regularizations are required ”to obtain regularized interpolating kernels” such

98 as in [27]. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2017 doi:10.20944/preprints201705.0175.v1

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99 Away from the generalized functions literature, we furthermore encounter regularizations in terms of

100 ”smoothings”, ”interpolations”, ”zero-paddings” or ”approximations” because they are not only applied

101 to generalized functions, they are also applied to ordinary functions, usually to obtain better ”regularity”

102 properties for functions, i.e. better differentiability. Regularity is also a topic discussed in [36], for

103 example. It is closely related to localization.

Deﬁnition 6 (Localization). Let ϕ ∈ S. Then for any tempered distribution f ∈ S′ we deﬁne another tempered distribution by

⊓ ϕf := ϕ · f (6)

′ ′ 104 which is a generalized function of rapid descent in OC ⊂ S . The operation ⊓ ϕ is called localization ′ ′ 105 or restriction of f by means of ϕ. It is a linear continuous operation ⊓ ϕ : S → OC , f 7→ ⊓ ϕf. If ′ 106 ϕ ∈ D ⊂ S, it is also called ﬁnitization. The result of ⊓ ϕ is called local function of f in S .

f ⊓ ϕf

⊓ ✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻ −−→ϕ ✻✻✻ ...... ✻ ✻ ✻ ✻ ✻ ✻ 107

Figure 2. The localization of generalized function f yields local function ⊓ ϕf.

′ ′ ′ 108 Local functions belong to OC because S · S ⊂ OC [4,14]. They add the ’rapid descent’ property ′ 109 of Schwartz functions ϕ ∈ S to f ∈ S . It follows that localized functions can always be periodized

110 according to Lemma 2.

111 The term ”local” and the treatment of localizations have a long history in mathematics. It culminated,

112 however, in the term ”localization operator”. It appears 1988 for the ﬁrst time (see [37], p.133 in

113 [38]) in Daubechies’ article [39] and later in Daubechies’ 1992 standard textbook [36]. Meanwhile,

114 ”localizations” occur in many textbooks [2,36,38–47], amongst others as ”localized trigonometric

115 functions” or ”localized sine basis” [36,41,48], as ”localized frames” [49], ”local trigonometric bases”,

116 as ”local representations” [9] or simply in terms of ”locally integrable” functions.

117 5. Calculation Rules

118 ”One of the basic principles in classical Fourier analysis is the impossibility to ﬁnd a function f being

119 arbitrarily well localized together with its Fourier transform Ff”[50]. This, in particular, can easily be

120 seen if one tries to localize the function that is constantly 1.

Lemma 7 (Localization Balance). Let ϕ ∈ S and let ϕˆ := Fϕ. Then

′ F( ∩ ϕδ) = ⊓ ϕˆ1 ∈ OC and (7) ′ F( ⊓ ϕˆ1) = ∩ ϕδ ∈ OM in S . (8)

121 In (8) we see that by localizing 1, we delocalize δ, i.e., 1 and its Fourier transform δ cannot be both

122 arbitrarily well localized. This phenomenon is known as Heisenberg’s uncertainty principle [9,10,12,

123 50–52]. Vice versa, in (7) we see that by regularizing δ we increasingly deregularize 1. The entity ∩ ϕδ Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2017 doi:10.20944/preprints201705.0175.v1

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124 is also known as an ”approximate identity” of δ, usually denoted as δǫ where ǫ is a parameter describing ′ 125 the proximity to δ (see e.g. [14] p.316, p.401 or [21] p.5). Convolving any f ∈ S with δǫ, it creates an

126 approximate identity fǫ of f which is a function in the ordinary sense being inﬁnitely differentiable.

