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Gauge Institute JournalH. Vic DannonPeriodic Delta Function,and Poisson Integral for AbelSummation of Fourier Series,H. Vic Dannonvic0@comcast.netJune, 2012Abstract The Poisson Integral Theorem supplies the conditionsunder which the Poisson Integral atr =1 −dr, associated with afunctionf ( x ) equals f ( x ).It is believed to hold in the Calculus of Limits under someconditions. In fact,The Theorem cannot be proved in the Calculus of Limitsunder any conditions,because it requires integration of the singular Poisson Kernel.In Infinitesimal Calculus, the Poisson Kerneldr(2 − dr)12 22(1 −dr)(1 −cos πξ [ − x]) + ( dr)is the Periodic Delta Function,δPeriodic( ξ− x) = ... + δξ ( − x + 2) + δξ ( − x) + δξ ( −x− 2) +...δ ( ξ − x)violates the Calculus of Limits ConditionsPeriodic1

Gauge Institute JournalH. Vic Dannon The Hyper-realδ( x), is not defined in the Calculus of Limits,andδ ( x)is not integrable in any bounded interval.1(2δ( x + 0) + δ( x − 0) ) = 0does not replaceδ( x)at itsdiscontinuity point, x = 0 .ButδPeriodic( ξ − x)equals the Poisson Integral associated with it atr= 1 −dr.The Poisson Integral associated with any hyper-real periodicintegrable f ( x ) atr = 1 −dr , equals f ( x ).Keywords: Infinitesimal, Infinite-Hyper-Real, Hyper-Real,infinite Hyper-real, Infinitesimal Calculus, Delta Function,Fourier Transform, Periodic Delta Function, Delta Comb, FourierSeries, Poisson Kernel, Poisson Integral2000 Mathematics Subject Classification 26E35; 26E30;26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15;46S20; 97I40; 97I30.2

Gauge Institute JournalH. Vic DannonContents0. The Origin of the Poisson Integral Theorem1. Divergence of the Poisson Kernel in the Calculus of Limits2. Hyper-real line.3. Integral of a Hyper-real Function4. Delta Function5. Periodic Delta Function, δ ( ξ − x)6. Convergent SeriesPeriodic7. Poisson Sequence and δ ( ξ − x)Periodic8. Poisson Kernel and δ ( ξ − x)Periodic9. Poisson Integral and δ ( ξ − x)Periodic10. Poisson Integral TheoremReferences3

Gauge Institute JournalH. Vic Dannon= + z(1 + z + z + ...)12= 1 +1z1 −z2=1 1+z21−z==1 12112+ re− reiϕiϕiϕ1+ re 1−reiϕ1−re1−re−iϕ−iϕ=21 r2 21− + 2irsinϕ1+ r −2rcosϕ21− rri2 2 21 sinϕ= +1+ r − 2r cosϕ1+ r −2rcosϕThe real part isThat is,1 2+1r cosϕ+ r cos 2 ϕ + ... =2 2 21 − r1+ r −2rcosϕ21 2 −2iϕ −iϕ iϕ 2 2iϕ1 1 − r2{ .. + r e + re + 1 + re + r e + ..}=2 2Thus, the Fourier Series may have Abel Sumξ=1∫ξ=−1f211 − r( ξ)lim r ↑12 12 2 cos π (dξ+ r − r x −ξ)21+ r −2rcosϕThe Poisson Integral associated with f ( x ) at r = 1− 0 + .6

