Complex Analysis of Real Functions II: Singular Schwartz Distributions
aa r X i v : . [ m a t h . C V ] F e b Complex Analysis of Real Functions
II: Singular Schwartz Distributions
Jorge L. deLyra ∗ Department of Mathematical PhysicsPhysics InstituteUniversity of S˜ao PauloMay 26, 2018
Abstract
In the context of the complex-analytic structure within the unit disk centered at theorigin of the complex plane, that was presented in a previous paper, we show thatsingular Schwartz distributions can be represented within that same structure, so longas one defines the limits involved in an appropriate way. In that previous paper itwas shown that essentially all integrable real functions can be represented within thecomplex-analytic structure. The infinite collection of singular objects which we analyzehere can thus be represented side by side with those real functions, thus allowing allthese objects to be treated in a unified way.
In a previous paper [1] we introduced a certain complex-analytic structure within the unitdisk of the complex plane, and showed that one can represent essentially all integrable realfunctions within that structure. In this paper we will show that one can represent withinthe same structure the singular objects known as distributions, loosely in the sense of theSchwartz theory of distributions [2, 3], which are also known as generalized real functions.All these objects will be interpreted as parts of this larger complex-analytic structure,within which they can be treated and manipulated in a robust and unified way.In Sections 2 and 3 we will establish the relation between the complex-analytic structureand the singular distributions. There we will show that one obtains these objects throughthe properties of certain limits to the unit circle, involving a particular set of inner analyticfunctions, which will be presented explicitly. Following what was shown in [1] for integrablereal functions, each singular distribution will be associated to a corresponding inner analyticfunction. In fact, we will show that the entire space of all singular Schwartz distributionsdefined within a compact domain is contained within this complex-analytic structure. InSection 4 we will analyze a certain collection of integrable real functions which are closelyrelated to the singular distributions, through the concept of infinite integral-differentialchains of functions.For ease of reference, we include here a one-page synopsis of the complex-analytic struc-ture introduced in [1]. It consists of certain elements within complex analysis [4], as wellas of their main properties. ∗ Email: [email protected] ynopsis: The Complex-Analytic StructureAn inner analytic function w ( z ) is simply a complex function which is analytic within theopen unit disk. An inner analytic function that has the additional property that w (0) = 0is a proper inner analytic function . The angular derivative of an inner analytic function isdefined by w · ( z ) = ı z dw ( z ) dz . (1)By construction we have that w · (0) = 0, for all w ( z ). The angular primitive of an inneranalytic function is defined by w − · ( z ) = − ı Z z dz ′ w ( z ′ ) − w (0) z ′ . (2)By construction we have that w − · (0) = 0, for all w ( z ). In terms of a system of polarcoordinates ( ρ, θ ) on the complex plane, these two analytic operations are equivalent todifferentiation and integration with respect to θ , taken at constant ρ . These two operationsstay within the space of inner analytic functions, they also stay within the space of properinner analytic functions, and they are the inverses of one another. Using these operations,and starting from any proper inner analytic function w · ( z ), one constructs an infinite integral-differential chain of proper inner analytic functions, n . . . , w − · ( z ) , w − · ( z ) , w − · ( z ) , w · ( z ) , w · ( z ) , w · ( z ) , w · ( z ) , . . . o . (3)Two different such integral-differential chains cannot ever intersect each other. There isa single integral-differential chain of proper inner analytic functions which is a constantchain, namely the null chain, in which all members are the null function w ( z ) ≡ w ( z ) at a point z onthe unit circle is a soft singularity if the limit of w ( z ) to that point exists and is finite.Otherwise, it is a hard singularity . Angular integration takes soft singularities to other softsingularities, and angular differentiation takes hard singularities to other hard singularities.Gradations of softness and hardness are then established. A hard singularity that be-comes a soft one by means of a single angular integration is a borderline hard singularity,with degree of hardness zero. The degree of softness of a soft singularity is the number ofangular differentiations that result in a borderline hard singularity, and the degree of hard-ness of a hard singularity is the number of angular integrations that result in a borderlinehard singularity. Singularities which are either soft or borderline hard are integrable ones.Hard singularities which are not borderline hard are non-integrable ones.Given an integrable real function f ( θ ) on the unit circle, one can construct from it aunique corresponding inner analytic function w ( z ). Real functions are obtained throughthe ρ → ( − ) limit of the real and imaginary parts of each such inner analytic function and,in particular, the real function f ( θ ) is obtained from the real part of w ( z ) in this limit. Thepair of real functions obtained from the real and imaginary parts of one and the same inneranalytic function are said to be mutually Fourier-conjugate real functions.Singularities of real functions can be classified in a way which is analogous to thecorresponding complex classification. Integrable real functions are typically associated withinner analytic functions that have singularities which are either soft or at most borderlinehard. This ends our synopsis. 2hen we discuss real functions in this paper, some properties will be globally assumed forthese functions, just as was done in [1]. These are rather weak conditions to be imposed onthese functions, that will be in force throughout this paper. It is to be understood, withoutany need for further comment, that these conditions are valid whenever real functionsappear in the arguments. These weak conditions certainly hold for any integrable realfunctions that are obtained as restrictions of corresponding inner analytic functions to theunit circle.The most basic condition is that the real functions must be measurable in the senseof Lebesgue, with the usual Lebesgue measure [5, 6]. The second global condition we willimpose is that the functions have no removable singularities. The third and last globalcondition is that the number of hard singularities on the unit circle be finite, and hencethat they be all isolated from one another. There will be no limitation on the number ofsoft singularities.The material contained in this paper is a development, reorganization and extension ofsome of the material found, sometimes still in rather rudimentary form, in the papers [7–11]. This is where we begin the discussion of inner analytic functions that have hard singularitieswith strictly positive degrees of hardness. Let us start by simply introducing a certainparticular inner analytic function of z . If z is a point on the unit circle, this function isdefined as a very simple rational function of z , w δ ( z, z ) = 12 π − π zz − z . (4)This inner analytic function has a single point of singularity, which is a simple pole at z .This is a hard singularity with degree of hardness equal to one. Our objective here is toexamine the properties of the real part u δ ( ρ, θ, θ ) of this inner analytic function, w δ ( z, z ) = u δ ( ρ, θ, θ ) + ı v δ ( ρ, θ, θ ) . (5)We will prove that in the ρ → ( − ) limit u δ ( ρ, θ, θ ) can be interpreted as a Schwartzdistribution [2,3], namely as the singular object known as the
