## Abstract

Viscoelastic materials have non-negative relaxation spectra. This property implies that viscoelastic response functions satisfy certain necessary and sufficient conditions. These conditions can be expressed in terms of each viscoelastic response function ranging over a cone. The elements of each cone are completely characterized by an integral representation. The 1:1 correspondence between the viscoelastic response functions is expressed in terms of cone-preserving mappings and their inverses. The theory covers scalar- and tensor-valued viscoelastic response functions.

## 1. Introduction

The set of material response functions of linear viscoelasticity is defined by a single requirement, the requirement of positive relaxation spectrum (PRS). Whether a Laplace-domain material response function satisfies the requirement of PRS can be checked by testing the properties of its analytic continuation (Hanyga & Seredyńska 2007). The requirement of PRS, however, determines the set of material response functions of each kind (stress relaxation, creep as well as their Laplace-domain counterparts) as well as the relaxation and retardation spectra. Testing for PRS can be thus reduced to testing the membership of the material response function in an appropriate set. In the new formulation material response functions range over appropriate sets of functions that are connected by bijective mappings. A similar framework can be developed for material response in polar dielectrics. An extension to general dielectrics however brings in new classes of functions (Hanyga & Seredyńska 2008).

Viscoelastic material response functions of a specified kind (e.g. relaxation moduli) are arbitrary members of a convex cone of functions. The response functions can be scalar or tensor valued. Members of a cone are parametrized by a few non-negative numbers (or tensors) and a positive Radon measure (or a positive tensor-valued Radon measure, respectively) satisfying a convergence criterion. Every function in the cone is thus expressed in terms of an integral representation. Using the Choquet theory (Choquet 1969) it can be shown that each member of the cone is a convex linear combination of the cone’s extremal elements—the indecomposable models of viscous response. The integral representation is an analytic expression of this relation. The extremal elements of the cone of relaxation moduli are known as the Debye relaxation moduli. The extremal elements of the cone of creep compliances are called the Kelvin–Voigt elements. The integral representations can also be viewed as explicit mappings of the cone of relaxation (retardation) spectra onto the cone of relaxation moduli (creep compliances, respectively) or their Laplace transforms. The cone of relaxation moduli and the cone of creep compliances are related by a nonlinear bijective mapping. A nonlinear bijective mapping connects the cones of relaxation and retardation spectra.

Each cone is closed with respect to pointwise convergence. The cones of scalar material response functions are also closed with respect to some nonlinear operations such as composition. Using these relationships it is possible to find non-trivial examples of elements of a cone with potential applications for viscoelastic constitutive equations.

In the paper the cones of relaxation moduli, creep compliances and their Laplace transforms are determined. With the help of integral representations, they are expressed in terms of relaxation/retardation spectra. The inverse mappings for the integral representations are also obtained. The bijective mapping of relaxation spectra onto retardation spectra and conversely is determined for purely absolutely continuous spectra.

For readers’ convenience all the notations are gathered in table 1.

## 2. Mathematical concepts

### (a) The tensor spaces *T* and

In addition to real (or complex)-valued material response functions, we shall consider rank 4 tensor-valued material functions with the tensorial properties of the elastic stiffness tensor. It is convenient to consider such tensors as symmetric operators acting on strain or stress.

The linear space of symmetric rank 2 tensors endowed with the inner product
2.1
will be denoted by *S*. The set of symmetric linear operators on *S* is denoted by *T*.

### Definition 2.1.

A∈*L*(*S*) *belongs to T if* 〈**e**,A**f**〉=〈A**e**,**f**〉

The identity operator in *T* is denoted by I.

### Definition 2.2.

*Let*A∈*T*.

A*is positive semi-definite*, A≥0, *if* 〈**e**,A**e**〉≥0 *for every***e**∈*S*.

*The set of positive semi-definite elements of T is denoted by T*_{+}.

Laplace-domain response functions take values in a complexification of *T*. is the complex-linear space of complex-linear symmetric operators on the space of complex rank 2 symmetric tensors. The inner product in is defined by the formula
2.2

A^{†} denotes the Hermitean conjugate of A:
2.3
The real and imaginary parts of a tensor A are defined as follows:
2.4
2.5

If , then A^{†}=A*.