Gauge Institute JournalH. Vic Dannonξ= 12ξ=1211−r11−rlim f ( ξ) dξ = f( ξ) limr ↑12 2 22ξ=− 1 ξ=−1r ↑1∫ ∫ dξ1+ r −2r cos π( x − ξ) 1+ r −2r cos π( x −ξ)Poisson Kernel.Asr ↑ 1, the Poisson Sequence becomes the Poisson Kernel, whichis singular, and diverges at any ξ − x = 2m.x − ξ = 2m ⇒ cos π( x − ξ ) = 1,21r12 221− 1+⇒ =1+ r −2r cos π( x −ξ)1 −rr,11+1⎯⎯⎯→ =r ↑1 20 +∞Thus, the Poisson Integral diverges in the Calculus of Limits.Avoiding the singularity atξ = x , by using the Cauchy PrincipalValue of the integral does not recover the Theorem, because at anyξ −x≠2m,2 211−r11−r=2 2 2 2 1221+ r −2r cos π( x −ξ) (1 − r) + 4r sin [ π(x −ξ)]0⎯⎯⎯→ =2sin [ π( x − ξ)]r ↑1 2 120That is, the Poisson Kernel vanishes, and the integral isidentically zero, for any function f ( x ).Plots of the Poisson Sequence confirm thatIn the Calculus of Limits,the Poisson Integral Kernel is either singular or zero9

Gauge Institute JournalH. Vic Dannon1.2 Plots of Poisson Sequence with1r = 1 −For k = 37,kkplots the spikes at x = 0 , x =−2, x = 2, x =−4, x = 4For k = 49,gives 7 spikes10

Gauge Institute JournalH. Vic DannonThus, the Poisson Integral Theorem does not hold in the Calculusof Limits.1.3 Infinitesimal Calculus SolutionBy resolving the problem of the infinitesimals [Dan2], we obtainedthe Infinite Hyper-reals that are strictly smaller than ∞ , andconstitute the value of the Delta Function at the singularity.The controversy surrounding the Leibnitz Infinitesimals derailedthe development of the Infinitesimal Calculus, and the DeltaFunction could not be defined and investigated properly.In Infinitesimal Calculus, [Dan3], we can differentiate over jumpdiscontinuities, and integrate over singularities.11

Gauge Institute JournalH. Vic DannonThe Delta Function, the idealization of an impulse in Radarcircuits, is a Discontinuous Hyper-Real function which definitionrequires Infinite Hyper-reals, and which analysis requiresInfinitesimal Calculus.In [Dan5], we show that in infinitesimal Calculus, the hyper-realω=∞1( x)e i ωδ = x ω2π∫ dω=−∞is zero for any x ≠ 0 ,it spikes atx = 0 , so that its Infinitesimal Calculusx =∞∫integral is δ( xdx ) = 1,andx =−∞1δ (0) = < ∞.dxHere, we show that in Infinitesimal calculus, the Poisson Kernel isa periodic hyper-real Delta Function: A periodic train of DeltaFunctions. And the Poisson Integral associated with a Hyper-realperiodic function f ( x ), atr = 1 −dr , equals f ( x ).12

Gauge Institute JournalH. Vic Dannon2.Hyper-real LineEach real number α can be represented by a Cauchy sequence ofrational numbers, ( r , r , r ,...) so that r → α .1 2 3The constant sequence ( ααα , , ,...) is a constant hyper-real.In [Dan2] we established that,1. Any totally ordered set of positive, monotonically decreasingnto zero sequencesinfinitesimal hyper-reals.( ι1, ι2, ι3,...)constitutes a family of2. The infinitesimals are smaller than any real number, yetstrictly greater than zero.1 1 13. Their reciprocals ( , , ,...ι 1ι 2ι 3) are the infinite hyper-reals.4. The infinite hyper-reals are greater than any real number,yet strictly smaller than infinity.5. The infinite hyper-reals with negative signs are smallerthan any real number, yet strictly greater than −∞.6. The sum of a real number with an infinitesimal is anon-constant hyper-real.7. The Hyper-reals are the totality of constant hyper-reals, afamily of infinitesimals, a family of infinitesimals with13