Dirac delta “function” , whichwe will denote by δ ( θ − θ ). This object is also known as a generalized real function , sinceit is not really a real function in the usual sense of the term. In the Schwartz theory ofdistributions this object plays the role of an integration kernel for a certain distribution.Note that w δ ( z, z ) can, in fact, be written explicitly as a function of ρ and θ − θ . Sincewe have that z = ρ exp( ı θ ) and that z = exp( ı θ ), we have at once that w δ ( z, z ) = 12 π − π ρ e ı ( θ − θ ) ρ e ı ( θ − θ ) − . (6)The definition of the Dirac delta “function” is that it is a symbol for a limiting process,which satisfies certain conditions. In our case here the limiting process will be the limit ρ → ( − ) from the open unit disk to the unit circle. The limit of u δ ( ρ, θ, θ ) represents thedelta “function” in the sense that it satisfies the conditions that follow.1. The defining limit of δ ( θ − θ ) tends to zero when one takes the ρ → ( − ) limit whilekeeping θ = θ . 3. The defining limit of δ ( θ − θ ) diverges to positive infinity when one takes the ρ → ( − ) limit with θ = θ .3. In the ρ → ( − ) limit the integral Z ba dθ δ ( θ − θ ) = 1 (7)has the value shown, for any open interval ( a, b ) which contains the point θ .4. Given any continuous integrable function g ( θ ), in the ρ → ( − ) limit the integral Z ba dθ g ( θ ) δ ( θ − θ ) = g ( θ ) (8)has the value shown, for any open interval ( a, b ) which contains the point θ .This is the usual form of this condition, when it is formulated in strictly real terms.However, we will impose a slight additional restriction on the real functions g ( θ ), byassuming that the limit to the point z on the unit circle that corresponds to θ , ofthe corresponding inner analytic function w γ ( z ), exists and is finite. This implies that w γ ( z ) may have at z a soft singularity, but not a hard singularity.Note that, although it is customary to list both separately, the third condition is in factjust a particular case of the fourth condition. It is also arguable that the second conditionis not really necessary, because it is a consequence of the others. We may therefore considerthat the only really essential conditions are the first and the fourth ones.The functions g ( θ ) are sometimes named test functions within the Schwartz theory ofdistributions [2, 3]. The additional part of the fourth condition, that the limit to the point z of the corresponding inner analytic function w γ ( z ) must exist and be finite, consists ofa weak limitation on these test functions, and does not affect the definition of the singulardistribution itself. This is certainly the case for our definition here, since we define thisobject through a definite and unique inner analytic function.In this section we will prove the following theorem. Theorem 1:
The ρ → ( − ) limit of the real part of the inner analytic function w δ ( z, z ) converges to the generalized function δ ( θ − θ ) . Before we attempt to prove this theorem, our first task is to write explicitly the real andimaginary parts of w δ ( z, z ). In order to do this we must now rationalize it, w δ ( z, z ) = 12 π − π z ( z ∗ − z ∗ )( z − z )( z ∗ − z ∗ )= 12 π − π ρ − ρ cos(∆ θ ) − ı ρ sin(∆ θ ) ρ − ρ cos(∆ θ ) + 1 , (9)where ∆ θ = θ − θ . We must examine the real part of this function, u δ ( ρ, θ, θ ) = 12 π − π ρ [ ρ − cos(∆ θ )]( ρ + 1) − ρ cos(∆ θ ) . (10)We are now ready to prove the theorem, which we will do by simply verifying all theproperties of the Dirac delta “function”. 4 roof 1.1 : If we take the limit ρ → ( − ) , under the assumption that ∆ θ = 0, we getlim ρ → ( − ) u δ ( ρ, θ, θ ) = 12 π − π − cos(∆ θ )2 − θ )= 0 , (11)which is the correct value for the case of the Dirac delta “function”. Thus we see that thefirst condition is satisfied.If, on the other hand, we calculate u δ ( ρ, θ, θ ) for ∆ θ = 0 and ρ < u δ ( ρ, θ , θ ) = 12 π − π ρ ( ρ − ρ − = 12 π − π ρρ − , (12)which diverges to positive infinity as ρ → ( − ) , as it should in order to represent the singularDirac delta “function”. This establishes that the second condition is satisfied.We now calculate the real integral of u δ ( ρ, θ, θ ) over the circle of radius ρ <
1, which isgiven by Z π − π dθ ρ u δ ( ρ, θ, θ ) = 12 π Z π − π dθ ρ (cid:26) − ρ [ ρ − cos(∆ θ )]( ρ + 1) − ρ cos(∆ θ ) (cid:27) = ρ π Z π − π d (∆ θ ) (cid:0) − ρ (cid:1) ( ρ + 1) − ρ cos(∆ θ )= (cid:0) − ρ (cid:1) π Z π − π d (∆ θ ) 1[( ρ + 1) / (2 ρ )] − cos(∆ θ ) , (13)since d (∆ θ ) = dθ . This real integral over ∆ θ can be calculated by residues. We introducean auxiliary complex variable ξ = λ exp( ı ∆ θ ), which becomes simply exp( ı ∆ θ ) on the unitcircle λ = 1. We have dξ = ı ξd (∆ θ ), and so we may write the integral on the right-handside as Z π − π d (∆ θ ) 1[(1 + ρ ) / (2 ρ )] − cos(∆ θ ) = I C dξ ı ξ ρ ) /ρ ] − ξ − /ξ = 2 ı I C dξ − [(1 + ρ ) /ρ ] ξ + ξ , (14)where the integral is now over the unit circle C in the complex ξ plane. The two roots ofthe quadratic polynomial on ξ in the denominator are given by ξ (+) = 1 /ρ,ξ ( − ) = ρ. (15)Since ρ <
1, only the simple pole corresponding to ξ ( − ) lies inside the integration contour,so we get for the integral Z π − π dθ ρ ) / (2 ρ )] − cos(∆ θ ) = 2 ı (2 π ı ) lim ξ → ρ ξ − /ρ = 4 π ρ (1 − ρ ) . (16)5t follows that we have for the real integral in Equation (13) Z π − π dθ ρ u δ ( ρ, θ, θ ) = (cid:0) − ρ (cid:1) π π ρ (1 − ρ )= ρ, (17)and thus we have that the integral is equal to 1 in the ρ → ( − ) limit. Once we have thisresult, and since according to the first condition the integrand goes to zero everywhere onthe unit circle except at ∆ θ = 0, which is the same as θ = θ , the integral can be changedto one over any open interval ( a, b ) on the unit circle containing the point θ , without anychange in its limiting value. This establishes that the third condition is satisfied.In order to establish the validity of the fourth and last condition, we consider an essentiallyarbitrary integrable real function g ( θ ), with the additional restriction that it be continuousat the point z . As was established in [1], it corresponds to an inner analytic function w γ ( z ) = u γ ( ρ, θ ) + ı v γ ( ρ, θ ) , (18)where we also assume that g ( θ ) is such that w γ ( z ) may have at z a soft singularity, butnot a hard singularity, so that its limit to z exists. We now consider the following realintegral over the circle of radius ρ < Z π − π dθ ρ u γ ( ρ, θ ) u δ ( ρ, θ, θ ) = 12 π Z π − π dθ ρ u γ ( ρ, θ ) (cid:26) − ρ [ ρ − cos(∆ θ )]( ρ + 1) − ρ cos(∆ θ ) (cid:27) = ρ π Z π − π d (∆ θ ) u γ ( ρ, θ ) (cid:0) − ρ (cid:1) ( ρ + 1) − ρ cos(∆ θ )= (cid:0) − ρ (cid:1) π Z π − π d (∆ θ ) u γ ( ρ, θ )[( ρ + 1) / (2 ρ )] − cos(∆ θ ) , (19)since d (∆ θ ) = dθ . This real integral over ∆ θ can be calculated by residues, exactly likethe one in Equation (13) which appeared before in the case of the third condition. Thecalculation is exactly the same except for the extra factor of u γ ( ρ, θ ) to be taken intoconsideration when calculating the residue, so that we may write directly that Z π − π d (∆ θ ) u γ ( ρ, θ )[( ρ + 1) / (2 ρ )] − cos(∆ θ ) = 2 ı (2 π ı ) lim ξ → ρ u γ ( ρ, θ ) ξ − /ρ = 4 π ρ (1 − ρ ) lim ξ → ρ u γ ( ρ, θ ) . (20)Note now that since ξ = λ exp( ı ∆ θ ), and since we must take the limit ξ → ρ , we in facthave that in that limit λ e ı ∆ θ = ρ, (21)which implies that λ = ρ and that ∆ θ = 0, and therefore that θ = θ . We must thereforewrite u γ ( ρ, θ ) at the point given by ρ and θ , thus obtaining Z π − π d (∆ θ ) u γ ( ρ, θ )[( ρ + 1) / (2 ρ )] − cos(∆ θ ) = 4 π ρ (1 − ρ ) u γ ( ρ, θ ) . (22) Post-publication note: it is important to observe here that, from its very beginning, this argument goesthrough without any change if we replace u γ ( ρ, θ ) directly by g ( θ ) in the integrand. Hence, the theorem istrue regardless of whether or not the real function g ( θ ) is represented by an inner analytic function.
