Journal of Inequalities in Pure and Applied Mathematics

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1 Journal of Inequalities in Pure Applied Mathematics Volume 5 Issue 1 Article ON THE HEISENBEG-PAULI-WEYL INEQUALITY JOHN MICHAEL ASSIAS PEDAGOGICAL DEPATMENT E. E. NATIONAL AND CAPODISTIAN UNIVESITY OF ATHENS SECTION OF MATHEMATICS AND INFOMATICS 4 AGAMEMNONOS ST. AGHIA PAASKEVI ATHENS 1534 GEECE. jrassias@primedu.uoa.gr eceived 0 January 003; accepted 04 March 003 Communicated by A. Babenko ABSTACT. In 197 W. Heisenberg demonstrated the impossibility of specifying simultaneously the position the momentum of an electron within an atom.the following result named Heisenberg inequality is not actually due to Heisenberg. In 198 according to H. Weyl this result is due to W. Pauli.The said inequality states as follows: Assume thatf : C is a complex valued function of a rom real variable x such that f L. Then the product of the second moment of the rom real x for f the second moment of the rom real ξ for ˆf is at least E f /4π where ˆf is the Fourier transform of f such that ˆf ξ e iπξx f x dx f x eiπξx ˆf ξ dξ i 1 E f f x dx. In this paper we generalize the afore-mentioned result to the higher moments for L functions f establish the Heisenberg-Pauli-Weyl inequality. Key words phrases: Pascal Identity Plancherel-Parseval-ayleigh Identity Lagrange Identity Gaussian function Fourier transform Moment Bessel equation Hermite polynomials Heisenberg-Pauli-Weyl Inequality. 000 Mathematics Subject Classification. Primary: 6Xxx; Secondary: 4Xxx 60Xxx 6Xxx. 1. INTODUCTION In 197 W. Heisenberg 9 demonstrated the impossibility of specifying simultaneously the position the momentum of an electron within an atom. In 1933 according to N. Wiener a pair of transforms cannot both be very small. This uncertainty principle was stated in 195 by Wiener according to Wiener s autobiography 3 p in a lecture in Göttingen. In 199 J.A. Wolf 4 in 1997 G. Battle 1 established uncertainty principles for Gelf pairs wavelet states respectively. In 1997 according to Foll et al. 6 in 001 according to Shimeno 14 the uncertainty principle in harmonic analysis says: ISSN electronic: c 004 Victoria University. All rights reserved. We thank the computer scientist C. Markata for her help toward the computation of the data of Table 9.1 for the last twenty four cases p We greatly thank Prof. Alexer G. Babenko for the communication the anonymous referee for his/her comments

2 JOHN MICHAEL ASSIAS A nonzero function its Fourier transform cannot both be sharply localized. The following result of the Heisenberg Inequality is credited to W. Pauli according to H. Weyl 0 p.77 p In 198 according to Pauli 0 the following proposition holds: the less the uncertainty in f the greater the uncertainty in ˆf is conversely. This result does not actually appear in Heisenberg s seminal paper 9 in 197. According to G.B. Foll et al. 6 in 1997 Heisenberg 9 gave an incisive analysis of the physics of the uncertainty principle but contains little mathematical precision. The following Heisenberg inequality provides a precise quantitative formulation of the above-mentioned uncertainty principle according to Pauli 0. In what follows we will use the following notation to denote the Fourier transform of f x : F f x f x ˆ ξ Second Order Moment Heisenberg Inequality 3 6. For any f L f : C such that f f x dx E f any fixed but arbitrary constants x m ξ m for the second order moments variances µ f σ x x f m f x dx µ ˆf σ ˆf the second order moment Heisenberg inequality ξ ξ m ˆf ξ dξ H 1 µ f µ ˆf E f 16π holds where f x ˆf ξ e iπξx f x dx e iπξx ˆf ξ dξ i 1. Equality holds in H 1 iff if only if the Gaussians f x c 0 e πixξm e cx xm c 0 e cx +cx m+iπξ mx cx m hold for some constants c 0 C c > 0. We note that if x m 0 ξ m 0 then f x c 0 e cx xm c 0 C c > 0. Proof. Let x m ξ m 0 that the integrals in the inequality H 1 be finite. Besides we consider both the ordinary derivative d dx f e f f the Fourier differentiation formula F f ξ f x ˆ ξ πiξ ˆf ξ. Then we get that the finiteness of the integral ξ ˆf ξ dξ 1 4π f x dx J. Inequal. Pure Appl. Math. 51 Art

