Frequency-domain modelling is a core tool for the analysis of linear time-invariant structures. In a process that has been unclear, additional Dirac delta distributions can arise in the frequency- domain transfer functions of certain structures, beyond those seemingly given by the structural model—for instance, in the mechanical impedance of a linear spring. Previous analyses have manually append these "hidden deltas" to the relevant transfer functions in order to ensure that they remain causal, but questions remain as to their exact origin, and behaviour in in non-causal models. Here, we demonstrate that these hidden deltas arise from the theory of distributions, and the solution of the distributional division equation. We demonstrate a rigorous and reliable method for deriving these hidden deltas in which the role of causality constraints are made clear. Furthermore, we demonstrate that the appropriate frequency-domain conditions for causality in such systems are generalized—not, classical—Hilbert transform relations, and that the process of appendingdeltadistributionsisrelatedtotheanalysisofcausalityviathesegeneralizedrelations. INTRODUCTION Several decades of research (Titchmarsh 1948; Makris and Efthymiou 2020) have shed sig- nificant light on the relationship between frequency-domain models of structural phenomena, and the causality of these phenomena: their relationship to the directional nature of time, and whether they respect it. The constraints of causality provide insight into the behaviour of viscoelastic 1 Pons, March 22, 2024 constitutivemodels—includinginexactorapproximaterate-independentdampingmodels(Keivan etal.2017;Makris1997a;Pons2023),fractional-ordermodels(EnelundandOlsson1999;Makris and Efthymiou 2020), and power-law media (Gulgowski and Stefański 2021; Kelly and McGough 2009). They also allow the identification of viscoelastic loss moduli based only on storage moduli behaviour (Madsen et al. 2008). Analyses of structural model causality are relevant to seismology (Meza-Fajardo and Lai 2007; Deng and Morozov 2018), seismic analysis (Keivan et al. 2017), rheology (Shanbhag and Joshi 2022; Makris and Efthymiou 2020); biomechanics (Kelly and Mc- Gough2009;Madsenetal.2008;Pons2023);hydrodynamicwave-energyconversion(Faedoetal. 2017); aeroelasticity (Park et al. 2014); and the study of metamaterials (Srivastava 2021). It has also been known for several decades (Crandall 1991; Makris 1997b) that under certain conditions, an unusual phenomenon can arise within these lines of analysis. In certain simple, causal, models, the well-established derivation of frequency-domain transfer functions leads to model formulations that are non-causal—a contradiction with the known behaviour of the model, andanapparenterrorinestablishedderivations. Thewidespreadconventionalapproach(Crandall 1991; Falnes 1995; Makris 1997b, 2017, 2018; Faedo et al. 2017) is to manually append Dirac