6t follows that we have for the real integral in Equation (19) Z π − π dθ ρ u γ ( ρ, θ ) u δ ( ρ, θ, θ ) = (cid:0) − ρ (cid:1) π π ρ (1 − ρ ) u γ ( ρ, θ )= ρu γ ( ρ, θ ) . (23)Finally, we may now take the ρ → ( − ) limit, since w γ ( z ) and thus u γ ( ρ, θ ) are well definedat z in that limit, and thus obtainlim ρ → ( − ) Z π − π dθ ρ u γ ( ρ, θ ) u δ ( ρ, θ, θ ) = u γ (1 , θ ) ⇒ Z π − π dθ u γ (1 , θ ) (cid:20) lim ρ → ( − ) u δ ( ρ, θ, θ ) (cid:21) = u γ (1 , θ ) ⇒ Z π − π dθ g ( θ ) (cid:20) lim ρ → ( − ) u δ ( ρ, θ, θ ) (cid:21) = g ( θ ) , (24)since u γ ( ρ, θ ) converges to g ( θ ), in the ρ → ( − ) limit, almost everywhere on the unitcircle. Just as before, once we have this result, and since according to the first conditionthe integrand goes to zero everywhere on the unit circle except at ∆ θ = 0, which is thesame as θ = θ , the integral can be changed to one over any open interval on the unit circlecontaining the point θ , without any change in its value. This establishes that the fourthand last condition is satisfied.Having established all the properties, we may now write symbolically that δ ( θ − θ ) = lim ρ → ( − ) u δ ( ρ, θ, θ ) . (25)This concludes the proof of Theorem 1.It is important to note that, when we adopt as the definition of the Dirac delta “function”the ρ → ( − ) limit of the real part of the inner analytic function w δ ( z, z ), the limitationsimposed on the test functions g ( θ ) and on the corresponding inner analytic functions w γ ( z )become irrelevant. In fact, this definitions stands by itself, and is independent of any set oftest functions. Given any integrable real function f ( θ ) and the corresponding inner analyticfunction w ( z ) with real part u ( ρ, θ ), we may always assemble the real integral over a circleof radius ρ < Z π − π dθ ρ u ( ρ, θ ) u δ ( ρ, θ, θ ) , (26)which is always well defined within the open unit disk. It then remains to be verified onlywhether or not the ρ → ( − ) limit of this integral exists, in order to define the correspondingintegral Z π − π dθ f ( θ ) δ ( θ − θ ) . (27)This limit may exist for functions that do not satisfy the conditions imposed on the testfunctions. In fact, one can do this for the real part of any inner analytic function, regardlessof whether or not it corresponds to an integrable inner analytic function, so long as the ρ → ( − ) limit of u ( ρ, θ ) exists almost everywhere. Whenever the ρ → ( − ) limit of theintegral exists, it defines the action of the delta “function” on that particular real object.7t is also interesting to observe that the Dirac delta “function”, although it is notsimply a conventional integrable real function, is in effect an integrable real object, evenif it corresponds to an inner analytic functions that has a simple pole at z , which isa non-integrable hard singularity, with degree of hardness equal to one. This apparentcontradiction is explained by the orientation of the pole at z = z . If we consider thereal part u δ ( ρ, θ ) of the inner analytic function w δ ( z, z ), although it is not integrable alongcurves arriving at the singular point from most directions, there is one direction, that of theunit circle, along which one can approach the singular point so that u δ ( ρ, θ ) is identicallyzero during the approach, which allows us to define its integral using the ρ → ( − ) limit .The same is not true, for example, for the imaginary part v δ ( ρ, θ ) of the same inner analyticfunction, which generates the Fourier-conjugate function to the delta “function”, and thatdiverges to infinity as 1 / | z − z | when one approaches the singular point along the unitcircle, thus generating a non-integrable real function in the ρ → ( − ) limit.In the development presented in [1] the real functions were represented by their Fouriercoefficients, and the inner analytic functions by their Taylor coefficients. We can easily dothe same here, if we observe that the inner analytic function w δ ( z, z ) in Equation (4) isthe sum of a geometric series, w δ ( z, z ) = 12 π + 1 π z/z − z/z = 12 π + 1 π ∞ X k =1 (cid:18) zz (cid:19) k = 12 π + 1 π ∞ X k =1 [cos( kθ ) − ı sin( kθ )] z k . (28)This power series is the Taylor series of w δ ( z, z ) around the origin, and therefore it followsthat the Taylor coefficients of this inner analytic function are given by c = 12 π ,c k = cos( kθ ) π − ı sin( kθ ) π , (29)where k ∈ { , , , . . . , ∞} . Since according to the construction presented in [1] we havethat c = α / c k = α k − ı β k , we have for the Fourier coefficients of the delta“function” α = 1 π ,α k = cos( kθ ) π ,β k = sin( kθ ) π , (30)where k ∈ { , , , . . . , ∞} . Note that these are in fact the results one obtains via theintegrals defining the Fourier coefficients [12], Post-publication note: this characterizes the inner analytic function w δ ( z, z ) associated to the Diracdelta “function” as an irregular inner analytic function, since it is not integrable around its singular point,which is a hard singular point with degree of hardness 1, while the corresponding real object is. k = 1 π Z π − π dθ cos( kθ ) δ ( θ − θ ) ,β k = 1 π Z π − π dθ sin( kθ ) δ ( θ − θ ) , (31)by simply using the fundamental property of the delta “function”.Having established the representation of the Dirac delta “function” within the structureof the inner analytic functions, in sequence we will show that the Dirac delta “function”is not the only singular distribution that can be represented by an inner analytic function.As we will see, one can do the same for its first derivative, and in fact for its derivativesof any order. This is an inevitable consequence of the fact that the proper inner analyticfunction w · δ ( z, z ) associated to w δ ( z, z ) is a member of an integral-differential chain. The derivatives of the Dirac delta “function” are defined in a way which is similar to thatof the delta “function” itself. The first condition is the same, and the second and thirdconditions are not really required. The crucial difference is that the fourth condition in thedefinition of the Dirac delta “function” is replaced by the second condition in the list thatfollows. The “function” δ n ′ ( θ − θ ) is the n th derivative of δ ( θ − θ ) with respect to θ if itsdefining limit ρ → ( − ) satisfies the two conditions that follow.1. The defining limit of δ n ′ ( θ − θ ) tends to zero when one takes the ρ → ( − ) limit whilekeeping θ = θ .2. Given any integrable real function g ( θ ) which is differentiable to the n th order, in the ρ → ( − ) limit the integral Z ba dθ g ( θ ) δ n ′ ( θ − θ ) = ( − n g n ′ ( θ ) (32)has the value shown, for any open interval ( a, b ) which contains the point θ , where g n ′ ( θ ) is the n th derivative of g ( θ ) with respect to θ .This is the usual form of this condition, when it is formulated in strictly real terms.However, we will impose a slight additional restriction on the real functions g ( θ ), byassuming that the limit to the point z on the unit circle that corresponds to θ , ofthe n th angular derivative of the corresponding inner analytic function w γ ( z ), existsand is finite. Since these proper inner analytic functions are all in the same integral-differential chain, this implies that the limits to z of all the inner analytic functions w m · γ ( z ) exist, for all 0 ≤ m ≤ n .The second condition above is, in fact, the fundamental property of each derivative of thedelta “function”, including the “function” itself in the case n = 0. Just as in the case ofthe delta “function” itself, the additional