### (b) Positive Radon measures and positive *T*-valued Radon measures

### Definition 2.3.

*A positive Radon measure is a continuous linear functional on the space**of compactly supported continuous functions, endowed with the supremum norm*.

Let *T*′ denote the conjugate space of *T*.

### Definition 2.4.

*A positive T-valued Radon measure is a continuous linear functional on the space**of compactly supported continuous T′-valued functions endowed with the sup norm*.

### Theorem 2.5.

Hanyga & Seredyńska 2007*A positive T-valued Radon measure can be decomposed into a product of a T-valued function H and a positive Radon measure m*:
*where**and*H(*r*) *is positive semi-definite for m-almost all*.

The Radon measure *m* can be chosen as the trace of the *T*-valued Radon measure *M*_{klkl}; in this case, I−H(*r*)≥0 for *m*-almost all *r*≥0.

In viscoelasticity, the measure *m* is a distribution function of the relaxation or retardation spectrum while the tensor-valued density represents effective anisotropic properties for a given relaxation time (Hanyga & Seredyńska 2007).

### (c) Completely monotone and Bernstein functions

### Definition 2.6.

*Let f be a real or T-valued function on*.

*We shall say that f is* completely monotone (CM) *if it is infinitely differentiable and if for every non-negative integer n*

The set of CM functions will be denoted by . A CM function *f* is infinitely differentiable; hence, it is integrable on every compact interval contained in . A CM function *f* is locally integrable if it is integrable on some right neighbourhood of 0, in particular if it is integrable on ]0,1]. We shall be mostly interested in *locally integrable complex monotone* (LICM) functions. The set of LICM functions (real or *T*-valued, depending on the context) will be denoted by .

The set of LICM functions is a cone. Indeed, if and *a*,*b*≥0, then .

### Definition 2.7.

*A real or T-valued function f on**is a* Bernstein function, (BF), *if f*≥0 *and the derivative*.

*The set of real* (*T*-*valued*) *BFs is denoted by*, *respectively*).

A BF *f* always has a finite limit . This is obvious for a real BF *f* because *f* is non-negative, continuous and non-decreasing. For a *T*-valued function it is clear that each function 〈**w**,*f*(*x*)**w**〉, where **w**∈*S*, has a limit *F*(**w**) at *x*=0. By polarization,
has a limit at *x*=0 that depends linearly on **w** and **v**. Define the components of the tensor *f*_{0}∈*T* with respect to an arbitrary basis **w**^{n}, *n*=1,…,*d*, in *S*
*f*_{0} is the limit of *f*(*x*) at 0. The function *f*(*x*) is extended by continuity to by setting *f*(0)=*f*_{0}.

### (d) Integral representation theorems for LICM and Bernstein functions

The following theorem, associated with the name of Bernstein, is a fundamental theorem for CM and related functions (Widder 1946).

### Theorem 2.8

(Bernstein). *A real function f is CM if and only if there is a positive Radon measure m such that*
2.6*for x*>0.

*The Radon measure m is uniquely determined by the function f*.

The uniqueness part follows from the uniqueness of inversion of Laplace–Stieltjes transforms (Widder 1946).

### Theorem 2.9

*A T-valued function f is CM if and only if there is a positive T-valued Radon measure*M*such that*
2.7*for x*>0.

*The Radon measure*M*is uniquely determined by the function f*.

Theorem 2.9 is a corollary of theorem 2.8. It is proved for matrix-valued functions in Gripenberg *et al.* (1990) and for *T*-valued functions in Hanyga & Seredyńska (2007).

### Theorem 2.10.

*A real function f is LICM if and only if equation (2.6) is satisfied with a positive Radon measure m satisfying the inequality*
2.8

### Proof.

By the Fubini theorem it follows from equation (2.6) that From the series representation of the exponential function, it is easy to see that and therefore , Hence implies equation (2.8).