deltadistributionstothesetransferfunctionssoastoensurecausality—the"hiddendeltas"(Makris 1997b). Thisprocessresolvesthecausalityviolation,butitraisesseveralquestions. Whyismanual correction required? What is missing in the analysis such that these deltas do not arise naturally? And, will similar hidden deltas arise in more complex transfer functions? Here,weusethetheoryofdistributions,asperSchwartz(1957),toresolvethesequestions. We demonstratethatthehiddendeltasarisefromthesolutionofthedistributionaldivisionequation: a rigorous basis for these terms that predicts their presence in general transfer functions. Distribu- tion division connects these hidden deltas in causal structural models with the non-unique deltas that are observed in non-causal models (Makris 1997b): both arise from the non-uniqueness of distributional division, with causality a constraint forcing uniqueness. In addition, we show how distributional division is closely connected to frequency-domain causality analysis. Distributional analogues of Titchmarsh’s theorem and the Kramers-Kronig relations allow causality analysis in 2 Pons, March 22, 2024 the frequency domain, for a restricted space of distributional transfer functions; but we identify thatthegeneralisationofthistheoremduetoBeltramiandWohlers(1966)significantlyextendsthe space—including,tothecaseofconstantorimpropertransferfunctionsthattypicallypresentchal- lenges for frequency-domain causality analysis (Carcione et al. 2019; Makris 2018; Waters et al. 2000). Inthisway,distribution-theoreticprinciplesnotonlyelucidateaspectsoffrequency-domain structural analysis that have previously been opaque; but provide new analysis routes for the study of causality in frequency-domain structural systems. TRANSFER FUNCTIONS AND HIDDEN DELTAS FollowingMakris(1997b,2017),consideroneofthesimplestconceivablestructures—alinear spring, in the time ( 𝑡) domain: 𝐹(𝑡)=𝑘𝑥(𝑡), (1) with force output 𝐹proportional to displacement input 𝑥via stiffness 𝑘. Note that if we redefined the input variable 𝑥to be velocity or acceleration, we would have a linear damper or inerter, respectively (Makris 2017): these structures can all be analysed along the same lines. Taking the Fourier transformF{·}—that is, F{𝑓(𝑡)}=ˆ𝑓(𝜔)=∫∞ −∞𝑓(𝑡)𝑒−𝑖𝜔𝑡𝑑𝑡, F−1{ˆ𝑓(𝜔)}=1 2𝜋∫∞ −∞ˆ𝑓(𝜔)𝑒𝑖𝜔𝑡𝑑𝜔,(2) —of Eq. 1, we obtain the transfer function (TF) between force and displacement: ˆ𝐹(𝜔)=𝑘ˆ𝑥(𝜔). (3) Eq. 3 defines the dynamic stiffness of the spring as ˆ𝑄0(𝜔)=ˆ𝐹/ˆ𝑥=𝑘. Note that certain reference works, notably Nussenzveig (1972), reverse the sign of 𝜔in the Fourier transform, and so are sign-flipped with respect to this analysis. Based on Eq. 1-3, we pose a pair of apparently simple questions. What is the mechanical impedance of the spring—the TF between force and velocity? (Findeisen 2000) And, the TF between force and acceleration? To define these TFs, we have the well-established Fourier transform of a derivative: 3 Pons, March 22, 2024 F{¤𝑥(𝑡)}=𝑖𝜔ˆ𝑥(𝜔). (4) Applying