′ ′ Proof. According to (4O), δ ∈ C can be convolved with ϕ ∈ S ⊂ S and, equivalently, 1 ∈ OM can be multiplied with ψ ∈ S ⊂ S′, hence

F( ∩ ϕδ) = F(ϕ ∗ δ) = Fϕ ·Fδ =ϕ ˆ · 1 = ⊓ ϕˆ1

′ 127 holds in S . The second formula is shown in an analogous manner. It is moreover interesting to observe that in analogy to the Dirac comb identity [19]

△△△δ ≡ III ≡ ⊥⊥⊥1

the following identity, let’s say a ”localization balance”

∩ ϕδ ≡ Ω ≡ ⊓ ϕ1

′ 128 holds a balance in S if ϕ ≡ ϕˆ is satisﬁed for ϕ ∈ S, which obviously is the best achievable compromise −πt2 129 in localizing 1 and thereby delocalizing δ. It is true for the Gaussian Ω(t) ≡ e and herewith expresses t −t 130 Hardy’s uncertainty principle [53]. But it is also true for the Hyperbolic secant Ω(t) ≡ 2/(e + e ), see

131 e.g. [12], and for every fourth Hermite function H, i.e., all H satisfying FH ≡ H. A connection between

132 Gaussians and Hyperbolic Secants is that both belong to a class of ”Polya´ frequency functions” [54,55].

133 Gaussians, Hyperbolic Secants and Hermite functions are treated in [56,57] for example. Hyperbolic

134 Secants may also replace Gaussians in Gabor systems, see e.g. Janssen and Strohmer [58]. A link

135 between Gaussians and Hyperbolic Secants is furthermore known in soliton physics where the ”initial

136 Gaussian beam reshapes to a squared hyperbolic secant proﬁle” [59]. Studying ﬁxpoints Ω of the Fourier

137 transform in S is therefore worthwhile goal.

138 Another calculation rule we need to prove the theorem below is the following. It holds in analogy to

139 already shown properties of discretizations and periodizations [19].

′ ′ ′ Lemma 8. Let ϕ ∈ S, α ∈ OM , g ∈ OC and f ∈ S . Then αf and g∗f exist in S and

′ α · ( ⊓ ϕf) = ⊓ ϕ(αf) = ( ⊓ ϕα) · f ∈ OC and (9) ′ g ∗ ( ∩ ϕf) = ∩ ϕ(g∗f) = ( ∩ ϕg) ∗ f ∈ OM in S . (10)

′ Proof. We may allow that at most one of the operands in ϕ ∗ g ∗ f is no element in OC . This is indeed ′ ′ ′ ′ true as ϕ ∈ S ⊂ OC , g ∈ OC and f is an arbitrary element in S . It follows that ϕ ∗ g ∗ f exists in S ′ ′ ′ and, hence, operands may be interchanged arbitrarily. Using OC ∗ S ⊂ S twice and (5), we obtain

g ∗ ( ∩ ϕf) = g ∗ (ϕ ∗ f) = ϕ ∗ g ∗ f = ∩ ϕ(g ∗ f)

′ 140 in S . The other half of this equation results from the fact that the roles of f and g can be exchanged due

141 to commutativity. The second formula is then shown in an analogous manner. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2017 doi:10.20944/preprints201705.0175.v1

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1 426. A Regularization-Localization Duality

143 The interaction between regularizations and localizations is ubiquitous in the literature today, for

144 example as ”regularization” and multiplication with smooth ”cutoff functions” in Hormander¨ [2], as ′ 145 ”two components of the approximation procedure” in S , see Strichartz [6], or as ”approximation by

146 cutting and regularizing” in Treves` [15], p.302, or in terms of ”cutting out” one period of f and

147 applying ”(quasi-)interpolation” [47]. Detailed studies of the interaction of both, regularizations and

148 localizations, can be found for example in [34,37,38,60] and in engineering literature, we encounter

149 these interactions in terms of the interplay between ”windowing” on one hand and ”interpolation” on the

150 other. Another equivalent is the so-called ”zero-padding” technique found in engineering textbooks as a

151 way to implement interpolations. It corresponds to the regularization of a discrete function by embedding

152 it into a higher-dimensional space where it is smooth.