Conversely, equation (2.8) implies that , while for some *θ*, 0≤*θ*≤1. On the other hand,
and . Hence, equation (2.8) implies that . ▪

### Theorem 2.11.

*A T-valued function f is LICM if and only if equation (2.7) is satisfied with a Radon measure M satisfying the inequality*
2.9

The set of positive Radon measures *m* satisfying equation (2.8) will be denoted by . The set of positive *T*-valued Radon measures M satisfying equation (2.9), respectively, will be denoted by .

### Theorem 2.12.

*A real function f on**is a BF if and only if there are two non-negative numbers c,d and a positive Radon measure n on**such that*
2.10*and*
2.11*The triple (c,d,n) is uniquely defined by the function f*.

The set of positive Radon measures satisfying equation (2.11) will be denoted by .

We shall prove the corresponding theorem for *T*-valued BFs.

### Theorem 2.13.

*A T-valued function f on**is a BF if and only if there are two positive semi-definite operators*C,D*and a positive T-valued Radon measure*N*such that*
2.12*and*
2.13*The triple* (C,D,N) *is uniquely defined by the function f*.

### Proof.

Let *f*(0)=C. The function *f* is the indefinite integral of the LICM function *f*′:
Substituting
where M satisfies equation (2.9), and applying the Fubini theorem, one gets
Equation (2.12) and the inequality (2.13) are obtained upon setting N(d*q*)=M(d*q*)/*q*.

The triple (C,D,N) in the *T*-valued case is uniquely determined by the function *f*. The tensor C is equal to the limit . Indeed,
2.14
In order to prove this fact, we first show that
The integrand is bounded by *x**y*:1−e^{−xy}=*x**y*e^{−θxy}≤*x**y*, for some 0≤*θ*≤1. On the other hand, equation (2.13) implies the inequality
For the remainder of the integral we note that 1−*e*^{−xy}≤1 and
The Lebesgue Dominated Convergence Theorem implies equation (2.14). Concerning the tensor D and the Radon measure N, we note that
where M({0})=D and M(d*y*)=*y*N(d*y*). The Radon measure M is uniquely defined by *f*′, hence D and N are uniquely defined by *f*. ▪

The set of positive *T*-valued Radon measures satisfying equation (2.11) will be denoted by .

### (e) Stieltjes functions

The Laplace transform of an LICM function *g* is a particular example of a Stieltjes function. Using the Bernstein and Fubini theorems, one easily proves that the Laplace transformation
applied to a *T*-valued LICM function *f* yields
2.15
where M satisfies the inequality (2.9).

The set of *T*-valued *Stieltjes functions**f* (Hirsch 1972) is defined by the integral representation
2.16
where M satisfies the inequality (2.9). The subset of such that will be denoted by .

The contribution of the point 0 is *x*^{−1}M({0}). Introducing a new parameter B=M({0})≥0 results in an equivalent integral representation
2.17
The Radon measure M needs to be defined on only.

For easy reference, we shall spell out the definition of real Stieltjes functions.

### Definition 2.14.

*f is a Stieltjes function if there are two non-negative real numbers a,b and a positive Radon measure m on**satisfying equation (2.8) such that*
2.18

It follows easily from the integral representation (2.18) that Stieltjes functions are CM.

A Stieltjes function *g* can be continued to the upper half-plane and it then maps to :

### (f) Inversion of the Stieltjes transformation

We now show that the Stieltjes transformation can be inverted to calculate the constant *a* and the Radon measure *m*.

We begin with proving that . It suffices to prove that
But
For *x*≥1, the second factor is bounded by

The constant *b* can be recovered from the formula . It suffices to prove that
This, however, follows from the inequality 1/(1+*y*/*x*)≤1/(1+*y*) for *x*≤1, equation (2.8), and the Lebesgue Dominated Convergence Theorem.

The Radon measure *m* can now be recovered from the Stieltjes inversion formula
2.19
which follows from equation (A9).

The case of *T*-valued Stieltjes functions is entirely analogous.

### (g) Complete BFs

An important subclass of BFs is obtained by substituting in the integral representation (2.12) a Radon measure with an LICM density Applying the Fubini theorem and working out the inner integral, one obtains equation (2.23).

### Definition 2.15.

*A real BF f defined by the integral representation (2.10) with*
2.20*is called* a complete BF (*CBF*).

*The set of CBFs is denoted by*.