Gauge Institute JournalH. Vic Dannonnegative sign, a family of infinite hyper-reals, a family ofinfinite hyper-reals with negative sign, and non-constanthyper-reals.8. The hyper-reals are totally ordered, and aligned along aline: the Hyper-real Line.9. That line includes the real numbers separated by the nonconstanthyper-reals. Each real number is the center of aninterval of hyper-reals, that includes no other real number.10. In particular, zero is separated from any positive realby the infinitesimals, and from any negative real by theinfinitesimals with negative signs, −dx .11. Zero is not an infinitesimal, because zero is not strictlygreater than zero.12. We do not add infinity to the hyper-real line.13. The infinitesimals, the infinitesimals with negativesigns, the infinite hyper-reals, and the infinite hyper-realswith negative signs are semi-groups withrespect to addition. Neither set includes zero.14. The hyper-real line is embedded in , and is not ∞homeomorphic to the real line. There is no bi-continuousone-one mapping from the hyper-real onto the real line.14

Gauge Institute JournalH. Vic Dannon15. In particular, there are no points on the real line thatcan be assigned uniquely to the infinitesimal hyper-reals, orto the infinite hyper-reals, or to the non-constant hyperreals.16. No neighbourhood of a hyper-real is homeomorphic toan nball. Therefore, the hyper-real line is not a manifold.17. The hyper-real line is totally ordered like a line, but itis not spanned by one element, and it is not one-dimensional.15

Gauge Institute JournalH. Vic Dannon3.Integral of a Hyper-real FunctionIn [Dan3], we defined the integral of a Hyper-real Function.Let f () x be a hyper-real function on the interval [ ab] , .The interval may not be bounded.f () x may take infinite hyper-real values, and need not bebounded.At eacha≤x≤b,there is a rectangle with basedx dx[ x − , x + 2], height f () x , and area2f ( xdx. )We form the Integration Sum of all the areas for the x ’s thatstart at x = a, and end at x = b,∑ f ( xdx ) .x∈[ a, b]If for any infinitesimal dx , the Integration Sum has the samehyper-real value, then f () x is integrable over the interval [ ab] , .Then, we call the Integration Sum the integral of f () x from x = a,to x= b, and denote it by16

Gauge Institute JournalH. Vic Dannonx=b∫ f ( xdx ) .x=aIf the hyper-real is infinite, then it is the integral over [, ab] ,If the hyper-real is finite,x=b∫ fxdx ( ) = real part of the hyper-real . x=a3.1 The countability of the Integration SumIn [Dan1], we established the equality of all positive infinities:We proved that the number of the Natural Numbers,Card , equals the number of Real Numbers,2 Card Card = , andwe have2 Card2Card Card = ( Card) = .... = 2 = 2 = ... ≡ ∞.In particular, we demonstrated that the real numbers may bewell-ordered.Consequently, there are countably many real numbers in theinterval [ ab] , , and the Integration Sum has countably many terms.While we do not sequence the real numbers in the interval, thesummation takes place over countably many f ( xdx. )The Lower Integral is the Integration Sum where f ( x ) is replaced17

Gauge Institute JournalH. Vic Dannonby its lowest value on each interval3.2∑x∈[ a, b]⎛⎜⎝dx dx2 2[ x − , x + ]⎞inf f ( t)dx⎠⎟x− ≤t≤ x+dx dx2 2The Upper Integral is the Integration Sum where f ( x ) is replacedby its largest value on each interval3.3∑x∈[ a, b]⎛⎜⎝dx dx2 2[ x − , x + ]⎞ sup f ( t)dx⎠⎟x− ≤t≤ x+dx dx2 2If the integral is a finite hyper-real, we have3.4 A hyper-real function has a finite integral if and only if itsupper integral and its lower integral are finite, and differ by aninfinitesimal.18