this relation to Eq. 3 in a normal manner leads to the TFs in: force/velocity: ˆ𝑄1(𝜔)=ˆ𝐹 𝑖𝜔ˆ𝑥=−𝑖𝑘 𝜔, force/acceleration: ˆ𝑄2(𝜔)=ˆ𝐹 −𝜔2ˆ𝑥=−𝑘 𝜔2,(5) in which we observe a problem. ˆ𝑄1(𝜔)and ˆ𝑄2(𝜔)areapparently non-causal : theydonotrespectthedirectionalityoftime,and theprinciple thateffect shouldfollowcause. Thiscan beobserved directlyintheir inverseFourier transforms, which represent the structure’s time-domain response to an impulse in the associated variable. Representing an impulse input with a Dirac delta distribution at 𝑡=0,𝛿(𝑡)—which we use without, as of yet, considering any deeper properties of distributions—then F{𝛿(𝑡)}=1, and via the inverse Fourier transform, we compute the time-domain responses, 𝑄(𝑡), to: a velocity impulse: 𝑄1(𝑡)=1 2𝑘sgn(𝑡), an acceleration impulse: 𝑄2(𝑡)=1 2𝑘𝑡sgn(𝑡),(6) where sgn(𝑡)is the signum function. As per Makris (1997b, 2018, 2017), these responses are non-causal: the impulse occurs at 𝑡=0; whereas nonzero response occurs back to 𝑡→−∞. Where did the well-established analysis of Eq. 1-5 go wrong? Previous studies have not addressed this question directly, but instead have manually modified the TFs of Eq. 1 to maintain causality (Crandall 1991; Falnes 1995; Makris 1997b, 2017, 2018; Faedo et al. 2017). With the arguments that one can add an impulse, 𝛿(𝑡), into the singularity of the TF without "an observer noticing" (Crandall 1991); and a motivation based on the derivative of the logarithm (Makris 1997b); these studies append additional distributional terms: ˆ𝑄1,mod(𝜔)=−𝑖𝑘 𝜔+𝜋𝑘𝛿(𝜔), ˆ𝑄2,mod(𝜔)=−𝑘 𝜔2+𝑖𝜋𝑘𝛿(1)(𝜔),(7) where𝛿(1)(𝜔)isthedistributionalfirstderivativeoftheDiracdelta. Theseappendedtermsarethe "hidden deltas" (Makris 1997b), specifically formulated to solve the causality violation: 4 Pons, March 22, 2024 𝑄1,mod(𝑡)=𝑘𝐻(𝑡), 𝑄2,mod(𝑡)=𝑘𝑡𝐻(𝑡),(8) for Heaviside step function 𝐻(𝑡). In practical terms, this modification restores causality—though it does not elucidate the error over Eq. 1-5, nor indicate whether these hidden deltas might appear in other transfer functions. Interestingly, in the case of Eq. 7, these deltas may also be derived from a loose application of Titchmarsh’s theorem (the Kramers-Kronig relations) (Makris 1997b; Nussenzveig 1972), which expresses conditions for causality in a square-integrable TF in terms of the Hilbert transform. However, as Beltrami and Wohlers (1966) allude to, TFs such as Eq. 7 are neither square integrable ( 1/𝜔), nor ordinary functions ( 𝛿,𝛿(1)), and thus are not admissible to a classicalanalysis,despiteitscorrectresults. Theprevalenceofdistributions( 𝛿,𝐻,sgn)throughout this process suggests that distribution-theoretic principles are at work—to these we now turn. HIDDEN DELTAS AND THE DISTRIBUTIONAL DIVISION EQUATION Properties and spaces of distributions InEq.6-7,whenweintroducedthedeltadistribution, 𝛿(𝑡),wedidsoblithely. Distributions,in