*A T-valued BF f defined by the integral representation (2.12) with*N(d*y*)=L(*y*) d*y*, , *is called a T*-valued CBF (*TCBF*).

*The set of TCBFs is denoted by*.

### Theorem 2.16.

*A real BF f on**is a CBF if there are two non-negative real numbers a,b and a positive Radon measure m satisfying equation (2.8) such that*
2.21

### Proof.

There is a positive Radon measure *ρ* such that
Substituting equation (2.20) in equation (2.10), one formally obtains equation (2.21) with *m*(d*r*):=*ρ*(d*r*)/*r*.

We now prove that equation (2.11) is satisfied if and only if *m* satisfies equation (2.8). Equation (2.11) is equivalent to the inequality
2.22
A simple scaling argument implies the asymptotics for and for . Hence, equation (2.22) is equivalent to the pair and . Both integrands are majorized by 2*ρ*(d*r*)/[*r*(1+*r*)] and minorized by *ρ*(d*r*)/[*r*(1+*r*)]. Equation (2.11) is thus equivalent to the inequality
Hence, equation (2.11) is satisfied if and only if *m* satisfies equation (2.8). ▪

The proof of the *T*-valued counterpart of theorem 2.16 is identical.

### Theorem 2.17.

*f is a TCBF if and only if there are two positive semi-definite operators*A*and*B*and a positive T-valued Radon measure*R*on**such that*
2.23*and*
2.24

### Corollary 2.18.

*If**then g*(*x*):=*f*(*x*)/*x is a Stjelties function*.

*If*, *then f*(*x*):=*x**g*(*x*) *is a CBF*.

### Examples

, for 0≤*α*,*β*≤1;

is a Stieltjes function for 0≤*α*,*β*≤1.

### Corollary 2.19.

, 0≤*α*≤1, *implies that*, .

### Example

*p*^{α−1}(1+*p*^{α})^{−1} is known to be the Laplace transform of an LICM function *E*_{α}(−*t*^{α}) if 0<*α*<1; hence, *p*^{α}(1+*p*^{α})^{−1} and *p*^{αβ}(1+*p*^{α})^{−β} are CBFs for 0<*β*<1; thus, *p*^{αβ−1}(1+*p*^{α})^{−β} is the Laplace transform of an LICM function (Podlubny 1998).

### Corollary 2.20.

*If*, *g*(*x*):=*f*(*x*)/*x is a T-valued Stjelties function*.

*If g is a T-valued Stieltjes function, f*(*x*):=*x**g*(*x*) *is a TCBF*.

### Example

The function *x*^{−α}≡*x*^{1−α}/*x* is a Stieltjes function for 0≤*α*≤1.

A less trivial result is due to Berg (1980): are non-zero, 0≤*α*≤1, implies that (log-convexity of ). It follows immediately that and 0≤*α*≤1 imply that . Furthermore, for 0≤*α*≤*β*≤1.

### Corollary 2.21.

*A real function f on**belongs to CBF if and only if there is a BF g such that*.

### Proof.

Let ,
where *n* satisfies equation (2.11). The Laplace transform of *g* is then given by
Hence, has the form (2.21) with *m*(d*y*):=*y**n*(d*y*) satisfying equation (2.8). ▪

### Corollary 2.22.

*A T-valued function f on**belongs to**if and only if there is a T-valued BF g such that*.

Let D denote the differentiation operator and *X* a pointwise multiplication by *x*, i.e. the mapping . , known as the Laplace–Carson transformation maps time-domain moduli and compliances to the Laplace domain. Hence, the cones and merit some examination. The image of under D is : . is isomorphic to , by the mapping . The Laplace transform , *f*(0)≥0, hence .

CBFs can also be defined in terms of the properties of their analytic continuations into the upper complex half-plane . Analytic properties of CBFs are of crucial importance for constructing the inverse mapping for the integral representation. They will also be used to derive some nonlinear closure relations.

### Theorem 2.23.

*A real function f on**is a CBF if and only if f has a regular analytic continuation F to**and*, .