3 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 3 implies f L. Integration by parts yields a+ a+ x e f x f x dx x d dx f x dx x f x a a a+ a + a a f x dx < a < a + <. Since f xf f L the integrals in this equality are finite as a or a + thus both limits L lim a f a L + lim a + f a + are finite. These two limits a a+ L ± are equal to zero for otherwise f x would behave as 1 for big x meaning that f / x L leading to contradiction. Therefore for the variances about the origin m f s x f x dx f one gets π f 4 4π E f 1 16π 1 16π m ˆf s ˆf x d f x dx dx f x dx ξ ˆf ξ dξ 1 x e f x f 16π x 1 1 4π 1 xf x f x dx 4π 1 4π dx xf x f x + xf xf x x f x dx f x dx dx s f s. ˆf Equality in these inequalities holds iff the differential equation f x cxf x of first order holds for c > 0 or if the Gaussians f x c 0 e cx hold for some constants c 0 C c > 0. Assuming any fixed but arbitrary real constants x m ξ m employing the transformation f xmξ m x e πixξm f x x m x x x m 0 we establish the formula ˆf xmξ m ξ e πixmξ ξm ˆf ξ ξm e πixmξm ˆf ξm x m ξ. Therefore the map f f xmξ m preserves all L p p N norms of f ˆf while shifting the centers of mass of f ˆf by real x m ξ m respectively. Therefore equality holds in H 1 for any fixed but arbitrary constants x m ξ m iff the general formula f x c 0 e πixξm e cx xm J. Inequal. Pure Appl. Math. 51 Art

4 4 JOHN MICHAEL ASSIAS holds for some constants c 0 C c > 0. One can observe that this general formula is the complete general solution of the following a-differential equation f a c x x m f a by the method of the separation of variables where a πξ m i f a e ax f. We note that In fact or or f a df a dx df a dx dx dx df a dx x x x m. ln f a c x x m e ax f x f a x c 0 e cx xm f x c 0 e ax e cx xm. This is a special case for the equality of the general formula 8.3 of our theorem in Section 8 on the generalized weighted moment Heisenberg uncertainty principle. Therefore the proof of this fundamental Heisenberg Inequality H 1 is complete. We note that if f L the L -norm of f is f 1 ˆf then f ˆf are both probability density functions. The Heisenberg inequality in mathematical statistics Fourier analysis asserts that: The product of the variances of the probability measures f x dx ˆf ξ dξ is larger than an absolute constant. Parts of harmonic analysis on Euclidean spaces can naturally be expressed in terms of a Gaussian measure; that is a measure of the form c 0 e c x dx where dx is the Lebesgue measure c c 0 > 0 constants. Among these are: Logarithmic Sobolev inequalities Hermite expansions. Finally one 14 observes that: if f L σ f σ ˆf 1 4 f 4 ˆf ξ 1 π f x 1 π e iξx f x dx e iξx ˆf ξ dξ where the L -norm f is defined as in H 1 above. In 1999 according to Gasquet et al. 8 the Heisenberg inequality in spectral analysis says that the product of the effective duration x the effective bwidth ξ of a signal cannot be less than the value 1π 4 H Heisenberg lower bound where x σ /E f f / ξ σ ˆf E ˆf σ /E ˆf f with f : C ˆf : C defined as in H1 E f ˆf ξ dξ E ˆf. In this paper we generalize the Heisenberg inequality to the higher moments for L functions f establish the Heisenberg-Pauli-Weyl inequality. J. Inequal. Pure Appl. Math. 51 Art

5 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 5. PASCAL TYPE COMBINATOIAL IDENTITY We state prove the new Pascal type combinatorial identity. Proposition.1. If 0 k k k i C k i i is the greatest integer k then + k 1 k i k + 1 k i i 1 k i + 1 k i + 1 i holds for any fixed but arbitrary k N {1...} 0 i k for i N0 {0 1...} such that k 1 0. Note that the classical Pascal identity is k i k i k i i i 1 i 0 i Proof. It is clear that k k i k 1 i k i k + 1. Thus k k i + k 1 k i k i i k i i 1 k k i! k i i! k i! + k 1 k i! k i i 1! k i + 1! k k i k i+1! k i+1 i! k i+1! k i+1 + k 1 k i i! i k i+1! k i+1 k i + 1! k. k i + 1! 1 k k i k 1 i i! k i + 1! k i k i + 1 k i + 1 k + 1 i k i + 1 completing the proof of this identity. Note that all of the three combinations: k i i numbers if 1 i k for i N. k i i 1 k i+1 i exist are positive 3. GENEALIZED DIFFEENTIAL IDENTITY We state prove the new differential identity. Proposition 3.1. If f : C is a complex valued function of a real variable x 0 k is the greatest integer k f j dj f is the conjugate of then dx j * f x f k x + f k x f k x 1 i k k i k i i i0 d k i dx k i f i x holds for any fixed but arbitrary k N {1...} such that 0 i k for i N0 {0 1...}. J. Inequal. Pure Appl. Math. 51 Art

6 6 JOHN MICHAEL ASSIAS Note that for k 1 we have f x f 1 x + f 1 x f x d f x dx f x d 1 0 f 0 x dx 1 0 If we denote G k f ff k + f k f f i k i d k i dx k i f i i N 0 then * is equivalent to i0 dk i dx k i 0 i k k ** G k f 1 i k k i k i i d i dx f i for k N {1...} f i k i. Proof. For k 1 ** is trivial. Assume that ** holds for k claim that it holds for k + 1. In fact 1 G 1 k f ff k + f k f 1 ff k + f k 1 f f 1 f k + ff k+1 + f k+1 f + f k f 1 f 1 f k + f k f 1 + ff k+1 + f k+1 f f 1 f 1 k 1 + f 1 k 1 f 1 + G k+1 f or the recursive sequence G k 1 f 1 + G k+1 f G k+1 f G 1 k f G k 1 f 1 for k N {1...} with G 0 f 1 f 1 G 1 f f 1. From the induction hypothesis the recursive relation the fact that j+1 j λ i+1 λ i 1 i 1 1 i for i N 0 i0 k 1 k k i1 1 k v for v 1... k 1 k λ + 1 for λ J. Inequal. Pure Appl. Math. 51 Art