thesenseofSchwartz(1957),donotmapvaluesinthesenseofanordinaryfunction( e.g.,R→R). Instead,theyapproximatethismappingviaanintegralonaspaceoftestfunctions—inthemanner of a weak formulation. For details, see Pandey (2011) and Friedlander and Joshi (1998). By convention, we write distributions as functions, e.g.,𝛿(𝑥), but they do not inherit all properties of ordinaryfunctions—notably,indistributionaldifferentiation( 𝐷𝑛),whichcanbeappliedtosingular functions;andinmultiplicationanddivision,whicharenotalwaysdefined,andmayproducenon- unique results. Various well-behaved functions, such as 𝑥,𝑥2,etc., themselves define equivalent distributions; but the space of distributions also involves objects that do not correspond to any function—notably, 𝛿(𝑥)and𝐷𝑛𝛿(𝑥)=𝛿(𝑛)(𝑥). Various singular or discontinuous functions can be given greater utility via distributional formulation: 𝐻(𝑥);sgn(𝑥); and 1/𝑥—the latter, with integration defined via Cauchy principal value, defines the distribution denoted p.v.(1/𝑥). To analyse the causality of distributional TFs, we must define several spaces of distributions: 5 Pons, March 22, 2024 •The space of all distributions— D′(Nussenzveig 1972) or 𝐷′(Beltrami and Wohlers 1966). •Thespaceof tempered distributions —L′⊂D′(Nussenzveig1972),or 𝑆′(BeltramiandWohlers 1966; Pfaffelhuber 1971). Tempered distributions are continuous in a distributional sense, which permits certain singularities; grow no faster than polynomial as 𝑥→∞; and are the natural domain of the Fourier transform: Fmaps a tempered distribution to another tempered distribution. Within L′are:𝛿(𝑥),𝐻(𝑥),p.v.(1/𝑥𝑛)for all𝑛, all polynomials, and all 𝐿𝑝- integrable functions with 𝑝≥1(King 2009). •The space of summable distributions —D′ 𝐿1⊂L′(Beltrami and Wohlers 1966; Pandey 2011), orD′ 𝐿(Nussenzveig 1972), D∗ 𝐿1 0(Ishikawa 1987). Summable distributions can be expressed as a finite sum of the distributional derivatives of ordinary integrable ( 𝐿1) functions—in analogy withtheSobolevspace 𝑊𝑛,1forsome𝑛. Theycanalsobedefinedinother 𝐿𝑝norms: wefollow Nussenzveig’s (1972) treatment of 𝐿1; but Beltrami and Wohlers (1966) and Ishikawa (1987) provide generalisations. D′ 𝐿1contains𝛿(𝑥)and any continuous function that decays at least as fast asO(𝑥−2). It does not contain p.v.(1/𝑥),𝐻(𝑥), or a constant ( 𝑐). •Asetofspaces—D′(𝑛+1) 𝐿1⊂L′forinteger𝑛≥0(Nussenzveig1972),containinganydistribution 𝑔(𝑥)that satisfies: 𝑔(𝑥)(1+𝑥2)−𝑛+1 2∈D′ 𝐿1. (9) It follows that 𝑔(𝑥)is now allowed to show growth of O(𝑥𝛼),𝛼 <𝑛. Distributions within D′(1) 𝐿1 (i.e.𝑛=0) can always be convolved with with p.v.(1/𝑥), and so always have a well-defined Hilbert transform (Nussenzveig 1972)—though generalisations outside this space are possible (Pandey 2011). Practically, 𝑛can be assessed for a given 𝑔(𝑥)by incrementally testing whether 𝑔(𝑥)(1+𝑥2)−𝑛+1 2is integrable. As structurally-relevant examples, cf.Faedo et al. (2017) and Keivanetal.(2017): atminimum 𝑛,p.v.(1/𝑥)isin𝐷′(1) 𝐿1,𝐻(𝑥);sgn(𝑥)andaconstant( 𝑐)arein 𝐷′(2) 𝐿1; and a polynomial of order 𝑚is in𝐷′(𝑚+2) 𝐿1. 