Gauge Institute JournalH. Vic Dannon4.Delta FunctionIn [Dan5], we have defined the Delta Function, and established itsproperties1. The Delta Function is a hyper-real function defined from thehyper-real line into the set of two hyper-reals⎧⎪ 1 ⎫⎨0, ⎪⎬⎪⎩dx ⎪ . The⎭hyper-real0 is the sequence 0, 0, 0,... . The infinite hyperreal1dxdepends on our choice of dx .2. We will usually choose the family of infinitesimals that isspanned by the sequences1n , 12n,1n3,… It is asemigroup with respect to vector addition, and includes allthe scalar multiples of the generating sequences that arenon-zero. That is, the family includes infinitesimals withnegative sign. Therefore,1dxwill mean the sequence n .Alternatively, we may choose the family spanned by thesequences12 n ,13 n ,14 n ,… Then, 1dxwill mean the19

Gauge Institute JournalH. Vic Dannonsequence 2 n. Once we determined the basic infinitesimaldx , we will use it in the Infinite Riemann Sum that definesan Integral in Infinitesimal Calculus.3. The Delta Function is strictly smaller than ∞4. We define,1χ δ ( x) ≡ dx ( ),dx xdx⎡ ⎤ ,⎢−⎣ 2 2 ⎥⎦whereχ ⎡⎢−⎣dx,dx2 2⎧ dx dx1, x ∈ ⎡−, ⎤( x)= ⎪ ⎢ 2 2 ⎥⎨ ⎣ ⎦ .⎪⎪ 0, otherwise⎩⎤⎥⎦5. Hence, for x < 0 , δ ( x) = 0 at fordxx =− , δ( x)jumps from 0 to2dx dx⎢ ⎣,2 2 ⎥ ⎦ , 1( x)x ∈ ⎡−⎤δ = .dx1dx , at x = 0 ,δ (0) =1dx atdxx = , δ( x)drops from2 for x > 0 , δ ( x) = 0. xδ ( x) = 01dx to 0.6. If dx =1, ( x) = 1 1( x),2 1 1( x),3 1 1( x )...n[ − , ] [ − , ] [ − , ]δ χ χ χ2 2 4 4 6 67. If dx =2,n1 2 3δ ( x) = , , ,...2 2 22 cosh x 2 cosh 2x 2 cosh 3x20

Gauge Institute JournalH. Vic Dannon8. If dx =1,n− x − 2x − 3x[0, ∞) [0, ∞) [0, ∞)δ( x) = e χ ,2 e χ , 3 e χ ,...x =∞∫9. δ( xdx ) = 1.x =−∞In [Dan6], we obtained10.k =∞1 −ik( ξ−x)δξ ( − x)= e2π∫ dkk =−∞In [Dan8], we defined the Periodic Delta Function, and obtained11.δ Periodic( x ) = ... + δ ( x + 4) + δ ( x + 2) + δ ( x ) + δ ( x − 2) + δ ( x − 4) + ...1 −inπx 1 −i πx 1 1 iπx 1 inπxe e e e2 2 2 2 2= ... + + ... + + + + ... + + ...21

Gauge Institute JournalH. Vic Dannon5.Periodic Delta Function δ ( ξ − x)5.1 Periodic Delta FunctionPeriodicδPeriodic( ξ− x) = ... + δξ ( − x + 2) + δξ ( − x) + δξ ( −x− 2) +...is a periodic hyper-real Delta function, with period T = 2 .In [Dan8], we obtained1 inπξ( x) 1 iπξ( x)2 2− − − −δPeriodic( ξ − x) = ... + e + ... + e1 1 i πξ ( − x ) 1 in πξ ( − x )ee2 2 2+ + + ... + + ...22

Gauge Institute JournalH. Vic Dannon6.Convergent SeriesIn [Dan11], we defined convergence of infinite series inInfinitesimal Calculus6.1 Sequence Convergence to a finite hyper-real aa → a iff a − a = infinitesimal .nn6.2 Sequence Convergence to an infinite hyper-real Aa → A iff anrepresents the infinite hyper-real A.n6.3 Series Convergence to a finite hyper-real sa1 + a2 + ... → s iff a1 + ... + an− s = infinitesimal .6.4 Series Convergence to an Infinite Hyper-real Sa1 + a2 + ... → Siffa1+ ... +anrepresents the infinite hyper-real S .23