part of the second condition, involving the inneranalytic function w γ ( z ), consists of a weak limitation on the test functions g ( θ ), and doesnot affect the definition of the singular distributions themselves. This is certainly the casefor our definitions here, since we define each one of these objects through a definite andunique inner analytic function.In this section we will prove the following theorem.9 heorem 2: For every strictly positive integer n there exists an inner analytic function w δ n ′ ( z, z ) whose real part, in the ρ → ( − ) limit, converges to δ n ′ ( θ − θ ) . Before we attempt to prove this theorem, let us note that the proof relies on a propertyof angular differentiation, which was established in [1], namely that angular differentiationis equivalent to partial differentiation with respect to θ , at constant ρ . When we take the ρ → ( − ) limit, this translates to the fact that taking the angular derivative of the inneranalytic function w ( z ) within the open unit disk corresponds to taking the derivative withrespect to θ , on the unit circle, of the corresponding real object.If this derivative cannot be taken directly on the unit circle, then one can define itby taking the angular derivative of the corresponding inner analytic function and thenconsidering the ρ → ( − ) limit of the real part of the resulting function. Since analyticfunctions can be differentiated any number of times, the procedure can then be iteratedin order to define all the higher-order derivatives with respect to θ on the unit circle.Equivalently, one can just consider traveling along the integral-differential chain indefinitelyin the differentiation direction.Consider therefore the integral-differential chain of proper inner analytic functions thatis obtained from the proper inner analytic function associated to w δ ( z, z ), that is, theunique integral-differential chain to which the proper inner analytic function w · δ ( z, z ) = w δ ( z, z ) − π = − π zz − z (33)belongs. Consider in particular the set of proper inner analytic functions which is obtainedfrom w · δ ( z, z ) in the differentiation direction along this chain, for which we have w n · δ ( z, z ) = u n ′ δ ( ρ, θ, θ ) + ı v n ′ δ ( ρ, θ, θ )= ∂ n ∂θ n u δ ( ρ, θ, θ ) + ı ∂ n ∂θ n v δ ( ρ, θ, θ ) , (34)for all strictly positive n , where we recall that w δ ( z, z ) = u δ ( ρ, θ, θ ) + ı v δ ( ρ, θ, θ ) . (35)We will now prove that in the ρ → ( − ) limit we have δ n ′ ( θ − θ ) = lim ρ → ( − ) u n ′ δ ( ρ, θ, θ ) , (36)for n ∈ { , , , . . . , ∞} , or, equivalently, that we have for the inner analytic function w δ n ′ ( z, z ) associated to the derivative δ n ′ ( θ − θ ) w δ n ′ ( z, z ) = w n · δ ( z, z ) , (37)for n ∈ { , , , . . . , ∞} . We are now ready to prove the theorem, as stated in Equation (36).Let us first prove, however, that the first condition holds for all the derivatives of the delta“function”. 10 roof 2.1 : Since w δ ( z, z ) has a single singular point at z , the same is true for all its angular deriva-tives. Therefore the ρ → ( − ) limit of all the angular derivatives exists everywhere withinthe open interval of the unit circle that excludes the point θ . Since u δ (1 , θ, θ ) is identi-cally zero within this interval, and since angular differentiation within the open unit diskcorresponds to differentiation with respect to θ on the unit circle, so that we have u n ′ δ (1 , θ, θ ) = ∂ n ∂θ n u δ (1 , θ, θ ) , (38)for n ∈ { , , , . . . , ∞} , it follows at once that u n ′ δ (1 , θ, θ ) = 0 ⇒ lim ρ → ( − ) u n ′ δ ( ρ, θ, θ ) = 0 , (39)for n ∈ { , , , . . . , ∞} , everywhere but at the singular point θ , for all values of n . Thisestablishes that the first condition holds.Let us now prove that the second condition, which relates directly to the singular point,holds, leading to the result as stated in Equation (36). Proof 2.2 :
In order to do this, we start with the case n = 1, and consider the following real integralon the circle of radius ρ <
1, which we integrate by parts, noting that the integrated termis zero because we are integrating on a circle, Z π − π dθ u γ ( ρ, θ ) (cid:20) ∂∂θ u δ ( ρ, θ, θ ) (cid:21) = − Z π − π dθ (cid:20) ∂∂θ u γ ( ρ, θ ) (cid:21) u δ ( ρ, θ, θ ) , (40)where w γ ( z ) = u γ ( ρ, θ ) + ı v γ ( ρ, θ ) is the inner analytic function associated to g ( θ ). Notethat the partial derivatives involved certainly exist, since both u δ ( ρ, θ, θ ) and u γ ( ρ, θ ) arethe real parts of inner analytic functions. If we now take the ρ → ( − ) limit, on theright-hand side we recover the Dirac delta “function” on the unit circle, and therefore wehave Z π − π dθ g ( θ ) (cid:20) lim ρ → ( − ) ∂∂θ u δ ( ρ, θ, θ ) (cid:21) = − Z π − π dθ (cid:20) ddθ g ( θ ) (cid:21) δ ( θ − θ )= ( − g ′ ( θ ) , (41)so long as g ( θ ) is differentiable, were we used the fundamental property of the Dirac delta“function”. We thus obtain the relation for the derivative of the delta “function”, Z π − π dθ g ( θ ) δ ′ ( θ − θ ) = ( − g ′ ( θ ) , (42)where δ ′ ( θ − θ ) = lim ρ → ( − ) ∂∂θ u δ ( ρ, θ, θ ) . (43)We may therefore write that 11 ′ ( θ − θ ) = lim ρ → ( − ) u ′ δ ( ρ, θ, θ ) . (44)In this way we have obtained the result for δ ′ ( θ − θ ) by using the known result for δ ( θ − θ ).We may now repeat this procedure to obtain the result for δ ′ ( θ − θ ) from the result for δ ′ ( θ − θ ), and therefore from the result for δ ( θ − θ ). We simply consider the following realintegral on the circle of radius ρ <
1, which we again integrate by parts, recalling that theintegrated term is zero because we are integrating on a circle, Z π − π dθ u γ ( ρ, θ ) (cid:20) ∂∂θ u ′ δ ( ρ, θ, θ ) (cid:21) = − Z π − π dθ (cid:20) ∂∂θ u γ ( ρ, θ ) (cid:21) u ′ δ ( ρ, θ, θ ) . (45)If we now take the ρ → ( − ) limit, on the right-hand side we recover the first derivative ofthe Dirac delta “function” on the unit circle, and therefore we have Z π − π dθ g ( θ ) (cid:20) lim ρ → ( − ) ∂∂θ u ′ δ ( ρ, θ, θ ) (cid:21) = − Z π − π dθ (cid:20) ddθ g ( θ ) (cid:21) δ ′ ( θ − θ )= ( − g ′ ( θ ) , (46)so long as g ( θ ) is differentiable to second order, were we used the fundamental property ofthe first derivative of the Dirac delta “function”. We thus obtain the relation for the secondderivative of the delta “function”, Z π − π dθ g ( θ ) δ ′ ( θ − θ ) = ( − g ′ ( θ ) , (47)where δ ′ ( θ − θ ) = lim ρ → ( − ) ∂ ∂θ u δ ( ρ, θ, θ ) . (48)We may therefore write that δ ′ ( θ − θ ) = lim ρ → ( − ) u ′ δ ( ρ, θ, θ ) . (49)Clearly, this procedure can be iterated n times, thus resulting in the relation δ n ′ ( θ − θ ) = lim ρ → ( − ) u n ′ δ ( ρ, θ, θ ) , (50)for n ∈ { , , , . . . , ∞} . Note that all the derivatives with respect to θ involved in theargument exist, for arbitrarily high orders, since both u δ ( ρ, θ, θ ) and u γ ( ρ, θ ) are the realparts of inner analytic functions, and thus are infinitely differentiable on both arguments.We may now formalize the proof using finite induction. We thus assume the results forthe case n − δ ( n − ′ ( θ − θ ) = lim ρ → ( − ) u ( n − ′ δ ( ρ, θ, θ ) , Z ba dθ g ( θ ) δ ( n − ′ ( θ − θ ) = ( − n − g ( n − ′ ( θ ) , (51)and proceed to examine the next case. We consider therefore the following real integralon the circle of radius ρ <
