Delta Function and the Poisson Summation Formula H. Vic Dannon [email protected] July, 2015
Abstract In the Calculus of Limits, the Poisson Summation
Formula holds under unnecessary conditions. Using Delta function, and Periodic Delta Functions, we present here an Infinitesimal Calculus proof, that is free of unnecessary conditions. Then, we apply the formula to a particular series.
Keywords: Infinitesimal, Infinite-Hyper-Real, Hyper-Real, infinite Hyper-real, Infinitesimal Calculus, Delta Function, Fourier Transform, Periodic Delta Function, Delta Comb, Fourier Series, Dirichlet Kernel,
2000 Mathematics Subject Classification 26E35; 26E30;
26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15; 46S20; 97I40; 97I30.
1 Contents
0. Calculus of Limits Limitations on the Poisson Summation Formula 1. Hyper-real Line 2. Integral of a Hyper-real Function 3. Delta Function
4. Periodic Delta Function dxPeriodic ()- x 5. The Poisson Summation Formula in Infinitesimal Calculus 6. An Inquiry 7. Applying the Formula to 6. References
2 Calculus of Limits Limitations on the Poisson Summation Formula
In the Calculus of Limits, the Poisson Summation Formula is claimed to hold under restricting conditions: For instance, the version presented in [Apostol, p.332]: Theorem 11.24
If f ()x ³ 0, for -¥ x =¥ ò f ()xdx exists as an improper Riemann Integral x =-¥ f increases on -¥ m=¥ fm()++ fm ( -) Then å , m=-¥ 2 and m=¥ x =¥ å ò f ()xe-2pimx dx m=-¥ x =-¥ are absolutely convergent to the same limit. In fact, it is likely that even under these restrictions, the formula does not hold. However, since no scientist may ever care about these restrictions, we will skip an examination that will render them meaningless, and establish the Formula in Infinitesimal Calculus, free of restricting conditions 3 For a Hyper-real Function in Infinitesimal Calculus, we obtain under no limitations mm=¥ =¥ ååf ()mF= (2)pm. mm=-¥ =-¥ 4 1. Hyper-real Line The minimal domain and range, needed for the definition and analysis of a hyper-real function, is the hyper-real line. Each real number a can be represented by a Cauchy sequence of rational numbers, (,rrr123, ,...) so that rn a . The constant sequence (aaa, , ,...) is a constant hyper-real. In [Dan2] we established that, 1. Any totally ordered set of positive, monotonically decreasing to zero sequences (iii123 , , ,...) constitutes a family of infinitesimal hyper-reals. 2. The infinitesimals are smaller than any real number, yet strictly greater than zero. 3. Their reciprocals (11, ,1 ,...) are the infinite hyper-reals. ii123i 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 smaller than any real number, yet strictly greater than -¥. 6. The sum of a real number with an infinitesimal is a non-constant hyper-real. 5 7. The Hyper-reals are the totality of constant hyper-reals, a family of infinitesimals, a family of infinitesimals with negative sign, a family of infinite hyper-reals, a family of infinite hyper-reals with negative sign, and non-constant hyper-reals. 8. The hyper-reals are totally ordered, and aligned along a line: the Hyper-real Line. 9. That line includes the real numbers separated by the non- constant hyper-reals. Each real number is the center of an interval of hyper-reals, that includes no other real number. 10. In particular, zero is separated from any positive real by the infinitesimals, and from any negative real by the infinitesimals with negative signs, -dx . 