Gauge Institute JournalH. Vic Dannon7.Poisson Sequence and δ ( ξ − x)Periodic7.1 Poisson Sequence DefinitionLetr = 1 − , k = 1, 2, 3,...k1kThe Sequence of Poisson Integrals at r = 11 − ,kkξ=1∫ξ=−1f211 − r() ξdξ2 12+ r − 2 r cos π ( x −ξ)r =− 11k==ξ=1∫ξ=−1k − 1f () ξdξ.2( kk− 1) + 1 − 2( kk− 1)cos πξ ( −x)Poisson Sequencegives rise to the Poisson SequenceP ( ξ − x)=kk − 12( kk− 1) + 1−2( kk−1)cos πξ ( − x).7.2 Poisson Sequence is a Periodic Delta Sequence andrepresents the Periodic Delta Function,δPeriodic( ξ− x) = ... + δξ ( − x + 2) + δξ ( − x) + δξ ( −x− 2) +...24

Gauge Institute JournalH. Vic DannonEachP ( x)k=k − 12( kk− 1) + 1−2( kk−1)cosπx, k = 1, 2, 3,...1. has the sifting property on each interval,x =−3∫.. P ( x) dx = 1; P ( x) dx = 1; P ( x) dx = 1..x =−52. is a continuous functionkx =−1∫x =−3kx = 1∫x =−13. peaks on each of these interval to lim P ( ξ − x ) = k −1.Proof of (1)x= 1 x=1ξ−x→2m1Pk( x) dx = ( k −1)dx2( kk− 1) + 1−2( kk−1)cosπx∫ ∫ x=− 1 x=−1By [Spiegel, p.78],p−q2⎡ p − q= ( k −1) arctan tana ( p − q)( p + q)⎢⎣p + qk12a⎤ax⎥⎦kx = 1x =−121= ( k −1) arctan[ 4 k( k − 1) + 1tan πx]2π 4( kk− 1) + 12k−11/ πx = 1x =−11 1= 1 {arctan[(2 k − 1)tan π ] − arctan[(2 1)tan( )]}2 k − ππ − 2= 1.Proof of (3)1π−12 2π25

Gauge Institute JournalH. Vic DannonAs ξ −x→ 2m,k −1 k −1→ =2 kk ( − 1) + 1−2 kk ( −1)cosπx 2 kk ( − 1) + 1−2 kk ( −1)k− 1.7.3 Poisson Sequence Represents δ ( ξ − x)PeriodicδPeriodic( ξ − x)=k − 12( kk−1)[1−cos πξ ( − x)] + 126

Gauge Institute JournalH. Vic Dannon8.Poisson Kernel and δ ( ξ − x)Periodic8.1 Poisson Kernel in the Calculus of LimitsThe Sequence of Poisson Integrals at1r = 1 −kkξ=1 211 − rk∫ f ξdξk→∞ 2 2r1krkπ x ξξ =−+ − −r =− 11lim ( )1 2 cos ( )k k==ξ=1∫ξ=−1k − 1f () ξdξ.2( kk− 1) + 1 − 2( kk− 1)cos π ( x−ξ)Poisson Sequencegives rise to a Poisson Sequence.Alternatively, we could use r = 1 − 1, r = 1 − 1,…23kkkkNote that the hyper-realr =1 −drincludes all such sequences.The limit of the Poisson Sequence is the Poisson Kernelk − 1klim→∞ 2( kk− 1)[1 − cos π ( x− ξ )] + . 18.2 In the Calculus of Limits, the Poisson Kernel does not haveProof:the sifting property27