Let the operator denote the composition with *x*^{−1}: . The composition *f*°*g* of two BFs *f*,*g* is a BF. The composition *f*°*g* of a CM function *f* with a BF *g* is CM (Berg & Forst 1975; Hanyga & Seredyńska 2007). The function 1/*f*, where *f* is a BF, is a composition of the CM function 1/*x* with the BF *f*, hence it is a CM. Furthermore,

### Theorem 2.24.

;

;

.

### Proof.

*Ad* (*i*) If , *f*≢0, then its analytic continuation *F* to satisfies the inequality Im *F*(*z*)≥0. The function *G*(*z*):=1/*F*(*z*^{−1}) has the same property and *G*(*x*)≥0 for *x*>0. Hence has the integral representation
*c*,*d*≥0. Setting *ζ*=1/*z* and *u*=1/*q*, we get
with a positive Radon measure ν(d*u*):=*u**ρ*(d*u*^{−1}), for *u*>0, ν({0}):=*d*. Thus, .

*Ad* (*ii*) Let and let *F*,*G* denote their analytic continuations to . The analytic function *G* maps and to itself and is regular there. By the integral representation of CBFs, equation (2.21), the function *G* maps to itself. Hence, the range of *G* does not intersect the cut. *F*°*G* is thus an analytic continuation of *f*°*g* to . Applying the integral representation of *F*,
with *a*,*b*,*ρ*≥0. If Im *z*>0, then Im *G*(*z*)>0 and therefore Im *F*(*G*(*z*))≥0. Consequently .

*Ad* (*iii*). The proof is similar to the previous one, except that Im *F*(*z*)≤0 in ,
and
▪

## 3. Viscoelastic material response functions

### (a) Real response functions

Constitutive equations of linear viscoelasticity have the form
3.1
where denotes the strain rate. In particular, Newtonian viscosity corresponds to , or *G*=*b**δ*, where *δ* denotes the Dirac delta.

We assume that the relaxation modulus is a linear superposition of the Newtonian viscosity *b**δ*(*t*) and the Debye elements with non-negative coefficients:
3.2*G*(*t*)=0 for *t*<0, with *b*≥0, *a*=*m*({0})≥0 and a positive Radon measure *m* satisfying equation (2.8). The creep compliance *J*(*t*) for *t*>0 can be obtained by solving the Volterra convolution equation
3.3
This equation has a unique solution (Hanyga & Seredyńska 2007). The solution of equation (3.3) defines an invertible nonlinear mapping . The Laplace transformation converts equation (3.3) to
3.4
or
3.5

The Carson–Laplace transform of a non-zero relaxation modulus *G* has the following form
The Carson–Laplace transforms vary over the entire cone . On account of equation (3.5) and theorem 2.24, the functions vary over . But we know that coincides with . Furthermore, by corollary 2.21,
Hence, finally, the Laplace transforms vary over the entire cone and thus the creep compliances *J* vary over the entire cone . This argument reproduces part of the results of Hanyga & Seredyńska (2007), with an important addition: a precise determination of the sets of functions and .

From the integral representation of it is possible to derive necessary and sufficient conditions for to be the Laplace transform of an LICM function *G*. Such conditions can be found in the book of Gripenberg *et al.* (1990).

The role of the linear creep element *b**t* is partly clarified by assuming a *pure* Newtonian viscosity: *G*(*t*)=*b**δ*(*t*), *b*>0, which yields *J*(*t*)=*d**t* with *d*=1/*b*. Apart from the Newtonian element the relaxation modulus is a superposition of the Debye relaxation elements , including a Debye element with an infinite relaxation time *r*=1/τ=0. The last element ensures that the equilibrium relaxation modulus does not necessarily vanish.

### (b) *T*-valued response functions

Linear relations between the cones carry over to the *T*-valued case without substantial modifications. A general relaxation modulus consists of an equilibrium modulus, a Newtonian viscosity term and a spectrum of Debye elements
3.6
The creep compliance assumes the form
3.7

Equations (3.3) and (3.5) assume the following form 3.8 and 3.9

### (c) Cone isomorphisms

The commutative diagram defines a nonlinear bijective mapping . In viscoelasticity maps the relaxation spectra to the corresponding retardation spectra.