1, which we integrate by parts, recalling once more that theintegrated term is zero because we are integrating on a circle,12 π − π dθ u γ ( ρ, θ ) (cid:20) ∂∂θ u ( n − ′ δ ( ρ, θ, θ ) (cid:21) = − Z π − π dθ (cid:20) ∂∂θ u γ ( ρ, θ ) (cid:21) u ( n − ′ δ ( ρ, θ, θ ) . (52)If we now take the ρ → ( − ) limit, on the right-hand side we recover the ( n − th derivativeof the Dirac delta “function” on the unit circle, and therefore we have Z π − π dθ g ( θ ) (cid:20) lim ρ → ( − ) ∂∂θ u ( n − ′ δ ( ρ, θ, θ ) (cid:21) = − Z π − π dθ (cid:20) ddθ g ( θ ) (cid:21) δ ( n − ′ ( θ − θ )= ( − n g n ′ ( θ ) , (53)so long as g ( θ ) is differentiable to order n , were we used the fundamental property of the( n − th derivative of the Dirac delta “function”. We thus obtain the relation for the n th derivative of the delta “function”, Z π − π dθ g ( θ ) δ n ′ ( θ − θ ) = ( − n g n ′ ( θ ) , (54)where δ n ′ ( θ − θ ) = lim ρ → ( − ) ∂ n ∂θ n u δ ( ρ, θ, θ ) . (55)We may therefore write that, by finite induction, δ n ′ ( θ − θ ) = lim ρ → ( − ) u n ′ δ ( ρ, θ, θ ) , (56)for n ∈ { , , , . . . , ∞} . We have therefore completed the proof of Theorem 2.It is important to note that, just as in the case of the Dirac delta “function”, when weadopt as the definition of the n th derivative of the delta “function” the ρ → ( − ) limit ofthe real part of the inner analytic function w n · δ ( z, z ), for n ∈ { , , , . . . , ∞} , the limitationsimposed on the test functions g ( θ ) and on the corresponding inner analytic functions w γ ( z )become irrelevant. In fact, these definitions stand by themselves, and are independent ofany set of test functions. Not only one can use them for any inner analytic functions derivedfrom integrable real functions, but one can do this for any inner analytic function w ( z ),regardless of whether or not it corresponds to an integrable real function, so long as the ρ → ( − ) limit of the corresponding real part u ( ρ, θ ) exists almost everywhere. Just as inthe case of the Dirac delta “function”, whenever the ρ → ( − ) limit of the real integral Z π − π dθ ρ u ( ρ, θ ) u n ′ δ ( ρ, θ, θ ) , (57)exists, it defines the action of the n th derivative of the delta “function” on that particularreal object.It is also interesting to observe that, just as in the case of the Dirac delta “function”,it is true that its derivatives of all orders, although they are not simply integrable realfunctions, are in fact integrable real objects, even if they are related to inner analyticfunctions with non-integrable hard singularities. Just as is the case for the inner analyticfunction associated to the delta “function” itself, the poles of the proper inner analyticfunctions associated to the derivatives are always oriented in such a way that one canapproach the singularities along the unit circle while keeping the real parts of the functions13qual to zero, a fact that allows one to define the integrals on θ of the real parts via the ρ → ( − ) limit . Just as in the case of the delta “function”, the Fourier-conjugate functionsof the derivatives are simply non-integrable real functions. This fact provides the first hintthat there must be some relation of such non-integrable real functions with correspondinginner analytic functions.In the development presented in [1] the real functions were represented by their Fouriercoefficients, and the inner analytic functions by their Taylor coefficients. The same can bedone in our case here. Starting from the power series for w · δ ( z, z ) given in Equation (28),we can see that the definition of the angular derivative implies that we have for the inneranalytic functions associated to the derivatives of the delta “function”, w n · δ ( z, z ) = 1 π ∞ X k =1 ı n k n [cos( kθ ) − ı sin( kθ )] z k , (58)for n ∈ { , , , . . . , ∞} , so that the corresponding Taylor coefficients are given by c ( n )0 = 0and c ( n ) k = ı n k n π [cos( kθ ) − ı sin( kθ )] , (59)for n ∈ { , , , . . . , ∞} , and where k ∈ { , , , . . . , ∞} . The identification of the Fouriercoefficients α ( n ) k and β ( n ) k will now depend on the parity of n .Once we have the Dirac delta “function” and all its derivatives, both as inner analyticfunctions and as the corresponding real objects, we may consider collections of such singularobjects, with their singularities located at all the possible points of the periodic interval[ − π, π ], as well as arbitrary linear combinations of some or all of them. There is a well-knowntheorem of the Schwartz theory of distributions [2, 3] which states that any distributionwhich is singular at a given point θ can be expressed as a linear combination of the Diracdelta “function” δ ( θ − θ ) and its derivatives of arbitrarily high orders δ n ′ ( θ − θ ).Since, as was observed in [1], the set of all inner analytic functions forms a vector spaceover the field of complex numbers, it is immediately apparent that we may assemble suchlinear combinations within the space of inner analytic functions. Therefore, the set ofdistributions formed by the delta “functions” and all their derivatives, as defined here, withtheir singularities located at all possible points of the unit circle, constitutes a complete basisthat spans the space of all possible singular Schwartz distributions defined in a compactdomain. We may conclude therefore that the whole space of Schwartz distributions in acompact domain is contained within the set of inner analytic functions.It is interesting to note that, since we have the inner analytic function that correspondsto the delta “function” in explicit form, we are in a position to perform simple calculationsin order to obtain in explicit form the inner analytic functions that correspond to the firstfew derivatives of the delta “function”. For example, a few simple and straightforwardcalculations lead to the following proper inner analytic functions, w δ ′ ( z, z ) = − π ı zz ( z − z ) ,w δ ′ ( z, z ) = − π ı z ( z + z ) z ( z − z ) , Post-publication note: this characterizes all the inner analytic functions w n · δ ( z, z ) associated to thederivatives of the Dirac delta “function” as irregular inner analytic functions, since they are not integrablearound their singular points, which are hard singular points with degrees of hardness n + 1, while thecorresponding real objects are. δ ′ ( z, z ) = − π ı z (cid:0) z + 4 zz + z (cid:1) z ( z − z ) . (60)These proper inner analytic functions are all very simple rational functions of the complexvariable z , which can be written as functions of only z/z , and hence as functions of only ρ and θ − θ . Note that we can induce from these examples that the n th derivative of thedelta “function” is indeed represented by an inner analytic function with a pole of order n + 1 on the unit circle, which is thus a hard singularity with degree of hardness n + 1, asone would expect from the structure of the corresponding integral-differential chain. It is important to note that the Dirac delta “function” and all its derivatives, with theirsingularities located at a given point z on the unit circle, are all contained within a singleintegral-differential chain, making up, in fact, only a part of that chain, the semi-infinitechain starting from the delta “function” and propagating indefinitely in the differentiationdirection along the chain. However, the chain propagates to infinity in both directions. Inorder to complete its analysis, we must now determine what is the character of the realobjects in the remaining part of that chain, in the integration direction. In fact, they arejust integrable real functions, although they do have a specific character. They consist ofsections of polynomials wrapped around the unit circle, of progressively higher orders, andprogressively smoother across the singular point, as functions of θ , as one goes along theintegral-differential chain in the integration direction.Let us illustrate this fact with a few simple examples. Instead of performing angularintegrations of the inner analytic functions, we will do this by performing integrations onthe unit circle. As was established in [1], one can determine these functions by simpleintegration on θ , so long as one remembers two things: first, to make sure that the realfunctions or related objects to be integrated on θ have zero average over the unit circle, andsecond, to choose the integration constant so that the resulting real functions also have zeroaverage over the unit circle. For example, the integral of the zero-average delta “function” δ ′ ( θ − θ ) = δ ( θ − θ ) − π , (61)which is obtained from the real part of the proper inner analytic function in Equation (33),can be integrated by means of the simple use of the fundamental property of the delta“function”, thus yielding δ − ′ (∆ θ ) = 12 − ∆ θ π for ∆ θ > ,δ − ′ (∆ θ ) = − − ∆ θ π for ∆ θ < , (62)where ∆ θ = θ − θ . This is a sectionally linear function, with a single section consisting ofthe intervals [ − π,