Gauge Institute JournalH. Vic DannonFor ξ − x = 2m,k −1 k −1=2 kk ( −1)[1 −cos π( x− ξ)] + 1 2 kk ( −1)[1 − cos( π2 m)] + 1= k −1→ ∞.k→∞8.3 Hyper-real Poisson Kernel in Infinitesimal CalculusPoisson⎧ k 1 , ξ x 2( ξ x)⎪− − = m− = ⎨ .⎪ 0, ξ − x ≠ 2m⎪⎩Proof: At any ξ − x = 2m, the Kernel is an infinite hyper-real.At any ξ −x≠ 2m,k − 1 1= =2( kk−1)[1−cos πξ ( − x)] + 1 2[1 k −cos πξ ( − x)]+1k−1infinitesimal .8.4 Let1N = be an infinite Hyper-real, ThendrPoisson( ξ − x)==211 − r2 21−2r cos πξ ( − x)+ rr=−1 drdr(2 − dr)12 22(1 −dr)(1 −cos πξ [ − x]) + ( dr)= ... + δξ ( − x + 2) + δξ ( − x) + δξ ( −x− 2) + ...= δ ( ξ −x).periodicProof:28

Gauge Institute JournalH. Vic DannonPoisson( ξ −x)≡==211 − r2 21−2r cos πξ ( − x)+ rr=−1 dr211 −(1 −dr)2 21 −2(1 −dr)cos πξ [ − x] + (1 −dr)dr(2 − dr)12 22(1 −dr)(1 −cos πξ [ − x]) + ( dr)For x = 2m,Poisson(2 m)==dr(2 − dr)12 22(1 −dr)(1 − cos π2 m) + ( dr)dr(2 − dr)12 2( dr)For any x ≠ 2m,= 1 1dr− . 2Poisson( x)=dr(2 − dr)12 22(1 −dr)(1 − cos πx) + ( dr)≈122dr2(1 − cos πx)= infinitesimal .Therefore,Poisson⎧ 1 1 1, ξ − x = − 2 ⎧ , ξ = x ⎧, ξ − x = 2( ξ − x) = .. + dr dr dr⎨⎪ + ⎨⎪ + ⎨⎪+ ..⎪ 0, ξ −x ≠ −2 ⎪ 0, ξ ≠ x ⎪ 0, ξ −x≠ 2⎩ ⎩ ⎩29

Gauge Institute JournalH. Vic Dannon= ... + δξ ( − x + 2) + δξ ( − x) + δξ ( −x− 2) + ...= δ ( ξ − x).Periodic8.5.. + δξ ( − x + 4) + δξ ( − x + 2) + δξ ( − x) + δξ ( −x − 2) + δξ ( −x− 4) + .. ==dr(2 − dr)12 22(1 −dr)(1 −cos πξ [ − x]) + ( dr)30

Gauge Institute JournalH. Vic Dannon9.Poisson Integral and δ ( ξ − x)Periodic9.1 Poisson Integral of a Hyper-real Function f ( x )Let f ( x ) be a hyper-real function integrable on [ −1,1], so thatf (1) = f ( − 1) .Then, for eachn = ..., −3, −2, −1, 0,1,2, 3,... , the integrals12ξ=1∫ξ=−1−inπξf () ξ e dξ≡ c nexist, [Dan8], with finite, or infinite hyper-real values. Thec narethe Fourier Coefficients off ( x ).The Fourier Series associated with f ( x ) isi( −n) πx i( −1) πx i(1) πx i( n)πx−1 0 1n.. + c e + .. + c e + c + ce + .. + c e + .. =−nIt may equal The Poisson Integral associated with f ( x ) atr= 1 −dr,PoissonSξ=121 1 − r{()} ≡ ∫ () ξ2 21 1 + r − 2 r cos πξ ( −xξ=−)fx f dξr=−1 dr=ξ=1∫ξ=−1dr(2 − dr)f () ξ2(1 −dr )(1 − cos πξ [ − x ]) + ( dr )12 2dξFor each x , it may assume finite or infinite hyper-real values.31