## 4. Relation between the relaxation and retardation spectrum densities

The commutative diagram in §(c) shows that there is a bijective mapping between the cone of the triples (A,B,M) in equation (3.6) and the cone of the triples (C,D,N) in equation (2.12). In the scalar case, an explicit form of this relation will be obtained for absolutely continuous relaxation and retardation spectrum.

Conversion of relaxation spectra to retardation spectra and conversely plays an important role in accurate modelling of experimental data. Fully satisfactory conversion formulae can be derived only for scalar models with absolutely continuous spectra.

The Laplace transform of equation (3.6) is 4.1 If 4.2 then, in view of equation (A12), 4.3 hence, on , 4.4

But , where , hence
4.5
On the other hand, the creep compliance can be represented in the form (3.7). Hence,
4.6
and
4.7
The measure *r*N(d*r*) restricted to the set of smooth functions is a causal distribution. The integral on the right-hand side of equation (4.7) is a convolution of two distributions, *y*N(d*y*) and (*y*+iϵ)^{−1}. The distributional limit of the latter distribution for is vp*y*^{−1}−iπ*δ*(*y*), where vp denotes the principal value.

If, in particular, N(d*r*)=K(*r*) d*r*, then the limit for can be calculated explicitly
From equation (4.5) we then have

In the scalar case, . Hence, assuming that *n*(d*r*)=*K*(*r*) d*r*,
and, for *r*>0, , so that finally
4.8

We now calculate the inverse transformation in the scalar case. Applying equation (A10) to equation (4.6), we get
where *W* is the distribution
Substituting (4.4), we get for *r*>0,

Hence *n* does not have a discrete component. If *n*(d*r*)=*K*(*r*) d*r*, then the relaxation spectral density *H*(*r*) and the retardation spectral density *K*(*r*) satisfy the equation
4.9

Equations (4.8) and (4.9) are completed by the relations between the constants A,C in and D in equations (4.1) and (4.6). We begin with noting that A>0 implies that D=0. Indeed, the first inequality entails that for **w**≠0. Hence D=F(0)=0. If the stress does not relax to 0, then the creep compliance does not reach a linear growth rate. On the other hand, if A=0, then . If the integral is infinite, then D=0.

Let D=0. Passing to the limit in equation (3.9) yields the relation 4.10 while the limit gives 4.11

Analogous relations apply to the scalar case.

## 5. Summary

Each of the sets , , and is a convex cone closed in the topology of pointwise convergence. The integral representation states that each element of the cone is a convex linear combination of some basis elements of the cone. In view of the uniqueness of the integral representations, a basis element cannot be expanded in a convex linear combination of other basis elements. It is thus an extremal element in the convex cone. The integral representations are thus special cases of the Choquet theory (Choquet 1969). The extremal elements of consist of the elastic element 1 and the Debye stress relaxation functions , corresponding to the differential relaxation equation with the relaxation time τ=1/*r*. The extremal elements in the cone of creep compliances consist of 1, *t* and 1−e^{−rt}. The first one is purely elastic. The second one is the Newtonian creep. The third one is a Kelvin–Voigt element, i.e. a solution of the differential equation . The mapping is nonlinear and therefore the extremal elements in the cone of creep compliances are not images of the extremal elements of .

## Appendix A. Herglotz functions and inversion of integral representations

### Definition A.1.

*A complex analytic function F defined on**is called a Herglotz function if it is regular in**and maps**to itself*.

### Definition A.2.

*A**-valued function f is called a*-*valued Herglotz function if (i) f is regular analytic on**and* (*ii*) *Im f*(*z*)≥0 *for*.

This definition is an adaptation of the definition of complex- and matrix-valued Herglotz function (Gesztesy & Tsekanovskii 2000). The alternative names are Pick functions, Herglotz–Riesz, Herglotz–Nevanlinna functions.

The integral representation of a Herglotz function and its inversion is the main topic of this section.

We begin with an integral representation theorem.

### Theorem A.3.

*If f is a*-*valued Herglotz function then there are two tensors*, *with*B≥0, *and a positive T-valued Radon measure*R*on**such that*
A1*and*
A2

Equation (A2) should be interpreted in the following sense:

The inversion theorem below yields a unique triple (A,B,R) for a given Herglotz function *f*.