6 Pons, March 22, 2024 Distributional transfer functions ConsiderthenadistributionalrepresentationofEq.1-5—withinwhichwemayidentifytherole of delta distributions. If 𝑥(𝑡)∈L′, with the only practical restriction being polynomial growth as𝑡→∞, then𝐹(𝑡)∈L′. Under the Fourier transform, ˆ𝑥(𝜔),ˆ𝐹(𝜔)and the TF ˆ𝑄0=ˆ𝐹/ˆ𝑥=𝑘 are all inL′. Eq. 4, the Fourier transform of a derivative, is identical—but the final operation, the division by𝑖𝜔, is not. Division can only be defined for distributions in restricted cases, and may lead to non-unique solutions (Friedlander and Joshi 1998). In the case of division of 𝑘by(𝑖𝜔)𝑁, to determine the TF with respect to the 𝑁th derivative of 𝑥, we can guarantee that the quotient ˆ𝑄𝑁(𝜔)≜𝑘/(𝑖𝜔)𝑁exists, and we can compute it by solving the distributional division equation (Beltrami and Wohlers 1966; 1967; Nussenzveig 1972): (𝑖𝜔)𝑁ˆ𝑄𝑁(𝜔)=𝑘. (10) That is, to define division, we seek distributions which recover 𝑘under multiplication. Eq. 10 has a well-established non-unique solution (Beltrami and Wohlers 1966; 1967; Nussenzveig 1972): ˆ𝑄𝑁(𝜔)=𝑘 (𝑖𝜔)𝑁 +𝑁−1∑︁ 𝑚=1𝑏𝑚𝛿(𝑚)(𝜔)=𝑘 𝑖𝑁p.v.1 𝜔𝑁 +𝑁−1∑︁ 𝑚=1𝑐𝑚𝛿(𝑚)(𝜔), (11) where𝑏𝑚and𝑐𝑚arearbitrarycomplex-valuedconstants,representingthefactthat 𝜔𝑚𝛿(𝑚−1)(𝜔)= 0,andthusaddinganydeltaderivativeupto 𝛿𝑁−1(𝜔)toˆ𝑄𝑁(𝜔)willstillleadto 𝑘beingrecovered under multiplication (Eq. 10). The term [𝑘/(𝑖𝜔)𝑁]denotes the particular solution to the division equation, which we are here free to express as a factor of p.v.(1/𝜔𝑁). The𝑐𝑚𝛿(𝑚)(𝜔)of Eq. 11 are the hidden deltas of Makris (1997b), and the distributional division equation is the mechanism by which they arise. Distributional division formalizes the intuition of Crandall (1991), that the 𝛿(𝜔)is not "noticed" in p.v.(1/𝜔), though it also qualifies this intuition. These deltas are not specifically connected to the presence of a singularity in the quotient—they arise in any distributional division by 𝜔𝑁—but rather by the fact that this division is uniquely determined only up to 𝛿(𝑁−1)(𝜔). Makris (1997b) made a distinction between the causally-motivated hidden deltas, and the presence of a non-unique delta term in a non-causal 7 Pons, March 22, 2024 rate-independent damping model, but both arise from the same source: distributional division. However, there is an additional connection between these delta terms and causality. CAUSALITY IN DISTRIBUTIONS Causality constraints on the division equation As per ordinary TFs, the response of a distributional TF to an impulse 𝐷𝑁𝑥(𝑡)=𝛿(𝑡)is 𝑄𝑁(𝑡)=F−1{ˆ𝑄𝑁(𝜔)}, becauseF{𝛿(𝑡)}=1. For causality to be respected, 𝑄𝑁(𝑡)cannot represent a response prior to the impulse at 𝑡=0. Because a distribution acts on test functions rather than values, we require that its support ( supp{·})—the set of points around which the distributionmapsanytestfunctiontoanon-zerovalue(Nussenzveig1972)—belocatedin [0,∞): supp{𝑄𝑁(𝑡)}⊆[ 0,∞),for causality. (12) In a certain limited sense, the coefficients, 𝑐𝑚, of the hidden deltas in Eq. 11 determine whether 𝑄𝑁(𝑡)iscasual: theterms 𝛿(𝑚)(𝜔)transformtofactorsof 𝑡𝑚inthetimedomain. However,bythe uniquenessresultsofBeltramiandWohlers(1966)(Theorem1.37),weknowthatiftheoriginalTF ˆ𝑄0(𝜔)is causal, then, for any ˆ𝑄𝑁(𝜔), the set{𝑐𝑚}ensuring causality necessarily exists; whereas ifˆ𝑄0(𝜔)is not causal then no such set exists. Assessing causality can proceed in one of two ways. In cases such as the linear spring, we can use time-domain analysis directly. The inverse Fourier transform of Eq. 11 is (Kammler 2008): 𝑄𝑁(𝑡)=𝑘 2𝑡𝑁−1 (𝑁−1)!sgn(𝑡)+1 2𝜋𝑁−1∑︁ 𝑚=1𝑐𝑚𝑡𝑚 𝑖𝑚. (13) From Eq. 13 we determine that: ( i) to ensure𝑄𝑁(𝑡)is real-valued, then if 𝑚is even,𝑐𝑚must be purelyreal,andif 𝑚isodd,𝑐𝑚mustbepurelyimaginary. Wemayconvenientlydefinereal-valued coefficients𝑑𝑚as𝑖𝑚𝑑𝑚=𝑐𝑚to satisfy this condition. ( ii) To satisfy causality, only the highest- order term𝑡𝑁−1can be non-zero. Setting all coefficients to zero other than 𝑐𝑁−1, we can compute 8 Pons, March 22, 2024 this remaining coefficient as 𝑐𝑁−1=𝑖𝑁−1𝜋𝑘/(𝑁−1)!, and confirm: ˆ𝑄𝑁(𝜔)=𝑘 𝑖𝑁p.v.1 𝜔𝑁 +𝑖𝑁−1𝜋𝑘 (𝑁−1)!𝛿(𝑁−1)(𝜔), 𝑄𝑁(𝑡)=𝑘𝑡𝑁−1 (𝑁−1)!𝐻(𝑡),(14) which is causal. This is the solution for the hidden delta in any TF of the linear spring. For 𝑁=1 we recover𝜋𝑘𝛿(𝜔), and for𝑁=2,𝑖𝜋𝑘𝛿(1)(𝜔), as per Eq. 7. Time-domain causality analysis of this form is useful for assessing the causality of TFs defined a priori. However, in cases where we wish to identify TF properties or parameters that proceed from causality—e.g., loss moduli fromstoragemoduli(Madsenetal.2008),orcasualapproximationsofhystereticdamping(Makris 1997a)—then causality analysis directly in the frequency domain can be preferable. Causality analysis via the distributional Hilbert transform Frequency-domain causality analysis for distributional TFs rests on the finer spaces of distri- bution we have outlined previously. Initially, let us assume that ˆ𝑄𝑁(𝜔)∈D′(1) 𝐿1, which is true for 𝑁≥1in the linear spring, but notfor the original ˆ𝑄0(𝜔)=𝑘. Distributions inD′(1) 𝐿1can always be convolved (∗) with p.v.(1/𝑥): following Theorem 1.8.5 of Nussenzveig (1972), this allows us to construct the following relation: 𝔉−1 ˆ𝑄𝑁(𝜔)∗ p.v.1 𝜔 +𝑖𝜋𝛿(𝜔) =2𝜋𝑖𝐻(𝑡)𝑄𝑁(𝑡). (15) AsperEq.12, 𝐻(𝑡)𝑄𝑁(𝑡)iscausal,andwhen 𝑄𝑁(𝑡)itselfiscausal, 𝐻(𝑡)𝑄𝑁(𝑡)=𝑄𝑁(𝑡). Utilising this in Eq. 15 yields the fundamental theorem for causality in distributions (Beltrami and Wohlers 1965, Theorem 2; Nussenzveig 1972, Theorem 1.8.6): ˆ𝑄𝑁(𝜔)=1 𝑖𝜋ˆ𝑄𝑁(𝜔)∗p.v.1 𝜔 =1 𝑖Hˆ𝑄𝑁(𝜔) ,for causality. (16) H{·}denotes the distributional Hilbert transform, defined via the convolution in Eq. 16. Hilbert transforms can be evaluated via tabulated results (King 2009), or the Fourier transform of a convolution (Pandey 2011). Splitting ˆ𝑄𝑁(𝜔)into real and imaginary parts reveals that these parts mustbeHilberttransformspairs,butwewilloperatedirectlyon ˆ𝑄𝑁(𝜔). ApplyingEq.16toEq.14, 9 Pons, March 22, 2024 we confirm causality of the hidden delta solution. Given (King 2009): H p.v.1 𝜔𝑁 =(−1)𝑁𝜋 (𝑁−1)!