0) and (0 , π ], thus excluding the point ∆ θ = 0 where the singularity lies,and with a unit-height step discontinuity at that point. Note that it is an odd function of∆ θ . The next case can now be calculated by straightforward integration, which yields δ − ′ (∆ θ ) = − π θ − ∆ θ π for ∆ θ > ,δ − ′ (∆ θ ) = − π − ∆ θ − ∆ θ π for ∆ θ < . (63)15his is a sectionally quadratic function, this time a continuous function, again with thesame single section excluding the point ∆ θ = 0, but now with a point of non-differentiabilitythere. Note that it is an even function of ∆ θ . The next case yields, once more by straight-forward integration, δ − ′ (∆ θ ) = − π ∆ θ θ − ∆ θ π for ∆ θ > ,δ − ′ (∆ θ ) = − π ∆ θ − ∆ θ − ∆ θ π for ∆ θ < . (64)This is a sectionally cubic continuous and differentiable function, again with the same singlesection excluding the point ∆ θ = 0. Note that it is an odd function of ∆ θ . The trend isnow quite clear. All the real functions in the chain, in the integration direction startingfrom the delta “function”, are what we may call piecewise polynomials , even if we havejust a single piece within a single section of the unit circle, as is the case here. The n th integral is a piecewise polynomial of order n , which has zero average over the unit circle,and which becomes progressively smoother across the singular point as one goes along theintegral-differential chain in the integration direction.In order to generalize this analysis, we must now consider linear superpositions of delta“functions” and derivatives of delta “functions”, with their singularities situated at vari-ous points on the unit circle. A simple example of such a superposition, which we mayuse to illustrate what happens when we make one, is that of two delta “functions”, withsingularities at θ = 0 and at θ = ± π , added together with opposite signs, f ( θ ) = δ ( θ ) − δ ( θ − π ) , (65)that corresponds to the following inner analytic function, which this time is already a properinner analytic function, with two simple poles at z = ± w ( z ) = − π zz − π zz + 1= − π zz − . (66)Since we have now two singular points, one at z = 1 and another at z = −
1, correspondingrespectively to θ = 0 and θ = ± π , we have now two sections, one in ( − π,
0) and anotherin (0 , π ). The inner analytic functions at the integration side of the integral-differentialchain to which this function belongs are obtained by simply adding the correspondinginner analytic functions at the integration sides of the integral-differential chains of the twofunctions that are superposed. The same is true for the corresponding real objects withineach section of the unit circle. Since the real functions corresponding to each one of the twodelta “functions” that were superposed are zero-average piecewise polynomials, so are thereal functions corresponding to the superposition. For example, it is not difficult to showthat the first integral is the familiar square wave, with amplitude 1 / f − ′ ( θ ) = 12 for θ > ,f − ′ ( θ ) = −
12 for θ < , (67)which is a piecewise linear function with two sections, having unit-height step discontinuitieswith opposite signs at the two singular points θ = 0 and θ = ± π .We want to determine what is the character of the real functions, in the integrationside of the resulting integral-differential chain, in the most general case, when we consider16rbitrary linear superpositions of a finite number of delta “functions” and derivatives ofdelta “functions”, with their singularities situated at various points on the unit circle.From the examples we see that, when we superpose several singular distributions with theirsingularities at various points, the complete set of all the singular points defines a new setof sections. Given one of these singular points, since at least one of the distributions beingsuperposed is singular at that point, in general so is the superposition. Let there be N ≥ { θ , . . . , θ N } in the superposition. It follows that in general we end up witha set of N contiguous sections, consisting of open intervals between singular points, thatcan be represented as the sequence (cid:8) ( θ , θ ) , . . . , ( θ i − , θ i ) , ( θ i , θ i +1 ) , . . . , ( θ N , θ ) (cid:9) , (68)where we see that the sequence goes around the unit circle, and where we adopt the con-vention that each section ( θ i , θ i +1 ) is numbered after the singular point θ i at its left end. Inaddition to this, since for each one of the distributions being superposed the real functionson the integration side of the integral-differential chain of the corresponding delta “func-tion” are piecewise polynomials, and since the sum of any finite number of polynomialsis also a polynomial, so are the real functions of the integral-differential chain to whichthe superposition belongs, if we are at a point in that integral-differential chain where allsingular distributions have already been integrated out. Let us establish a general notationfor these piecewise polynomial real functions, as well as a formal definition for them. Definition 1 :
Piecewise Polynomial Real Functions
Given a real function f ( n ) ( θ ) that is defined in a piecewise fashion by polynomials in N ≥ N singular points θ i , with i ∈ { , . . . , N } , so that the polynomial P ( n i ) i ( θ ) at the i th section has order n i , we denotethe function by f ( n ) ( θ ) = n P ( n i ) i ( θ ) , i ∈ { , . . . , N } o , (69)where n is the largest order among all the N orders n i . We say that f ( n ) ( θ ) is a piecewisepolynomial real function of order n .Note that, being made out of finite sections of polynomials, the real function f ( n ) ( θ ) isalways an integrable real function. In fact, it is also analytic within each section, so thatthe N singularities described above are the only singularities involved. Since f ( n ) ( θ ) is anintegrable real function, let w ( z ) be the inner analytic function that corresponds to thisintegrable real function, as constructed in [1]. The ( n +1) th angular derivative of w ( z ) is theinner analytic function w ( n +1) · ( z ), which corresponds therefore to the ( n + 1) th derivativeof f ( n ) ( θ ) with respect to θ , that we denote by f ( n +1) ′ ( n ) ( θ ).In this section we will prove the following theorem. Theorem 3:
If the real function f ( n ) ( θ ) is a non-zero piecewise polynomial function oforder n , defined in N ≥ sections of the unit circle, with the exclusion of a finite non-emptyset of N singular points θ i , then and only then the derivative f ( n +1) ′ ( n ) ( θ ) is the superpositionof a non-empty set of delta “functions” and derivatives of delta “functions” on the unitcircle, with the singularities located at some of the points θ i , and of nothing else. roof 3.1 : In order to prove this, first let us note that the derivative f ( n +1) ′ ( n ) ( θ ) is identically zero withinall the open intervals defining the sections. This is so because the maximum order of allthe polynomials involved is n , and the ( n + 1) th derivative of a polynomial of order equalto or less than n is identically zero, f ( n +1) ′ ( n ) ( θ ) = 0 for all θ = θ i , i ∈ { , . . . , N } . (70)We conclude, therefore, that the real object represented by the inner analytic function w ( n +1) · ( z ) has support only at the N isolated singular points θ i , thus implying that it cancontain only singular distributions.Second, let us prove that the derivative cannot be identically zero over the whole unitcircle. In order to do this we note that one cannot have a non-zero piecewise polynomialreal function of order n , such as the one described above, that is also continuous anddifferentiable to the order n on the whole unit circle. This is so because this hypothesiswould lead to an impossible integral-differential chain.If this were possible, then starting from a non-zero real function f ( n ) ( θ ) that correspondsto a non-zero inner analytic function w ( z ), and after a finite number n of steps along thedifferentiation direction of the corresponding integral-differential chain, one would arriveat a real function that is continuous over the whole unit circle, that is constant withineach section and that has zero average over the whole unit circle. It follows that such afunction would have to be identically zero, thus corresponding to the inner analytic function w ( z ) ≡
0. But this is not possible, because this inner analytic function belongs to anotherchain, the one which is constant, all members being w ( z ) ≡
0, and we have shown in [1]that two different integral-differential chains cannot intersect.It follows that f ( n ) ( θ ) can be globally differentiable at most to order n −
1, so that the n th derivative is a discontinuous function, and therefore the ( n + 1) th derivative already givesrise to singular distributions. Therefore, every real function that is piecewise polynomial onthe unit circle, of order n , when differentiated n +1 times, so that it becomes zero within theopen intervals corresponding to the existing sections, will always result in the superpositionof some non-empty set of singular distributions with their singularities located at the pointsbetween two consecutive sections.We can also establish that only functions of this form give rise to such superpositions ofsingular distributions and of nothing else. The necessity of the fact that the real functionson integral-differential chains generated by superpositions of singular distributions mustbe piecewise polynomials comes directly from the fact that all such distributions and allsuch superpositions of distributions are zero almost everywhere, in fact everywhere but attheir singular points. Due to this, it is necessary that these real functions, upon a finite number n + 1 of differentiations, become zero everywhere strictly within the sections, thatis, within the open intervals between two successive singularities. Therefore, within eachopen interval the condition over the sectional real function at that interval is that d ( n +1) dθ ( n +1) f i ( θ ) ≡ , (71)and the general solution of this ordinary differential equation of order n + 1 is a polynomialof order n i ≤ n , containing at most n + 1 non-zero arbitrary constants, f i ( θ ) = P ( n i ) i ( θ ) . (72)18ince only polynomials have the property of becoming identically zero after a finite num-ber of differentiations, it is therefore an absolute necessity that these real functions bepolynomials within each one of the sections. This completes the proof of Theorem 3.Note that the inner analytic function w ( n +1) · ( z ) corresponding to f ( n +1) ′ ( n ) ( θ ) representstherefore the superposition of a non-empty set of singular distributions with their singular-ities located at the singular points. In other words, after n + 1 angular differentiations of w ( z ), which correspond to n +1 straight differentiations with respect to θ of the polynomials P ( n i ) i ( θ ) within the sections, one is left with an inner analytic function w ( n +1) · ( z ) whosereal part converges to zero in the ρ → ( − ) limit, at all points on the unit circle which arenot one of the N singular points.It is interesting to note that, since we have the inner analytic function that correspondsto the Dirac delta “function” in explicit form, it is not difficult to calculate directly its firstangular primitive. A few simple and straightforward calculations lead to w − · δ ( z, z ) = ı π Z z dz ′ z ′ z ′ − z = ı π ln (cid:18) z − zz (cid:19) . (73)This inner analytic function has a logarithmic singularity at z , which is a borderline hardsingularity. Note that, as expected, we have that w − · δ (0 , z ) = 0. In the Schwartz theory of distributions one important theorem states that it is not possibleto define, in a general way, the product of two distributions [13], which has the effect thatthe space of Schwartz distributions cannot be promoted from a vector space to an algebra.In this section we will interpret this important fact in the context of the representationof integrable real functions and singular distributions in terms of inner analytic functions.We start by noting that, although it is always possible to define the product of two inneranalytic functions, which is always an inner analytic function itself, this does not correspondto the product of the two corresponding real functions or related objects. If we have twoinner analytic functions given by w ( z ) = u ( ρ, θ ) + ı v ( ρ, θ ) ,w ( z ) = u ( ρ, θ ) + ı v ( ρ, θ ) , (74)the product of the two inner analytic functions is given by w ( z ) = [ u ( ρ, θ ) u ( ρ, θ ) − v ( ρ, θ ) v ( ρ, θ )]+ ı [ u ( ρ, θ ) v ( ρ, θ ) + v ( ρ, θ ) u ( ρ, θ )] , (75)whose real part is not just the product u ( ρ, θ ) u ( ρ, θ ). In fact, the problem of finding aninner analytic function whose real part is this quantity often has no solution. One can seethis very simply by observing that both u ( ρ, θ ) and u ( ρ, θ ) are always harmonic functions,and that the product of two harmonic functions in general is not a harmonic function. Sincethe real and imaginary parts of an inner analytic function are always harmonic functions,it follows that the problem posed in this way cannot be solved in general. The only simple19ase in which we can see that the problem has a solution is that in which one of the twofunctions being multiplied is a constant function.Let us state in a general way the problem of the definition of the product of two distri-butions. Suppose that we have two inner analytic functions such as those in Equation (74).The two corresponding real objects are u (1 , θ ) and u (1 , θ ), and their product, assumingthat it can be defined in strictly real terms, is simply the real object u (1 , θ ) u (1 , θ ). Theproblem of finding an inner analytic function that corresponds to this product is the prob-lem of finding an harmonic function u π ( ρ, θ ) whose limit to the unit circle results in thisreal object, u π (1 , θ ) = u (1 , θ ) u (1 , θ ) . (76)If one can find such a harmonic function, then it is always possible to find its harmonicconjugate function v π ( ρ, θ ) and therefore to determine the inner analytic function w π ( z ) = u π ( ρ, θ ) + ı v π ( ρ, θ ) , (77)which corresponds to the product of the two real objects. According to the constructionpresented in [1], this can always be done so long as the product u (1 , θ ) u (1 , θ ) is anintegrable real function of θ . However, if u (1 , θ ) and u (1 , θ ) are singular objects thatcan only be defined as limits from within the open unit disk, then the product may not bedefinable in strictly real terms, and it may not be possible to find an inner analytic functionsuch that the ρ → ( − ) limit of its real part results in this product, interpreted in someconsistent way. This is the content of the theorem that states that this cannot be done ingeneral.It is not too difficult to give examples of products which are not well defined. It sufficesto consider the product of any two singular distributions which have their singularities atthe same point on the unit circle. If one considers integrating the resulting object andusing for this purpose the fundamental property of any of the two distributions involved,one can see that the integral is not well defined in the context of the definitions given herefor the singular distributions. Although one cannot rule out that some other definition canbe found to include some such cases, we certainly do not have one at this time.We thus see that we are in fact unable to promote the whole space of integrable realfunctions and singular distributions to an algebra. However, there are some sub-spaceswithin which this can be done. Under some circumstances one can solve the problem ofdefining within the complex-analytic structure the product of two integrable real functions.This cannot be done for the whole sub-space of integrable real functions, because there is thepossibility that the product of two integrable real function will not be integrable. However,if we restrict the sub-space to those integrable real functions which are also limited, thenit can be done. This is so because the product of two limited integrable real functions isalso a limited real function, and therefore integrable. In this way, one can find the inneranalytic function that corresponds to the product, since according to the construction whichwas presented in [1], this can be done for any integrable real function. The resulting inneranalytic function will not, however, be related in a simple way to the inner analytic functionsof the two factor functions.One case in which the product can always be defined is that of an integrable real functionwith a Dirac delta “function”, so long as the real function is well defined at the singularpoint of the delta “function”. Given the nature of the delta “function”, this is equivalentto multiplying it by a mere real number, the value of the integrable real function at thesingular point of the delta “function”, 20 ( θ ) δ ( θ − θ ) = g ( θ ) δ ( θ − θ ) . (78)The corresponding inner analytic function is therefore given simply by g ( θ ) w δ ( z, z ). Sim-ilar statements are true, of course, for all the derivatives of the delta “function”. Therefore,in all such cases there is no difficulty in determining the inner analytic function that corre-sponds to the product.Note that this difficulty relates only to the definition of the product of two real objectson the unit circle. As was observed before, for all the singular distributions their definitionby means of inner analytic functions always provides the means to determine whether ornot they can be applied to a given real object, so long as it is represented by an inneranalytic function, and determines what results from that operation, if it is possible at all. We have extended the close and deep relationship established in a previous paper [1], be-tween integrable real functions and complex analytic functions in the unit disk centeredat the origin of the complex plane, to include singular distributions. This close relation-ship between, on the one hand, real functions and related objects, and on the other hand,complex analytic functions, allows one to use the powerful and extremely well-known ma-chinery of complex analysis to deal with the real functions and related objects in a veryrobust way, even if these objects are very far from being analytic. The concept of integral-differential chains of proper inner analytic functions, which we introduced in that previouspaper, played a central role in the analysis presented.One does not usually associate non-differentiable, discontinuous and unbounded realfunctions with single analytic functions. Therefore, it may come as a bit of a surprise that,as was established in [1], all integrable real functions are given by the real parts of certaininner analytic functions on the open unit disk when one approaches the unit circle. Thissurprise is now compounded by the fact that inner analytic functions can represent singulardistributions as well and, in fact, can represent what may be understood as a complete setof such singular objects.There are many more inner analytic functions within the open unit disk than thosethat were examined here and in [1], in relation to integrable real functions and singulardistributions. Therefore, it may be possible to further generalize the relationship betweenreal objects on the unit circle and inner analytic functions. For example, we have observed inthis paper that there are inner analytic functions whose real parts converge to non-integrablereal functions on the unit circle. Simple examples are the inner analytic functions given by¯ w δ n ′ ( z, z ) = − ı w δ n ′ ( z, z ) , (79)for n ∈ { , , , , . . . , ∞} , that correspond to the non-integrable real functions which arethe Fourier-conjugate functions of the Dirac delta “function” and its derivatives. Thissuggests that we consider the question of how far this can be generalized, that is, of whatis the largest set of non-integrable real functions that can be represented by inner analyticfunctions. This issue will be discussed in the fourth paper of this series.The singular distributions are integrable real objects associated to non-integrable sin-gularities of the corresponding inner analytic functions, a fact which is made possible bythe orientation of the singularities with respect to the direction along the unit circle. Thissuggests that the most general definition of such singular distributions may be formulatedin terms of the type and orientation of the singularities present on the unit circle. In this21ase one would expect that singular distributions would be associated to inner analyticfunctions with hard singularities that are oriented in a particular way, so that the integralsof their real parts can be defined via limits from the open unit disk to the unit circle .We believe that the results presented here establish a new perspective for the representa-tion and manipulation of singular distributions. It might also constitute a simpler and morestraightforward way to formulate and develop the whole theory of Schwartz distributionswithin a compact domain. Acknowledgments
The author would like to thank his friend and colleague Prof. Carlos Eugˆenio ImbassayCarneiro, to whom he is deeply indebted for all his interest and help, as well as his carefulreading of the manuscript and helpful criticism regarding this work.
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