11. Zero is not an infinitesimal, because zero is not strictly greater than zero. 12. We do not add infinity to the hyper-real line. 13. The infinitesimals, the infinitesimals with negative signs, the infinite hyper-reals, and the infinite hyper-reals with negative signs are semi-groups with respect 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-continuous one-one mapping from the hyper-real onto the real line. 6 15. In particular, there are no points on the real line that can be assigned uniquely to the infinitesimal hyper-reals, or to the infinite hyper-reals, or to the non-constant hyper- reals. 16. No neighbourhood of a hyper-real is homeomorphic to an n ball. Therefore, the hyper-real line is not a manifold. 17. The hyper-real line is totally ordered like a line, but it is not spanned by one element, and it is not one-dimensional. 7 2. Integral of a Hyper-real Function In [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 be bounded. At each ax££b, there is a rectangle with base [,xx-+dx dx ], height f ()x , and area 22 f ()xdx. We form the Integration Sum of all the areas for the x ’s that start at x= a, and end at xb= , å f ()xdx. xabÎ[,] If for any infinitesimal dx , the Integration Sum has the same hyper-real value, then f ()x is integrable over the interval [,ab]. Then, we call the Integration Sum the integral of f()x from xa= , to x= b, and denote it by xb= ò f ()xdx. xa= If the hyper-real is infinite, then it is the integral over [,ab], 8 If the hyper-real is finite, xb= ò fxdx( )= real part of the hyper-real . xa= 2.1 The countability of the Integration Sum In [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, Card = 2Card , and we have 22Card Card Card=====( Card ) .... 2 2 ... º¥. In particular, we demonstrated that the real numbers may be well-ordered. Consequently, there are countably many real numbers in the interval [,ab], and the Integration Sum has countably many terms. While we do not sequence the real numbers in the interval, the summation takes place over countably many f ()xdx. The Lower Integral is the Integration Sum where f ()x is replaced by its lowest value on each interval [,xx-+dx dx ] 22 æö ç ÷ 2.2 å ç inff (tdx )÷ ç xtx-££+dx dx ÷ xabÎ[,]èø22 9 The Upper Integral is the Integration Sum where f ()x is replaced by its largest value on each interval [,xx-+dx dx ] 22 æö ç ÷ ç supf (tdx )÷ 2.3 å ç ÷ ç xtx-££+dx dx ÷ xabÎ[,]èø22 If the integral is a finite hyper-real, we have 2.4 A hyper-real function has a finite integral if and only if its upper integral and its lower integral are finite, and differ by an infinitesimal. 10 3. Delta Function In [Dan5], we have defined the Delta Function, and established its properties 1. The Delta Function is a hyper-real function defined from the ïìü1 ï hyper-real line into the set of two hyper-reals íýïï0, . The îþïïdx hyper-real 0 is the sequence 0, 0, 0, ... . The infinite hyper- 1 real depends on our choice of dx . dx 2. We will usually choose the family of infinitesimals that is 1 1 1 spanned by the sequences , , ,… It is a n n2 n3 semigroup with respect to vector addition, and includes all the scalar multiples of the generating sequences that are non-zero. That is, the family includes infinitesimals with 1 negative sign. Therefore, will mean the sequence n . dx Alternatively, we may choose the family spanned by the 1 1 1 1 sequences , , ,… Then, will mean the 2n 3n 4n dx sequence 2n . Once we determined the basic infinitesimal 11 3. The Delta Function is strictly smaller than ¥ 1 4. We define, d()xxº céù-dx, dx (), dx ëêú22û ïì1,x Î-é dx , dx ù ï ëê 22ûú where céù-dx, dx ()x = í . ëûêú22 ï 0, otherwise îï 5. Hence, for x < 0 , d()x = 0 dx 1 at x =- , d()x jumps from 0 to , 2 dx 1 for x Î-éùdx, dx , d()x = . ëûêú22 dx 1 at x = 0 , d(0) = dx dx 1 at x = , d()x drops from to 0. 