7 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 7 such that k k if k N we find G k+1 f k 1 i i0 k 1 i0 { 1 0 i1 k k i k i i f i k+1 i 1 i k 1 k 1 i k 1 i i k k 0 k 0 0 k + 1 i k k i k i i k 1 +1 i1 f k+1 k + 1 i i1 k i k 1 k i k i i 1 i1 f k+1 k + 1 i i1 k + 1 i k 1 k i k i i 1 i1 f 0 k+1 0 f i k+1 i f i+1 k 1 i 1 i 1 k 1 k 1 i 1 k 1 i 1 i 1 k k i k i i f i k+1 i f i k+1 i k k i k i i f i k 1 i 1 f i k+1 i f i k+1 i + S f where S f 0 if k v for v k 1 +1 k 1 k k 1 k + 1 k 1 k f k 1 +1 k+1 k 1 +1 if k λ + 1 for λ J. Inequal. Pure Appl. Math. 51 Art

8 8 JOHN MICHAEL ASSIAS Thus G k+1 f f k+1 k { k + 1 i k i k i i where i1 + k 1 k i k i i 1 } f i k+1 i + S f 0 if k v for v 1... S f 1 k 1 1 k k 1 k 1 f k 1 k 1 k 1 k 1 +1 k 1 k 1 if k λ + 1 for λ if k v for v k 1 1 k k 1 k 1 k 1 f k+1 0 if k λ + 1 for λ if k v for v 1... f 1 k+1 k+1 if k λ + 1 for λ if k v for v k+1 k+1 k + 1 k+1 f k+1 k+1 k+1 k+1 k+1 k+1 if k λ + 1 for λ because k + 1 k + 1 if k λ + 1 for λ Besides we note that from the above Pascal type combinatorial identity C k if k v for v 1... one gets k+1 k { 1 k k k k k k k + k 1 k k } f k k k k k+1 k 1 f k k+1 k 1 k k + 1 k k k k k k+1 k + 1 k + 1 k+1 k + 1 k+1 k+1 f k+1 k+1 k+1 J. Inequal. Pure Appl. Math. 51 Art

9 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 9 if k v for v From these results one obtains G k+1 f 1 0 k + 1 k f k k+1 0 k+1 i1 k + 1 i k + 1 k + 1 i k + 1 i i i1 1 i k + 1 k + 1 i k + 1 i i f i k+1 i +S f f i k+1 i completing the proof of ** for k + 1 thus by the induction principle on k ** holds for any k N Special cases of **. G 1 f f 1 G f f f 1 G 3 f f 3 3 f 1 1 G 4 f f 4 4 f 1 + f G 5 f f 5 5 f f 1 G 6 f f 6 6 f f f 3 G 7 f f 7 7 f f 3 7 f 3 1 G 8 f f 8 8 f f 4 16 f 3 + f 4. We note that if one takes the above numerical coefficients of G i f i absolutely then one establishes the pattern a a a 3 9 b a 4 14 b a 5 0 b 3 16 c 1 with J. Inequal. Pure Appl. Math. 51 Art

10 10 JOHN MICHAEL ASSIAS a 1 b 1 a 1 c 1 b 1 a 1 a b a 1 + a 7 a b 3 a 1 + a + a 3 16 a a Following this pattern we get where G 9 f f 9 9 f a6 f 5 b4 f c f 4 1 Similarly one gets a 6 a b 4 b 3 + a c c 1 + b G 10 f f f a7 f 6 where a 7 a b 5 f c3 f 4 d1 f 5 b 5 b 4 + a c 3 c + b d 1 c 1 b 1 a Applications of the ecursive Sequence. G 4 f G 1 3 f G f 1 1 ff 3 + f 3 f f 1 f 1 + f 1 f 1 { f 4 f 3 1 } { f 1 } f f 4 4 f 1 + f G 5 f G 1 4 f G 3 f 1 { f 5 f f + } { 1 f 1 3 f 3 } 1 f 5 5 f f 1 yielding also the above generalized differential identity ** for k 3 k 4 respectively. J. Inequal. Pure Appl. Math. 51 Art

11 ON THE HEISENBEG-PAULI-WEYL INEQUALITY Generalization of the Identity **. We denote H kl f f l f k + f k f l. It is clear that H kl f G k l f l f l f l k l + f l k l f l G 0 f l f l G l k f k f k f k l k + f k l k f k if k > l if k l if k < l From these ** we conclude that k l 1 i k l f k l i i+l k l i if k > l k l i i i0 H kl f f l if k l l k 1 i i0 l k l k i l k i i For instance if k 3 > l then from this formula one gets H 3 f G 3 f In fact 3 f i+k l k i if k < l 1 i 3 3 i f 3 i i i+ 3 i i f f 1. H 3 f f f 3 + f 3 f f f 1 + f 1 f G 1 f f 1. Another special case if k 3 > 1 l then one gets H 31 f G 3 1 f i i f 3 1 i i i i i0 1 0 f f 1 f 1 f. J. Inequal. Pure Appl. Math. 51 Art