Gauge Institute JournalH. Vic Dannon9.2 P S{ x }Proof:δ ( ξ − ) = δ ( ξ −x)oisson Periodic Periodicξ = 11dr(2 − dr)P S{ δ ( ξ − x) } = ∫ δ ( ξ) dξ2 2(1 )(1 cos [ ]) ( )2−dr − πξ− x + droisson Periodic Periodicξ=−1ξ = 11dr(2 − dr)= ∫ {.. + δξ ( + 2) + δξ ( ) + δξ ( − 2) + ..} dξ2 2(1 )(1 cos [ ]) ( )2−dr − πξ− x + drξ=−11dr(2 − dr)= ... + +2 22(1 −dr)(1 − cos π[2 + x]) + ( dr)1dr(2 − dr)+ +2 22(1 −dr)(1 − cos πx) + ( dr)1dr(2 − dr)+ + ..2 22(1 −dr)(1 −cos π[2 − x]) + ( dr)= ... + .. + δ( x + 2) + δ( x) + δ( x − 2) + ..= δ Periodic( x ) . 32

Gauge Institute JournalH. Vic Dannon10.Poisson Integral TheoremThe Poisson Integral Theorem for a hyper-real function f ( x ), isthe Fundamental Theorem for Abel Summation of Fourier series.It supplies the conditions under which the Poisson Integralassociated with f ( x ) atr = 1 −dr , equals f ( x ).It is believed to hold in the Calculus of Limits under someConditions. In fact,The Theorem cannot be proved in the Calculus of Limitsunder any conditions,because it requires integration of the singular Poisson Kernel.10.1 Poisson Integral Theorem cannot be proved inthe Calculus of LimitsProof:In the Calculus of Limits,ξ= 12ξ=1211−r11−rlim f ( ξ) dξ = f( ξ) limr ↑12 2 22πξξ=− 1 ξ=−1r ↑1∫ ∫ dξ1+ r −2r cos ( − x) 1+ r −2r cos πξ ( −x)Poisson Kernel.Asr ↑ 1, the Poisson Sequence becomes the Poisson Kernel, whichis singular, and diverges at any ξ − x = 2m.33

Gauge Institute JournalH. Vic Dannonξ− x = 2m ⇒ cos π( ξ − x)= 1,21r12 221− 1+r⇒ =1+ r −2r cos πξ ( −x)1 − r,11+1⎯⎯⎯→ =r ↑1 20 +∞Thus, the Poisson Integral diverges in the Calculus of Limits.Avoiding the singularity atξ = x , by using the Cauchy PrincipalValue of the integral does not recover the Theorem, because at anyξ −x≠2m,2 211−r11−r=2 2 2 2πξ1221+ r −2r cos ( −x) (1 − r) + 4r sin [ πξ ( −x)]0⎯⎯⎯→ =2sin [ πξ ( − x)]r ↑1 2 120That is, the Poisson Kernel vanishes, and the integral isidentically zero, for any function f ( x ).Thus, the Poisson Integral Theorem does not hold in the Calculusof Limits.10.2 Calculus of Limits Conditions are irrelevant toPoisson Integral TheoremProof:The Poisson Conditions are34

Gauge Institute JournalH. Vic Dannon1. f ( x ) is integrable on [ c − L,c + L]2. f ( x ) is periodic with period T = 2L13. (2fx ( + 0) + fx ( − 0) ) replaces f ( x ) at a discontinuity point.It is clear from 10.1 that the Fejer conditions on f ( x ) do notresolve the singularity of the Fejer kernel, and are not sufficientfor the Fejer Summation Theorem. In Infinitesimal Calculus, by 8.4, the Poisson Kernel is thePeriodic Delta Function, and by 9.2, it equals its Poisson Integralat r = 1 −dr.Then, the Poisson Integral Theorem holds for any periodic Hyper-Real Function:10.3 Poisson Integral Theorem for Hyper-real f ( x )If f ( x ) is hyper-real function integrable on [ c − L, c + L], so thatf ( c − L) = f(c + L)Then, f ( x) = P S{ f( x )}Proof: Take L = 1, and c = 0.ξ = 1∫ξ =−1oisson{ }fx ( ) = f( ξ) ... + δ( ξ − x+ 2) + δ( ξ − x) + δ( ξ −x− 2) + ... dξ.δPeriodic( ξ−x), where the period of Delta is T=235