𝛿(𝑁−1)(𝜔), Hn 𝛿(𝑁−1)(𝜔)o =(−1)𝑁−1(𝑁−1)! 𝜋p.v.1 𝜔𝑁 ,(17) then: Hˆ𝑄𝑁(𝜔) =𝑘 𝑖𝑁H p.v.1 𝜔𝑁  +𝑖𝑁−1𝜋𝑘 (𝑁−1)!Hn 𝛿(𝑁−1)(𝜔)o =𝑖𝑁𝜋𝑘 (𝑁−1)!𝛿(𝑁−1)(𝜔)+𝑘 𝑖𝑁−1p.v.1 𝜔𝑁 =𝑖ˆ𝑄𝑁(𝜔)∴causal.(18) As noted by Beltrami and Wohlers (1966), Eq. 16, which is valid for distributions in D′ 𝐿1and thus ordinary functions in 𝐿1, is equivalent to Titchmarsh’s theorem for ordinary functions in 𝐿2. The distributionalformulationthusextendsthevalidityofanordinary-functionanalysisto 𝐿1,provided that certain statements are interpreted in a distributional sense. Causality analysis via the generalised Hilbert transform Nevertheless,therestrictionto D′(1) 𝐿1,including𝐿1and𝐿2,excludesarangeofrelevantstructural models. Constanttransferfunctions,suchasthedynamicstiffnessofthespring, ˆ𝑄0(𝜔)=𝑘,areone immediatecase(Carcioneetal.2019). Asanordinaryfunction, 𝑘∉𝐿2;asadistribution, 𝑘∉D′(1) 𝐿1; and we may confirm violation of Eq. 16: H{𝑘}/𝑖=0≠𝑘. Other more complex inadmissible transfer functions can be found in viscoelastic power-law media (Szabo 1994; Gulgowski and Stefański 2021; Waters et al. 2000). There is, however, an extension of the causality condition of Eq. 16 to a wider space of distributions, as derived by Beltrami and Wohlers (1966). For any distribution ˆ𝑄𝑁(𝜔)∈D′(𝑛+1) 𝐿1, we may define a generalised Hilbert transform: H(𝑛)ˆ𝑄𝑁(𝜔) =𝜔𝑛Hˆ𝑄𝑁(𝜔) 𝜔𝑛 =𝜔𝑛 𝜋ˆ𝑄𝑁(𝜔) 𝜔𝑛 ∗p.v.1 𝜔 , (19) andthen,asperBeltramiandWohlers(1966),Theorem3.13andNussenzveig(1972),Eq. 1.8.40: ˆ𝑄𝑁(𝜔)=1 𝑖H(𝑛)ˆ𝑄𝑁(𝜔) +P𝑛−1(𝜔),for causality. (20) 10 Pons, March 22, 2024 [ˆ𝑄𝑁(𝜔)/𝜔𝑛]again represents the particular solution of this distributional quotient, and P𝑛−1(𝜔) representsanarbitrarypolynomialoforder 𝑛−1in𝜔,accountingfordeltadistributionsintroduced by division ( cf.Eq. 11): these deltas become polynomial under convolution and multiplication by 𝜔𝑛. If Eq. 20 is satisfied for some P𝑛−1(𝜔), then ˆ𝑄𝑁(𝜔)is causal. UsingEq.20,ifweknowthatadistributionalTFisinsome D′(𝑛+1) 𝐿1,thenwemayrapidlyassess its causality by computing H(𝑛){·}/𝑖and observing whether this differs from the original TF by more thanP𝑛−1(𝜔). This allows a direct causality analysis of any improper or not strictly proper TF,withanumeratorofordergreaterthanorequaltothatofthedenominator. Forinstance,forthe dynamic stiffness of a spring, ˆ𝑄0(𝜔)=𝑘∈D′(2) 𝐿1, i.e.𝑛=1, we have: 1 𝑖H(1){𝑘}=𝜔 𝑖𝜋𝑘 𝜔 ∗p.v.1 𝜔 =𝑘𝜔 𝑖𝜋p.v.1 𝜔 ∗p.v.1 𝜔 =𝑘𝑖𝜔𝛿(𝜔)=0 =𝑘+P 0∴causal.(21) Thesameapproachisapplicabletothedynamicstiffnessesoflineardampers( ˆ𝑄0(𝜔)=𝑖𝑑𝜔,Makris 1997b) and inerters ( ˆ𝑄0(𝜔)=−𝑚𝜔2, Makris 2018), which are improper TFs. Indeed, several representation theorems—including Theorem 2 of Ishikawa (1987), Theorem 1.28 of Beltrami and Wohlers (1966), and Theorem 2 of Pfaffelhuber (1971)—indicate that anyTF inL′can be analysed via this method. For instance, we can directly confirm the non-causality of the classical rate-independent damper, with dynamic stiffness ˆ𝑄0(𝜔)=𝑖sgn(𝜔) ∈ D′(2) 𝐿1. Computing the convolution via Fourier transform ( 𝜔toΩ), and denoting the Euler–Mascheroni constant by 𝛾: 1 𝑖H(1){𝑖sgn(𝜔)}=𝜔 𝜋𝑖p.v.