12 1 JOHN MICHAEL ASSIAS In fact H 31 f f 1 f 3 + f 3 f 1 f 1 f 1 + f 1 f 1 G f 1 f 1 f. 4. GENEALIZED PLANCHEEL-PASEVAL-AYLEIGH IDENTITY We state prove the new Plancherel-Parseval-ayleigh identity. Proposition 4.1. If f f a : C are complex valued functions of a real variable x f a e ax f f a p dp f dx p a where a πξ m i with i 1 any fixed but arbitrary constant ξ m ˆf the Fourier transform of f such that ˆf ξ e iπξx f x dx with ξ real f x e iπξx ˆf ξ dξ as well as f f a p ξ ξ m p ˆf are in L then 4.1 ξ ξ m p 1 ˆf ξ dξ π p holds for any fixed but arbitrary p N 0 {0 1...}. Proof. Denote 4. g x e πixξm f x + x m f p a x dx for any fixed but arbitrary constant x m. From 4. one gets that 4.3 ĝ ξ e iπξx g x dx e iπξx e πixξm f x + x m dx e πixmξ+ξm e πiξ+ξmx f x dx e πixmξ+ξm ˆf ξ + ξm. Denote the Fourier transform of gp either by F g p ξ or F g p x ξ or also as g p x ˆ ξ. From this Gasquet et al. 8 p we find 4.4 ξ p ĝ ξ 1 πi p F g p ξ. Also denote 4.5 h p x g p x. From 4.5 the classical Plancherel-Parseval-ayleigh identity one gets ĥpξ dξ h p x dx J. Inequal. Pure Appl. Math. 51 Art

13 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 13 or 4.6 F g p ξ dξ Finally denote 4.7 µ p ˆf g p x dx. ξ ξ m p ˆf ξ dξ the p th moment of ξ for ˆf for any fixed but arbitrary constant ξ m p N 0. Substituting ξ with ξ + ξ m we find from that µ p ˆf ξ p ˆf ξ + ξm dξ ξ p e πix mξ+ξ m ˆf ξ + ξm dξ ξ p ĝ ξ dξ 1 F g p π p ξ dξ From this 4. we find 1 π p µ p ˆf 1 π p g p x dx. e πixξm f x + x m p dx. Placing x x m on x in this identity one gets µ p ˆf 1 π p e πix xmξm f x p dx 1 π p e ax f x p dx. Employing e ax f x f a x in this new identity we find µ p ˆf 1 f p π p a x dx completing the proof of the required identity THE p TH -DEIVATIVE OF THE PODUCT OF TWO FUNCTIONS We state outline a proof for the following well-known result on the p th -derivative of the product of two functions. Proposition 5.1. If f i : C i 1 are two complex valued functions of a real variable x then the p th -derivative of the product f 1 f is given in terms of the lower derivatives f m 1 f p m by p 5.1 f 1 f p p f m m 1 f p m m0 for any fixed but arbitrary p N 0 {0} N. J. Inequal. Pure Appl. Math. 51 Art

14 14 JOHN MICHAEL ASSIAS Proof. In fact for p 0 the formula 5.1 is trivial as f 1 f 0 f 0 1 f 0. When p 1 the formula 5.1 is f 1 f 1 f 1 1 f + f 1 f 1 which holds. Assume that 5.1 holds as well. Differentiating this formula we get p f 1 f p+1 p p f m+1 m 1 f p m p + f m m 1 f p+1 m m0 p+1 m1 p p p m 1 f p+1 1 f + p + 1 p + 1 p+1 m0 p + 1 m m0 f m 1 f p+1 m + m1 f p+1 1 f + p p m m0 p p p + m 1 m p p + 1 m m1 f m 1 f p+1 m as the classical Pascal identity p p P + m 1 m f m 1 f p+1 m f m 1 f p+1 m + p 0 f m 1 f p+1 m p p + 1 m f 1 f p+1 holds for m N : 1 < m p. Therefore by induction on p the proof of 5.1 is complete. f 1 f p+1 Employing the formula 5.1 with f 1 x fx f x e ax where a πξ m i i 1 ξ m fixed but arbitrary real placing f a x f 1 f x e ax fx one gets p p 5. f a p x e ax a m p m f m x. m0 Similarly from the formula 5.1 with f 1 x fx f x f x f ff p k m j we get the following formula 5.3 f k k k f j j f k j for the k th derivative of f. Note that from 5. with m k one gets the modulus of f a p to be of the form 5.4 f a p p p a k p k f k j0 k0 because e ax 1 by the Euler formula : e iθ cos θ + i sin θ with θ πξ m x. Also note that the p th moment of the real x for f is of the form 5.5 µ p f x x m p f x dx for any fixed but arbitrary constant x m p N 0. Placing x + x m on x in 5.5 we find µ p f x p f x + x m dx. J. Inequal. Pure Appl. Math. 51 Art