Gauge Institute JournalH. Vic DannonBy 8.4,δ ( ξ − x) = P ( ξ − x). Thus,Periodicoissonξ=1dr(2 − dr)f ( x) = ∫ f( ξ) 2(1 −dr )(1 − cos πξ [ − x ]) + ( dr )ξ=−112 2d ξ=ξ=1∫ξ=−1f211 − r() ξ2 12+ r − 2 r cos πξ ( −x)r =− 1 drdξ= P S {()} f x .oissonIn particular, the Periodic Delta Functionviolates the Poisson Conditions The Hyper-realδ( x), is not defined in the Calculus of Limits,andδ ( x)is not integrable in any bounded interval.1(2δ( x + 0) + δ( x − 0) ) = 0 does not replace δ( x)at itsdiscontinuity point, x = 0 .But by 9.2,Periodic( x )satisfies the Poisson Integral Theorem.δ36

Gauge Institute JournalH. Vic DannonReferences[Achieser] Achieser, N. I., Theory of Approximation, Ungar, 1956.[Carslaw] Carslaw, H. S., “Introduction to the Theory of Fourier Series andintegrals” Third Edition, Macmillan, 1930.[Dan1] Dannon, H. Vic, “Well-Ordering of the Reals, Equality of all Infinities,and the Continuum Hypothesis” in Gauge Institute Journal Vol. 6 No. 2, May2010;[Dan2] Dannon, H. Vic, “Infinitesimals” in Gauge Institute Journal Vol.6 No.4, November 2010;[Dan3] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge Institute JournalVol. 7 No. 4, November 2011;[Dan4] Dannon, H. Vic, “Riemann’s Zeta Function: the Riemann HypothesisOrigin, the Factorization Error, and the Count of the Primes”, in GaugeInstitute Journal of Math and Physics, Vol. 5, No. 4, November 2009.[Dan5] Dannon, H. Vic, “The Delta Function” in Gauge Institute Journal Vol.8, No. 1, February, 2012;[Dan6] Dannon, H. Vic, “Delta Function, the Fourier Transform, and theFourier Integral Theorem” in Gauge Institute Journal Vol. 8, No. 2, May,2012;[Dan7] Dannon, H. Vic, “Riemannian Trigonometric Series”, Gauge InstituteJournal, Volume 7, No. 3, August 2011.[Dan8] Dannon, H. Vic, “Periodic Delta Function and Dirichlet Summation ofFourier Series” posted to www.gauge-institue.org;[Dan9] Dannon, H. Vic, “Lebesgue Integration” Gauge Institute Journal Vol.7No. 1, February 2011;37

Gauge Institute JournalH. Vic Dannon[Dan10] Dannon, H. Vic,“Periodic Delta Function and Fejer-CesaroSummation of Fourier Series”, June 2012, posted to www.gauge-institue.org[Dan11] Dannon, H. Vic, “Infinite Series with Infinite Hyper-real Sum ” inGauge Institute Journal Vol. 8, No. 3, August, 2012;[Davis] Davis, Philip, “Interpolation and Approximation” Blaisdell, 1963.[Hardy] Hardy, G. H., Divergent Series, Chelsea 1991.[Natanson] Natanson, I. P., “Constructive Function Theory” Ungar, 1964.[Rogosinski] Rogosinski, Werner, “Fourier Series” Chelsea, 1950.[Tolstov] Tolstov, Georgi, “Fourier Series” Prentice-Hall,1962[Zygmund] Zygmund, A., “Trigonometric Series”, Second Edition, CambridgeUniversity Press, 1968.38