Equation (A1) assumes a simpler form if the Radon measure R has a finite mass: for all **w**∈*S*. Note that
in this case. The second term in the integrand of equation (A1) is needed for regularization of the integral in the more general case. If the Radon measure R has a finite total mass then the second term in the integrand of equation (A1) can be dropped at the expense of a redefinition of A.

### Theorem A.4.

*The triple* (A,B,R) *is given by*A = *Ref*(i), B = lim_{y→∞}[*f*(i*y*)/(i*y*)] *and*
A3
A4

Since , equation (A4) can be replaced by

The Radon measure R is more conveniently expressed in terms of a distribution function S, defined by the equations S(0)=0, R(]*a*,*b*])=S(*b*)−S(*a*). Let *δ*S(*y*):=S(*y*+0)−S(*y*−0)≡R({*y*}). In terms of the distribution function, equation (A3) assumes the form

The restriction of a Herglotz function to a segment ]*a*,*b*[ of the real axis such that is called a *Pick function on* ]*a*,*b*[. If ]*a*,*b*[ is non-empty and R vanishes on ]*a*,*b*[ then the function *f*(*z*) can be continued from to and the reflection principle *f*(*z*)*=*f*(*z**) holds. Analytic continuation across ]*a*,*b*[ is possible also if R is absolutely continuous on ]*a*,*b*[, R(d*y*)=K(*y*) d*y*. In this case
If , then B=0 and R is absolutely continuous.

We now turn to the case when the support of R does not intersect and for every **w**∈*S*. The Herglotz function *f* is then defined on . The integral in (A1) extends over . Setting *r*=−*y*, we have
A5
We now note that
The integrand tends to −1/[*r*(*r*^{2}+1)] and the integral tends to a finite limit
A6

if and only if

is finite for every **w**∈*S*. We now note that

while
Substituting equation (A6) in equation (A5) we have
A7
with a positive *T*-valued Radon measure Q(*d**r*):=*r*^{−1}R(−d*r*) satisfying the inequality
Hence, the restriction *f*(*x*) of *f*(*z*) to is a CBF. We have thus proved the following theorem:

### Theorem A.5.

*if and only if it has an analytic continuation to*, *which is a Herglotz function*.

According to theorem 2.24 can be expressed in the form *f*(*z*)=*z**g*(*z*), where .

### Corollary A.6.

*If**has the integral representation*
A8*and f*(*z*)=*z**g*(*z*) *then*
A9*and*
A10

### Proof.

Let be the distribution obtained by restricting the measure Q from to the space of smooth functions with compact support. The limit
in the distributions sense. Hence, , where ψ(*y*):=*y**q*(*y*) in the distributions sense, i.e.
for every . We now approximate the characteristic function χ_{[a,b]} by smooth functions
Note that
We now substitute *ϕ*_{η} for *ϕ* and pass to the limit . The Lebesgue Dominated Convergence Theorem yields equation (A10) upon changing the integration variable *y* to −*y* on both sides of the equation.

Equation (A9) is derived in a similar way from the identities

and Im *g*(*x*+iϵ)=−Im *g*(*x*−iϵ). ▪

### Corollary A.7.

*If**has an integral representation f*(*z*)=*z**g*(*z*) *where g is given by equation (A8) with a purely absolutely continuous measure*Q(d*r*)=U(*r*) d*r then*
A11

### Corollary A.8.

*If**has the integral representation (A8) with a purely absolutely continuous measure*Q(d*y*)=U(*y*) d*y*, *then*
A12

An alternative route to obtain inversion formulae leads through contour deformation in the inverse Laplace transformation formula applied to . Since the singularities of lie on the cut , the Bromwich contour can be deformed to the Hankel loop encircling the cut and a possible pole at 0. The inverse Laplace transform is thus transformed to a Laplace transform of a positive Radon measure R. The discontinuity of across the cut contributes the absolutely continuous spectrum while the poles of on contribute a discrete spectrum. The pole at 0 yields the instantaneous (elastic) modulus.

## Footnotes

- Received June 8, 2009.
- Accepted August 18, 2009.

- © 2009 The Royal Society