15 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 15 From this 4. one gets that 5.6 µ p f Besides 5.5 f a f yield 5.7 µ p f g x e πixξ m f x + x m f x + xm x p g x dx. x x m p f a x dx. 6. GENEALIZED INTEGAL IDENTITIES We state outline a proof for the following well-known result on integral identities. Proposition 6.1. If f : C is a complex valued function of a real variable x h : is a real valued function of x as well as w w p : are two real valued functions of x such that w p x x x m p wx for any fixed but arbitrary constant x m v p q 0 q p then i ii v 1 w p x h v x dx r0 1 r w p r x h v r 1 x + 1 v w p v x h x dx holds for any fixed but arbitrary p N 0 {0} N v N all r : r v 1 as well as holds if the condition w p x h v x dx 1 v r0 w p v x h x dx v 1 1 r lim w p r x h v r 1 x 0 x holds if all these integrals exist. Proof. The proof is as follows. i For v 1 the identity 6.1 holds because by integration by parts one gets wp h 1 w 1 x dx w p x dh x w p h x p h x dx. Assume that 6.1 holds for v. Claim that 6.1 holds for v + 1 as well. In fact by integration by parts from 6.1 we get wp h v+1 x dx w p x dh v x w p h v x w p 1 x h v x dx J. Inequal. Pure Appl. Math. 51 Art

16 16 JOHN MICHAEL ASSIAS w p h v x w p h v x w p h v x + v+1 1 r0 v 1 v r1 r0 v r1 1 r w r p w 1 r w p r+1 h v r 1 x + 1 v v+1 p h x dx w 1 r 1 w p r h v r 1 1 x 1 v v+1 p h x dx w 1 r w p r h v+1 r 1 x + 1 v+1 v+1 p h x dx h v+1 r 1 w x + 1 v+1 v+1 p h x dx which by induction principle on v completes the proof of the integral identity 6.1. ii The proof of 6. is clear from Special Cases of 6.: i If h x f l x v p q then from 6. one gets f 6.4 w p x l x p q dx 1 p q w p p q x f l x dx 6.5 for 0 l q p ii If h x e r qkj f k x f j x if 6.3 holds then from 6. we get p q w p x e r qkj f k x f x j dx 1 p q w p p q x e r qkj f k x f j x dx k+j q where r qkj 1 for 0 k < j q p v p q. 7. LAGANGE TYPE DIFFEENTIAL IDENTITY We state prove the new Lagrange type differential identity. Proposition 7.1. If f : C is a complex valued function of a real variable x f a e ax f where a βi with i 1 β πξ m for any fixed but arbitrary real constant ξ m as well as if p A pk β k p k 0 k p B pkj s pk p k p j β p j k 0 k < j p where s pk 1 p k 0 k p then p 7.1 f p A pk f k + a k0 0 k<j p B pkj e r pkj f k f j J. Inequal. Pure Appl. Math. 51 Art

17 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 17 holds for any fixed but arbitrary p N 0 r pkj 1 p k+j 0 k < j p e is the real part of. Note that s pk {±1} r pkj {±1 ±i}. Proof. In fact the classical Lagrange identity p p p 7. r k z k r k z k k0 k0 k0 {0} N where is the conjugate of 0 k<j p with r k z k C such that 0 k p takes the new form p p p r k z k r l z k r k z j k0 l0 k0 0 k<j p or the new identity 7.3 because 7.4 as well as k<j p 0 k<j p p r l l0 r j z k + 0 k<j p e r k r j z k z j p p r k z 0 + r l k 0 l0 p p r l r k z p + p r k z k k0 l0 k p p r k z k + k0 0 k<j p r k z j r j z k r k z j r j z k r k z j r j z k r k z j + 0 k j p r k z j r j z k p r k z 1 k 1 0 k<j p e r k r j z k z j r k z j + r j z k r k r j z k z j + r k r j z k z j r k z j + r j z k e r k r j z k z j 0 k<j p r k z j r j z k e r k r j z k z j r k z 0 + r k z r k z p. k 0 Setting p 7.6 r k k p a p k k k 1 k p βi p k i p k p k β p k one gets that 7.7 r k p k β p k A pk J. Inequal. Pure Appl. Math. 51 Art

18 18 JOHN MICHAEL ASSIAS 7.8 p p r k r j k j i p k j 1 p k p p k j p p r pkj s pk β k j p k j βi p k βi p j β p k j B pkj r pkj where A pk B pkj s pk 1 p k {±1} as well as r pkj i p k j k+j p 1 {±1 ±i}. Thus employing 5.4 substituting p z k f k r k k a p k 0 k p in 7.3 we complete from the proof of the identity 7.1. We note that Similarly we have s pk 1 if p kmod ; s pk 1 if p k + 1 mod. r pkj 1 if p k + jmod 4; r pkj i if p k + j + 1mod 4; r pkj 1 if p k + j + mod 4; r pkj i if p k + j + 3mod 4. Finally 7. may be called the Lagrange identity of first form 7.3 the Lagrange identity of second form Special cases of 7.1: 7.9 i because f 1 a e ax f A 10 f + A 11 f + B 101 e r 101 ff β f + f + β e iff β f + f + β Im ff 7.10 e iz Im z where Im z is the imaginary part of z C. J. Inequal. Pure Appl. Math. 51 Art