2 dx for x > 0 , d()x = 0 . xxd()= 0 1 6. If dx = , d()xxx= ccc[,]---11 (),2 [,]11 (),3 [,]11 ()...x n 22 44 66 12 3 7. If dx = 2 , d(x )= , , ,... n 2 cosh22xxx 2 cosh 2 2 cosh 2 3 8. If dx = 1 , dccc()xe= --xxx,2 e23 , 3 e - ,... n [0,¥¥¥ ) [0, ) [0, ) 12 x =¥ 9. ò d()xdx= 1. x =-¥ k=¥ 1 10. dx()-=xe--ik()x x dk 2p ò k=-¥ 13 4. Periodic Delta Function dxPeriodic()- x In [Dan5], we defined the Periodic Delta Function, and related it to the Dirichlet Kernel in Infinitesimal Calculus 1) Periodic Delta Function Definition dPeriodic() x-=+xxx...( dx -++ 2)() dx -+dx( --+x 2)... is a periodic hyper-real Delta function, with period T = 2. 2) Fourier Transform of dPeriodic()x -ii44pn pn {dPeriodic()x} = ...+e ++ 1 e +... 3) Fourier Integral Theorem for dPeriodic()x -1 {ddPeriodic()xx} = Periodic () 4) Dirichlet Sequence Definition The Fourier Series partial sums x =1 11--inpx() x --ipx( x)111ipx ()- x inpx()- x n {}f ()()xfe=+x{... + e++e+... + ed}x. ò 22222 x =-1 Dirichlet Sequence give rise to the Dirichlet Sequence Dxe()x - =111--inpx() x ++ ... e --ipx() x ++11 eipx ()- x ++...einpx()- x n 22222 14 =+1 cospx ( -xxn ) + cos2px ( - ) + ... + cospx ( -x ) 2 sin(nx+-1 )px ( ) = 2 , n = 0, 1, 2, .. 2sin1 px (- x ) 2 5) Dirichlet Sequence is a Periodic Delta Sequence sin(nx+ 1 )p Each Dx()= 2 , n = 0, 1, 2, 3, ... n 2sin1 px 2 has the sifting property on each interval, x =-3 x =-1 x =1 Dxdx()= 1 Dxdx()= 1 Dxdx()= 1 .. ò n ; ò n ; ò n .. x =-5 x =-3 x =-1 is a co ntinuous function 1 peaks on each of these intervals to limDxn ( ) =+ n . xm2 2 6) Dirichlet Sequence Represents dxPeriodic()- x sin 21n+ px( - x) dx()-=x 2 Periodic 2sin1 px (- x ) 2 =+1 cospx ( -+xxn ) cos2px ( -++ ) ... cospx ( -x ) 2 =++++++111eeee--inpx() x ... --ipx() x 11ipx ()- x ... inpx()-x . 22222 7) Hyper-real Dirichlet Kernel 15 ïì 1 +-=nx, x 2m Dx()x -=íï 2 irichlet ï 0, x -¹xm 2 îï 8) Let N = 1 be an infinite Hyper-real. Then dx sin(Nx+-1 )px ( ) Dx()x -= 2 irichlet 2sin1 px (- x ) 2 =+1 cospx ( -xxN ) + cos2 px ( - ) + ... + cos px ( -x ) 2 =++++++111eeee--iNpx() x .. --ipx() x 11ipx()- x .. iNpx(- x) 22222 =dx( -+xN 2 ) + ... + dx ( -+ x 2) + dx ( - x ) + dx ( -- x 2) + ... + dx ( -- xN 2 ) =-dxPeriodic()x 9) ...+-++-+--+dx (xxx 2) dx ( ) dx ( 2) ... = =+.. 111ee--ix2(px ) + -- ixpx ( ) ++11 ee ixpx(-- ) + ix 2(px ) +.. 22222 =+1 cos(px -+xx )cos2( px -+ )cos3( px -+x )... 2 10) ... +-++-+--+dq(f 2 p ) dq ( f ) dq ( f 2 p ) ... = =+.. 111ee--ii2(qf ) + -- ( qf ) ++11 ee ii(qf-- ) + 2( qf ) +.. 22222 =+1 cos()cos2()cos3()..qf -+ qf -+ qf -+. 2 16 5. The Poisson Summation Formula in Infinitesimal Calculus By [Dan4], in Infinitesimal Calculus, a Hyper-real function has a Fourier Transform {}()f ww= F ( ), without the Calculus of Limits conditions that in fact do not guarantee the Fourier Transform for any function. For such hyper-real function in Infinitesimal Calculus we need no Calculus of Limits conditions to prove the Poisson Summation Formula: 5.1 Poisson Summation Formula For a Hyper-real function f ()x in Infinitesimal Calculus ... +-+-+f ( 2)ffff ( 1) (0) + (1) + (2) + ... = =+-...(4)(2)(0)(2)(4).FFFFFpp +- + + pp + +.. Proof: To apply the Fourier Transform to the f ()n Summation, we define Sx()=+-+-++++++ ... fx ( 2) fx ( 1) fx () fx ( 1) fx ( 2) ... uu=¥ =¥ =+...òòf (ux )dd ( +- 2 udu ]) + fux ( ) ( +- 1 udu ) + uu=-¥ =-¥ 17 u=¥ +-ò f ()(uxudud ) + u =-¥ uu=¥ =¥ +--+--òòfu( )dd ( x 1 udu ) fu ( ) ( x 2 udu )+ ... uu=-¥ =-¥ u=¥ =ò fu(){.. +ddd ( x +-+ 2 u ) ( x +-+-+ 1 u ) ( x u ) u =-¥ +--+dd(xuxu 1 ) ( --+ 2 ) ..