153 However, we may summarize this regularization-localization duality in the following way.

Theorem 9 (Regularization vs. Localization). Let ϕ ∈ S, f ∈ S′ and let ϕˆ := Fϕ. Then

′ F( ∩ ϕf) = ⊓ ϕˆ(Ff) ∈ OC and (11) ′ F( ⊓ ϕˆf) = ∩ ϕ(Ff) ∈ OM in S . (12)

154 So, this duality asserts that regularizing a function means to localize its Fourier transform and, vice

155 versa, localizing a function means to regularize its Fourier transform. It is the one-to-one counterpart of ′ 156 a discretization-periodization duality in S , given in (1) and (2). Proof. Formally, according to the calculation rules shown above the following equalities hold

F( ∩ ϕf) = F ∩ ϕ(δ∗f) = F( ∩ ϕδ ∗ f) = F( ∩ ϕδ) ·Ff

= ⊓ ϕˆ1 ·Ff = ⊓ ϕˆ(1·Ff) = ⊓ ϕˆ(Ff)

′ ′ ′ 157 in S . We start using f = δ ∗ f where δ ∈ OC ⊂ S is the identity element with respect to convolution ′ ′ 158 in S . Then we apply Equations (10), (4), (7) and (9), in this order. Finally, with Ff = g ∈ S we use ′ ′ 159 g = 1 · g where 1 ∈ OM ⊂ S is the identity element with respect to multiplication in S . The second

160 formula is now shown in an analogous manner.

F◦ ∩ = F reg = ⊓ ◦ F ✻✻✻ ...... −−−−−−−−−−−−−→ ✻ ✻ F of regular functions ✻ ✻ ✻ ✻

F◦ ⊓ = Floc = ∩ ◦ F ✻✻✻ −−−−−−−−−−−−→ ✻ ✻ F of local functions ...... ✻ ✻ ✻ ✻

161 Figure 3. The Regularization-Localization Theorem.

162 An immediate consequence of the theorem is that f and its Fourier transform Ff cannot be both

163 arbitrarily well localized, a fact that is known as Heisenberg’s uncertainty principle. Also note that Floc , Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2017 doi:10.20944/preprints201705.0175.v1

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164 see ﬁgure above, is the Short-Time Fourier Transform (STFT) with window function ϕ ∈ S and it is the

165 Gabor transform if ϕ is a Gaussian. Consequently, the result of Gabor transforms will be smooth, i.e.,

166 they cannot be discrete for example. Its Fourier dual, the Fourier transform of regular functions F reg in

167 contrast to that, see ﬁgure above, corresponds to ﬁrst regularizing functions before Fourier transforming

168 them. Consequently, the result of such transforms will be local, i.e., they cannot be periodic for example.

169 Obviously, by looking at these interactions, one may think of discrete functions as the ’opposite’

170 of regular functions and, equivalently, one may think of periodic functions as the ’opposite’ of local

171 functions. This is examined more closely in the next section.

172 7. Four Subspaces

′ 173 Let ∁ OM be the complement of regular functions OM in S . It is the space of all ordinary or ′ 174 generalized functions in S which are not inﬁnitely differentiable in the ordinary functions sense. Let, ′ ′ ′ 175 furthermore, ∁ OC be the complement of local functions OC in S . It consists of all ordinary or ′ 176 generalized functions in S which do either not fade to zero as |t| increases (periodic functions for

177 example) or they fall to zero but too slowly (with polynomial decay rather than with exponential decay). ′ 178 Then, the following diagram holds in S .