19 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 19 ii 7.11 Also note that another way to find 7.9 is to employ directly only 5.4 as follows: f 1 af + f f a e ax f a iβf + f iβf + f iβf + f β f + f + β i ff ff β f + f + β Im ff. A 0 f + A 1 f + A f + e r 01 ff + r 0 ff + r 1 f f β 4 f + 4β f + f + e iβ 3 ff β ff iβf f β 4 f + 4β f + f + 4β 3 Im ff β e ff + 4β Im f f. Similarly from 5.4 we get also 7.11 as follows: f a a f + af + f β f iβf + f β f + iβf + f leading easily to ON THE HEISENBEG-PAULI-WEYL INEQUALITY We assume that f : C is a complex valued function of a real variable x w : a real valued weight function of x as well as x m ξ m any fixed but arbitrary real constants. Denote f a e ax f where a πξ m i with i 1 ˆf the Fourier transform of f such that ˆf ξ e iπξx f x dx Also we denote f x µ p w f e iπξx ˆf ξ dξ. w x x x m p f x dx the p th weighted moment of x for f with weight function w µ p ˆf ξ ξ m p ˆf ξ dξ the p th moment of ξ for ˆf. Besides we denote C q 1 q p p q p if 0 q the greatest integer p q q p I ql 1 p q w p p q x f l x p dx if 0 l q J. Inequal. Pure Appl. Math. 51 Art

20 0 JOHN MICHAEL ASSIAS where I qkj 1 p q w p p q x e r qkj f k x f j x dx k+j q r qkj 1 {±1 ±i} w p x x m p w. We assume that all these integrals exist. Finally we denote q D q A ql I ql + B qkj I qkj l0 0 k<j q if D q < holds for 0 q p where q A ql β l q l q q B qkj s qk k j with β πξ m s qk 1 q k if E pf < holds for p N. Besides we assume the two conditions: 8.1 p q 1 r0 for 0 l q p 8. p q 1 r0 1 r 1 r lim x wr p p/ E pf C q D q q0 β q j k if 0 k < j q lim x wr p x f l x p q r 1 0 p q r 1 x e r qkj f k x f x j 0 p for 0 k < j q p. From these preliminaries we establish the following Heisenberg- Pauli-Weyl inequality. Theorem 8.1. If f L then p 8.3 holds for any fixed but arbitrary p N. µ p w f p µp ˆf 1 π p Equality holds in 8.3 iff the a-differential equation p f a p x c p x x m p f a x E pf of p th order holds for constants c p > 0 any fixed but arbitrary p N. J. Inequal. Pure Appl. Math. 51 Art

21 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 1 In fact if c p k p > 0 k p {0} p N then this a-differential equation holds iff p 1 k f x e πixξm a j x j m+1 p x p m+1 Pp p+j Pp 4p+j Pp m+1p+j j0 m0 holds where x x x m 0 i 1 x m ξ m are any fixed but arbitrary constants a j j p 1 are arbitrary constants in C as well as P m+1p+j p m + 1 p + j m + 1 p + j 1 m + 1 p + j + 1 denote permutations for p N m N 0 j p 1. We note that if x m is the mean of x for f then x m x f x dx x f x f x dx. 0 Thus if f is either odd or even then f is even x m 0. Similarly if ξ m is the mean of ξ for ˆf then ξ m ξ ˆf ξ dξ. Also ξ m 0 if ˆf is either odd or even. We also note that the conditions may be replaced by the two conditions: 8.4 lim x wr p x f l x p q r 1 0 for 0 l q p 0 r p q 1 w p r x e 8.5 lim x for 0 k < j q p 0 r p q 1. r qkj f k x f j x p q r 1 0 Proof of the Theorem. In fact from the generalized Plancherel-Parseval-ayleigh identity 4.1 the fact that e ax 1 as a πξ m i one gets 8.6 M p µ p w f µ p ˆf w x x x m p f x dx ξ ξ m p ˆf ξ dξ 1 π p w x x x m p f a x dx f p a x dx. From 8.6 the Cauchy-Schwarz inequality we find 8.7 M p 1 π p wp x f a x f a p x dx where w p x x m p w f a e ax f. From 8.7 the complex inequality 8.8 ab 1 ab + ab J. Inequal. Pure Appl. Math. 51 Art