}du. =f (xxxxxx ) *++++++-+-+ {...ddddd ( 2) ( 1) ( ) ( 1) ( 2) ...} Periodic Delta Thus, Sx() is the convolution of f with a Periodic Delta Function, which was defined, and discussed extensively in [Dan5] By the Convolution Theorem, Sx() Fourier Transforms into the product of the Transforms of f , and the Periodic Delta {Sx ( )}=+ { fx ( )} {...dd (x+ 2)+ ( x+ 1) + ww ++-+-+dd()xx ( 1) d ( x 2) ..} w =++++Fxx(wd ){.. ( 2) d ( 1) + ++-+-+dd()xx ( 1) d ( x 2) ..} w x =¥ Since dd(2)xx+= (2) +e-ixwwdx =e2 i , w ò x =-¥ x =¥ dd(1)xx+= (1) +e-ixwwdx =e i , w ò x =-¥ x =¥ dd()xux=- ( )ed-ixw ue =0 =1, w ò x =-¥ 18 x =¥ dd(1)xx-= (1) -e--ixwwdx =e i , w ò x =-¥ x =¥ dd(2)xx-= (2) -e-ixwwdu =e2 i w ò x =-¥ {Su ( )}=+++++++ F (w ){... e22iiww e1 e-- i w e i w...} w By [Dan5, 8.4, and 8.5], the Hyper-real Dirichlet Kernel 22iiww-- i w i w ..++++++ee1 ee +... Dirichlet Kernel equals the Periodic Delta æöæöæöæöæöwwwww÷÷÷÷÷ ...++++++-+-+ 2dddddççççç 4÷÷÷÷ 2 2 2 2 2 2 4÷ ... èøèøèøèøèøçççççppppp÷÷÷÷÷ =2p {.. +++++ dw()()()()() 4 p dw2 p dw +-+-+ dw2 p dw4 p ..} Substituting the Periodic Delta into {()}Su , w {Sx ( )}=++++ F (wp )2 {... dw 4 p dw2 p+ w ()() ++-+-+dw() dw (2 p ) dw (4 p ) ...} =+-{..FF ( 4ppdw )2() + 4 p +- ( 2 ppdw )2 ( + 2 p)+ ++FF(0)2pd ( w ) (2 p )2 pd ( w -+- 2 p ) F (4 p )2 pd ( w 4 p )+ ...}. Since w=¥ -1{2pd() w+= 4 p } ò d () w +4 pedixw w = e i(4)- p x w=-¥ w=¥ -1{2pd() w+= 2 p } ò d () w +2 pedixwp w = e i(2)- x w=-¥ 19 w=¥ -10{2pd() w } ==ò d () wedixw w e=1 w=-¥ w=¥ -1{2pd() w-= 2 p } ò d () w -2 pedixwp w = e i2 x w=-¥ w=¥ -1{2pd() w-= 4 p } ò d () w -4 pedixwp w = e i4 x w=-¥ Sx()=+ .. F ( - 4)pp eix(4)--pp + F ( - 2) eix(2) + ++FFeFe(0) (2pp )ix24pp + (4 )ix + ... SFFFFF(0)=+- ... ( 4pp ) +- ( 2 ) + (0) + (2 pp ) + (4 )+ ... Therefore, ...+-+-+f ( 2)ffff ( 1) (0) + (1) + (2) + ... = S (0) = =+-...(4)(2)(0)(2)(4)...FFFFFpp +- + + pp + +. The Calculus of Limits proofs under their bizarre conditions, mask the fact that 5.2 the Formula does not indicate which terms in the f ()n summation contribute to whichever terms in the Fn() summation Many statements of the formula attempt to hint at 1.2 by using an index n for the f summation, and an index m for the F summation. This does not clarify the fact that the whole f summation equals the whole F summation. 20 6. An Inquiry Professor Thomas Radil inquired about applying the Formula to the summation of a series: 21 22 7. Applying Poisson Summation to 6. v p - 2pm Fm()1 2p = ed d v v 11pp- 2pm Fe(0)== lim d ddvm0 vd The rest of the series transforms into what you have 2222ppppvvvv 22pp----æö234 1 eddddç1++++= eee ...÷ ç ÷ 2pv vdèøç ÷ vd e d - 1 1 2pv - 1-e d Therefore, by the Poisson Summation Formula, æö 21p ç 1 ÷ S =+ç ÷. ç 2pv ÷ vd ç 2 ÷ èøe d - 1 23 References [Apostol] Tom Apostol, “Mathematical Analysis”, Second Edition, Addison Wesley, 1974. [Dan1] Vic Dannon, “Infinitesimals” in Gauge Institute Journal Vol.6 No 4, November 2010; [Dan2] Vic Dannon, “Infinitesimal Calculus” in Gauge Institute Journal Vol.7 No 4, November 2011; [Dan3] Vic Dannon, “The Delta Function” in Gauge Institute Journal Vol.8 No 1, February 2012; [Dan4] Vic Dannon, “Delta Function, the Fourier Transform, and Fourier Integral Theorem”, in Gauge Institute Journal Vol.8 No 2, May 2012. [Dan5] Vic Dannon, “Periodic Delta Function and Dirichlet Summation of Fourier Series” in Gauge Institute Journal Vol.9, No 2, May 2013. [Spiegel] Murray Spiegel, “Mathematical Handbook of Formulas and Tables”, Schaum’s Outline Series, McGraw-Hill, 1968. p.109. 24