△△△

′ ❄ ′

OM ∩ OC ✻ OM ∩ ∁ OC ⊓ ϕ ⊥⊥⊥ ∩

✻ ϕ ✻ ϕ ∩ ⊥⊥⊥ ❄ ❄

⊓ ϕ

′ ✻ ′

∁ OM ∩ OC ❄ ∁ OM ∩ ∁ OC △△△

′ Figure 4. Four subspaces in S , linked via operations ⊥⊥⊥, △△△, ∩ ϕ, ⊓ ϕˆ.

local global functions functions

△✲△△ ✛

regular periodic functions ⊓ ϕˆ ⊥⊥⊥ ϕ ∩ ∩ ✻ ϕ ✻

⊥⊥⊥ ❄ ❄

⊓ ✛ϕˆ ✲ discrete discrete functions

generalized △△△ periodic

Figure 5. Same as above, drawn in another fashion. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2017 doi:10.20944/preprints201705.0175.v1

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′ 179 Apparently, the Schwartz space S ≡ OM ∩ OC , the ”smooth world”, in some sense, is the ’opposite’ ′ 180 of ∁ OM ∩ ∁ OC , the ”discrete world”. One may also note that no additional information is used yet

181 beside pure operator deﬁnitions. There is also no statement yet on the reversibility of our operations ⊥⊥⊥ ′ 182 and △△△ and ∩ ϕ and ⊓ ϕ in S . Such inversions will be treated in a follow-on study.

183 8. Conclusions and Outlook

′ 184 It is shown that in analogy to a discretization-periodization duality in S there is also a ′ 185 regularization-localization duality in S . Proving these dualities even follows the same pattern. In

186 addition, the two dualities are inverses of each other in the sense that the ﬁrst one maps towards

187 discreteness and the latter one maps towards smoothness. A more detailed statement on the reversibility ′ 188 of discreteness in S will be given in a next study.

189 Acknowledgements

190 This paper is primarily based on studies conducted in the years 1995-1997 at the Institute of

191 Mathematics, Ludwig Maximilians University (LMU), Munich. The author is in particular very grateful

192 to Professor Otto Forster. The author would also like to thank his colleagues at the Microwaves and

193 Radar Institute, German Aerospace Center (DLR), for deep insights into signal processing and a great

194 cooperation for many years.

195 Conﬂicts of Interest

196 The author declares no conﬂicts of interest.

197 References

198 1. Wei, G.W.; Gu, Y. Conjugate ﬁlter approach for solving Burgers equation. Journal of

199 Computational and Applied mathematics 2002, 149, 439–456.

200 2. Hormander,¨ L. The analysis of linear partial differential operators I; Die Grundlehren der

201 mathematischen Wissenschaften, Springer, 1983.

202 3. Lighthill, M.J. An Introduction to Fourier Analysis and Generalised Functions; Cambridge

203 University Press, 1958.

204 4. Schwartz, L. Theorie´ des Distributions, Tome II; Hermann Paris, France, 1959.

205 5. Zemanian, A. Distribution Theory And Transform Analysis - An Introduction To Generalized

206 Functions, With Applications; McGraw-Hill, 1965.

207 6. Strichartz, R.S. Mock Fourier Series and Transforms Associated with Certain Cantor Measures.

208 Journal dAnalyse Mathematique´ 2000, 81, 209–238.

209 7. Woodward, P.M. Probability and Information Theory, with Applications to Radar; Pergamon

210 Press, 1953.

211 8. Benedetto, J.J. Harmonic Analysis and Applications; Vol. 23, CRC Press, 1996.

212 9. Feichtinger, H.G.; Strohmer, T. Gabor Analysis and Algorithms: Theory and Applications;

213 Springer, 1998. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2017 doi:10.20944/preprints201705.0175.v1

10 of 12

214 10. Gasquet, C.; Witomski, P. Fourier Analysis and Applications: Filtering, Numerical Computation,

215 Wavelets; Vol. 30, Springer Science & Business Media, 1999.

216 11. Brandwood, D. Fourier Transforms in Radar and Signal Processing; Artech House, 2003.

217 12. Kammler, D.W. A ﬁrst course in Fourier analysis; Cambridge University Press, 2007.

218 13. Bracewell, R.N. Fourier Transform and its Applications; McGraw-Hill Education, 1986.

219 14. Horvath,´ J. Topological vector spaces and distributions; Addison-Wesley Publishing Company,

220 1966.