22 JOHN MICHAEL ASSIAS with a w p x f a x b f p a 8.9 M p 1 π p 1 x we get w p x f a x f p a x + f a xf a p x dx. From 8.9 the generalized differential identity * one finds p/ M p w p+1 π p p x d p q C q f q dx p q a x dx From 8.10 the Lagrange type differential identity 7.1 we find p/ 1 q 8.11 M p w p+1 π p p x d p q C q A dx p q ql f l x q0 l0 q0 + 0 k<j q B qkj e r qkj f k x f x dx j. From the generalized integral identity 6. from f L the two conditions or or from that all the integrals exist one gets w p x dp q f l x 8.1 dx 1 p q w p q dx p q p x f l x dx as well as 8.13 I ql w p x dp q dx e r p q qkj f k x f j x 1 p q w p p q x e r qkj f k x f j x I qkj. From we find the generalized p th order moment Heisenberg uncertainty inequality for p N p/ 1 q M p C p+1 π p q A ql I ql + H p B qkj I qkj q0 l0 0 k<j q where 1 p+1 π p E pf H p p/ E pf C q D q if E pf < q0 holds or the general moment uncertainty formula 8.14 p M p 1 π p p E pf. Equality holds in 8.3 iff the a-differential equation f a p x c p x p f a x of p th order with respect to x holds for some constant c p 1 k p > 0 k p {0} any fixed but arbitrary p N.. J. Inequal. Pure Appl. Math. 51 Art

23 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 3 We consider the general a-differential equation d p y a p dx + p k px p y 0 of p th order where x x x m 0 k p 0 y f a x e ax f x a πξ m i p N the equivalent -differential equation p because d p y dx p + k px p y 0 dy dx dy dx dx dx dy d x x m dx dx dy dx d p y dx dp y p dx p p N. In order to solve equation p one may employ the following power series method in p. In fact we consider the power series expansion y n0 a nx n about x 0 0 converging absolutely in x < ρ lim a n x 0. Thus n a n+1 k px p y d p y dx p n0 np k pa n x n+p P n p a n x n p with permutations Pp n n n 1 n n p + 1 Pp n+p a n+p x n+p with n + p on n above P n+p n+pp or n p p n + p n + p 1 n + p n + p Pp n+p a n+p x n+p + Pp n+p a n+p x n+p n p n0 Pp n+p a n+p x n+p with a n+p 0 for all n { p p } or equivalently a p a p+1 a p 1 0. Therefore from these equation p one gets the following recursive relation or p a n+p k pa n n0 P n+p p a n+p + k pa n 0 Pp n+p n N 0 p N. J. Inequal. Pure Appl. Math. 51 Art

24 4 JOHN MICHAEL ASSIAS From the null condition p we get a p 0 a p a p 1 0 a 3p a 5p a 7p 0 a 3p+1 a 5p+1 a 7p a 4p 1 a 6p 1 a 8p 1 0 respectively. From p n pm with m N 0 one finds the following p sequences a pm+p a pm+p+1 a pm+p+... a pm+3p 1 with fixed p N all m N 0 such that a p k p Pp p a 0 a 4p k p Pp 4p a p k4 p Pp p Pp 4p a pm+p 1 m+1 kp m+ Pp p Pp 4p Pp pm+p are the elements of the first sequence a pm+p in terms of a 0 a p+1 k p Pp p+1 a 1 a 4p+1 k p Pp 4p+1 a p+1 a 0... a 0 k 4 p P p+1 p Pp 4p+1 a pm+p+1 1 m+1 kp m+ Pp p+1 Pp 4p+1 Pp pm+p+1 a 1 are the elements of the second sequence a pm+p+1 in terms of a 1... ap+p 1 a 3p 1 k p ap+3p 1 a 5p 1 k p P 5p 1 p a 3p 1 P 3p 1 p a p 1 kp Pp 3p 1 Pp 5p 1 a 1... a p 1... apm+3p 1 a m+3p 1 1 m+1 kp m+ a Pp 3p 1 Pp 5p 1 P p m+3p 1 p 1 are the elements of the last sequence a pm+3p 1 in terms of a p 1. Therefore we find the following p solutions or equivalently the y 0 y 0 x 1 + y 1 y 1 x x 1 + y p 1 y p 1 x x p 1 y j y j x x j 1 m+1 m0 1 m+1 m m+1 m0 1 m+1 m0 P p p Pp p+1 Pp 4p+1 k p x p m+1 Pp 4p Pp m+1p k p x p m+1 Pp m+1p+1 P 3p 1 p k p x p m+1... Pp 5p 1 P p m+3p 1 k p x p m+1 Pp p+j Pp 4p+j Pp m+1p+j J. Inequal. Pure Appl. Math. 51 Art

25 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 5 for all j { p 1} of the differential equation p in the form of power series converging absolutely by the ratio test. Thus an arbitrary solution of p of a p is of the form p 1 f x e ax a j y j x x 0 j0 with arbitrary constants a j j p 1. Choosing a 0 1 a 1 0 a 0... a p 0 a p 1 0; a 0 0 a 1 1 a 0... a p 0 a p 1 0; ; a 0 0 a 1 0 a 0... a p 0 a p 1 1 one gets that y i i { p 1} are partial solutions of p satisfying the initial conditions y y y p ; y y y p ; ; y p y p y p 1 p If Y p y 0 y 1... y p 1 then the Wronskian at x 0 is W Y p 0 y 0 0 y 1 0 y p 1 0 y 0 0 y 1 0 y p 1 0 y p y p y p 1 p yielding that these p partial solutions y 0 y 1... y p 1 of p are linearly independent. Thus the above formula y f a x p 1 j0 a jy j gives the general solution of the equation p also of a p. We note that both the above-mentioned differential equations a p p are solved completely via well-known special functions for p 1 with Gaussian functions for p with Bessel functions via functions in terms of power series converging in for p 3. Therefore the proof of our theorem is complete. We claim that if p 1 the function f : C given explicitly in our Introduction with c c 1 k1/ > 0 k 1 {0} satisfies the equality of H 1. In fact the corresponding a-differential equation a 1 dy dx + k 1x y 0 J. Inequal. Pure Appl. Math. 51 Art