𝑖 |𝜔| ∗p.v.1 𝜔 =𝜔 𝜋F−1 F p.v.1 |𝜔| F p.v.1 𝜔 =2𝑖𝜔F−1{ln|Ω|sgn(Ω)+𝛾sgn(Ω)} =2 𝜋ln|𝜔|≠𝑖sgn(𝜔)+P 0(𝜔)∴non-causal.(22) We may verify with a few further steps that the addition of this residual term ( 2/𝜋ln|𝜔|) to the rate-independent damping model causes it to become causal—as per Makris (1997a). Indeed, 11 Pons, March 22, 2024 Eq. 22 elucidates one final paradoxical causality result in the literature. Makris (1997a) derives a causal rate-independent damping model with dynamic stiffness: ˆ𝑄0(𝜔)=𝑖sgn𝜔 𝜀 +2 𝜋ln 𝜔 𝜀 , (23) where𝜀is an arbitrary positive constant. The real and imaginary parts of Eq. 23 are an exact generalised Hilbert transform pair ( cf.Eq. 22) and thus the model is causal for all 𝜀. However, while sgn(𝜔/𝜀)=sgn(𝜔)always, sgn(𝜔)and2/𝜋ln|𝜔/𝜀|arenotexact generalised Hilbert transform pairs. The difference is, indeed, a polynomial P0, asln|𝜔/𝜀|=ln|𝜔|−ln𝜀, and thus forany𝜀apolynomialresidual P0willexistinEq.22,satisfyingcausality. Inthisway,thehidden deltas allow us to derive the equivalent simplified model: ˆ𝑄0(𝜔)=𝑖sgn(𝜔)+2 𝜋ln 𝜔 𝜀 =𝑖sgn(𝜔)+2 𝜋ln|𝜔|+𝑐, (24) and confirm that it is causal for all 𝜀and all𝑐. Eq.20isapowerfulconditiontoassesscausalityinlinearsystems,butitalsohasakeyphysical connection. The generalised Hilbert transform involves dividing a transfer function ˆ𝑄(𝜔)by𝜔𝑛, convolving it, and then multiplying again by 𝜔𝑛. This is equivalent to integrating the impulse response𝑄(𝑡)𝑛times, multiplying by a step function to force causality, and then differentiating 𝑛 timesbackagain. Thiscausalityassessmentworksbecause,ifasystemiscausalwithrespecttoany kinematicvariable,thenitiscausalwithrespecttoanyderivativeorintegralofthisvariable. Itisnot possibleviadifferentiationorintegrationtopropagatecasualsignalstobefore 𝑡=0,andthusweare freetochoosethedifferential/integralorder( 𝑛)atwhichtoperformthecausalityanalysis—wehave onlytochoose 𝑛toreachD′(1) 𝐿1. ThisprocessunderpinstheimplicitchoiceofMakris(1997b,2017) to analyse the spring’s causality in mechanical impedance ( 𝑛=1) rather than dynamic stiffness (𝑛=0): it is here that we reach D′(1) 𝐿1and the fundamental theorem (Eq. 16) is applicable. The representationtheoremsofIshikawa(1987)andothersfurthersupportthischoicebyindicatingthat a suitable𝑛exists for anydistribution inL′, including any slowly-growing function. Distribution- theoretic principles not only provide the basis for the presence of the hidden deltas, but also 12 Pons, March 22, 2024 justification for choices made by current studies to analyse causality in specific higher-derivative transfer functions, such as mechanical impedance. Distributional analysis predicts exactly which higherderivativeisrequiredforconventionalcausalityanalysistobevalid,providesadirectmethod forassessingcausalityatanyinitialderivativeorderwhichdoesnotrequirecomputationofhidden deltas.