221 15. Treves,` F. Topological Vector Spaces, Distributions and Kernels: Pure and Applied Mathematics;

222 Vol. 25, Dover Publications Inc, Mineola, New York, 1967.

223 16. Grubb, G. Distributions and Operators; Vol. 252, Springer Science & Business Media, 2008.

224 17. Gracia-Bond´ıa, J.M.; Varilly, J.C. Algebras of Distributions Suitable for Phase-Space Quantum

225 Mechanics. I. Journal of Mathematical Physics 1988, 29, 869–879.

226 18. Dubois-Violette, M.; Kriegl, A.; Maeda, Y.; Michor, P.W. Smooth*-Algebras. arXiv preprint

227 math/0106150 2001.

228 19. Fischer, J.V. On the duality of discrete and periodic functions. Mathematics 2015, 3, 299–318.

229 20. Nguyen, H.Q.; Unser, M.; Ward, J.P. Generalized Poisson Summation Formulas for Continuous

230 Functions of Polynomial Growth. Journal of Fourier Analysis and Applications 2016, pp. 1–20.

231 21. Walter, W. Einfuhrung¨ in die Theorie der Distributionen; BI-Wissenschaftsverlag,

232 Bibliographisches Institut & FA Brockhaus, 1994.

233 22. Hunter, J.K.; Nachtergaele, B. Applied Analysis; World Scientiﬁc, 2001.

234 23. Beals, R. Advanced mathematical analysis: periodic functions and distributions, complex

235 analysis, Laplace transform and applications; Vol. 12, Springer Science & Business Media,

236 2013.

237 24. Estrada, R.; Kanwal, R.P. Regularization, pseudofunction, and Hadamard ﬁnite part. Journal of

238 mathematical analysis and applications 1989, 141, 195–207.

239 25. Wei, G. Wavelets generated by using discrete singular convolution kernels. Journal of Physics

240 A: Mathematical and General 2000, 33, 8577.

241 26. Wei, G. Discrete singular convolution for the solution of the Fokker–Planck equation. The

242 Journal of chemical physics 1999, 110, 8930–8942.

243 27. Wei, G. Quasi wavelets and quasi interpolating wavelets. Chemical Physics Letters 1998,

244 296, 215–222.

245 28. Strichartz, R.S. A Guide to Distribution Theorie and Fourier Transforms; World Scientiﬁc, 2003.

246 29. Friedrichs, K.O. On the differentiability of the solutions of linear elliptic differential equations.

247 Communications on Pure and Applied Mathematics 1953, 6, 299–326.

248 30. Schechter, M. Modern Methods in Partial Differential Equations, An Introduction, 1977.

249 31. Yosida, K. Functional analysis. 123, 1980.

250 32. Gaffney, M.P. A special Stokes’s theorem for complete Riemannian manifolds. Annals of

251 Mathematics 1954, pp. 140–145.

252 33. Ni, L.; Markenscoff, X. The self-force and effective mass of a generally accelerating dislocation

253 I: Screw dislocation. Journal of the Mechanics and Physics of Solids 2008, 56, 1348–1379. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2017 doi:10.20944/preprints201705.0175.v1

11 of 12

254 34. Ashino, R.; Desjardins, J.S.; Heil, C.; Nagase, M.; Vaillancourt, R. Pseudo-differential operators,

255 microlocal analysis and image restoration. In Advances in Pseudo-Differential Operators;

256 Springer, 2004; pp. 187–202. ∞ 257 35. Boyd, J.P. Asymptotic Fourier Coefﬁcients for a C Bell (Smoothed-Top-Hat) & the Fourier

258 Extension Problem. Journal of Scientiﬁc Computing 2006, 29, 1–24.

259 36. Daubechies, I. Ten Lectures on Wavelets; Vol. 61, SIAM, 1992.

260 37. Cordero, E.; Tabacco, A. Localization operators via time-frequency analysis. In Advances in

261 Pseudo-Differential Operators; Springer, 2004; pp. 131–147.

262 38. Ashino, R.; Boggiatto, P.; Wong, M.W. Advances in pseudo-differential operators; Vol. 155,

263 Birkhauser,¨ 2012.

264 39. Daubechies, I. Time-frequency localization operators: a geometric phase space approach. IEEE

265 Transactions on Information Theory 1988, 34, 605–612.

266 40. Oberguggenberger, M.B. Multiplication of distributions and applications to partial differential

267 equations; Vol. 259, Longman Scientiﬁc & Technical, Harlow, Essex, U.K., 1992.

268 41. Coifman, R.R.; Wickerhauser, M.V. Entropy-based algorithms for best basis selection. IEEE

269 Transactions on information theory 1992, 38, 713–718. s s 270 42. Triebel, H. A localization property for B pq and F pq spaces. Studia Mathematica 1994,

271 109, 183–195.

272 43. Feichtinger, H.G.; Grochenig,¨ K. Gabor frames and time-frequency analysis of distributions.

273 journal of functional analysis 1997, 146, 464–495.

274 44. Skrzypczak, L. Heat and Harmonic Extensions for Function Spaces of Hardy–Sobolev–Besov

275 Type on Symmetric Spaces and Lie Groups. Journal of approximation theory 1999, 96, 149–170.

276 45. Feichtinger, H.G. Modulation spaces: looking back and ahead. Sampling Theory in Signal and

277 Image Processing 2006, 5, 109.

278 46. Nguyen, H.Q.; Unser, M. A sampling theory for non-decaying signals. Applied and

279 Computational Harmonic Analysis 2015.

280 47. Feichtinger, H.G. Thoughts on Numerical and Conceptual Harmonic Analysis. In New Trends in

281 Applied Harmonic Analysis; Springer, 2016; pp. 301–329.

282 48. Daubechies, I. The Wavelet Transform, Time-Frequency Localization and Signal Analysis.

283 Information Theory, IEEE Transactions on 1990, 36, 961–1005.

284 49. Grochenig,¨ K. Localization of frames, Banach frames, and the invertibility of the frame operator.

285 Journal of Fourier Analysis and Applications 2004, 10, 105–132.

286 50. Wilczok, E. New uncertainty principles for the continuous Gabor transform and the continuous

287 wavelet transform. Documenta Mathematica 2000, 5, 201–226.

288 51. Mallat, S. A Wavelet Tour of Signal Processing; Academic press, 1999.

289 52. Higgins, J.R. Sampling theory in Fourier and signal analysis: foundations; Oxford University

290 Press on Demand, 1996.

291 53. Grochenig,¨ K.; Zimmermann, G. Hardy’s theorem and the short-time Fourier transform of

292 Schwartz functions. Journal of the London Mathematical Society 2001, 63, 205–214.

293 54. Schoenberg, I.J. On totally positive functions, Laplace integrals and entire functions of the

294 Laguerre-Polya-Schur´ type. Proceedings of the National Academy of Sciences 1947, 33, 11–17. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2017 doi:10.20944/preprints201705.0175.v1

12 of 12

295 55. Schoenberg, I. On variation-diminishing integral operators of the convolution type. Proceedings

296 of the National Academy of Sciences 1948, 34, 164–169.

297 56. Boyd, J.P. Asymptotic coefﬁcients of Hermite function series. Journal of Computational Physics

298 1984, 54, 382–410.

299 57. Boyd, J.P. Chebyshev and Fourier spectral methods; Courier Corporation, 2001.

300 58. Janssen, A.; Strohmer, T. Hyperbolic secants yield Gabor frames. Applied and Computational

301 Harmonic Analysis 2002, 12, 259–267.

302 59. Fazio, E.; Renzi, F.; Rinaldi, R.; Bertolotti, M.; Chauvet, M.; Ramadan, W.; Petris, A.; Vlad, V.

303 Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides.

304 Applied physics letters 2004, 85, 2193–2195.

305 60. Boggiatto, P. Localization operators with L p symbols on modulation spaces. In Advances in

306 Pseudo-differential Operators; Springer, 2004; pp. 149–163.