26 6 JOHN MICHAEL ASSIAS where x x x m 0 y f a x e ax f x a πξ m i is satisfied by y 0 y 0 x m m0 m0 by the power series method because or k 1x m+1 P 1 P 4 1 P m m+1 c 1 x m+1 m + 1! P 1 P 4 1 P m m + 1 m+1 m + 1! y m1 1 m cx m m! 1 + e cx 1 e cx because m1 1m t m m! e t 1. Therefore the general solution of the differential equation a 1 is of the form y a 0 y 0 with arbitrary constant c 0 a 0 or e ax f x c 0 e cx xm or f x c 0 e πixξm e cx xm. However one may establish this f much faster by the direct application of the method of separation of variables to the differential equation a 1. Analogously to the proof of H 1 in the Introduction we prove the following more general inequality H Fourth Order Moment Heisenberg Inequality. For any f L f : C any fixed but arbitrary constants x m ξ m the fourth order moment Heisenberg inequality H µ 4 f µ 4 ˆf 1 64π 4 E f holds if µ 4 f x 4 f x dx µ 4 ˆf ξ 4 ˆf ξ dξ with x x x m ξ ξ ξ m are the fourth order moments ˆf ξ e iπξx f xdx f x e iπξx ˆf ξdξ i 1 as well as E f 1 4π ξmx f x x f x 4πξ m x Im f x f x dx if E f < holds where Im denotes the imaginary part of. Equality holds in H iff the a-differential equation f a x c x f a x of second order holds for a πξ m i y f a x e ax f x a constant c 1 k > 0 k {0} x x x m 0 or equivalently a d y dx + k x y 0 J. Inequal. Pure Appl. Math. 51 Art

27 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 7 holds iff f x x e πixξm c 0 J 1/4 1 k x + c 1 J 1/4 1 k x holds for some constants c 0 c 1 C with J ±1/4 the Bessel functions of the first kind of orders ± We note that if x 4 m 0 ξ m 0 then f x 1 1 x c 0 J 1/4 k x + c 1 J 1/4 k x. We claim that the above function f in terms of the Bessel functions J ±1/4 is the general solution of the said a-differential equation a of second order. In fact the -differential equation d y dx is equivalent to the following Bessel equation of order r 1 4 with u equation is z d u dz + z du dz + y x / u z y x + k x y 0 z 1 u 0 4 z 1 k x. But the general solution of this Bessel c 0 J 1/4 z + c 1 J 1/4 z c 0 J 1/4 1 k x + c 1 J 1/4 1 k x for some constants c 0 c 1 C. In fact if we denote 8.15 S sgn x then one gets d u dz Thus we establish But or d dx du dz du / dz dx dx / du dz dz dx { 1 x > 0 1 x < 0 / S dy 1 y k x 3/ dx x 5/ / S d y S dy + 5 y k x 5/ dx x 7/ dx 4. x 9/ z d u dz + z du dz 1 4 x 3/ d y dx d y dx k x y u. x 3/ d y k dx x 4 y x 4z u. Therefore the above-mentioned Bessel equation holds. J. Inequal. Pure Appl. Math. 51 Art

28 8 JOHN MICHAEL ASSIAS However dy dx dy dx dx dx dy d x x m dx dx d y dx d dy d dy dx dx dx dx dy dx dx dx d y. dx Therefore the above two equations a are equivalent. Thus one gets y f a x e ax f x 1 1 x c 0 J 1/4 k x + c 1 J 1/4 k x or f x 1 1 x e c ax 0 J 1/4 k x + c 1 J 1/4 k x establishing the function f in terms of the two Bessel functions J ±1/4. However z z n 4 J 1/4 z 1 1 n n!γ z > n + 1 n0 4 Thus if z 1 k x > 0 then 1 x J 1/4 k x S Γ 4 k 1 x k 4 5 x5 + 4 because 5 Γ Γ Γ Γ... 4 S sgn x x 0 such that x Sx. Similarly z 1 z n 4 J 1/4 z 1 n n!γ z > n Therefore because or Γ 3 4 x J 1/4 1 k x Γ 1 Γ 4 3 Γ 4 π Γ Γ 1 4 n0 Γ 1 4 π 1 4 k 1 k 3 4 x4 + 1 Γ 1 1 π 4 4 sin 1π π 4 7 Γ Γ 4 3 π 4Γ A direct way to find the general solution of the above -differential equation is by applying the power series method for. In fact consider two arbitrary constants a 0 c 0 a 1 c 1 such that y a n x n about x 0 0 converging absolutely in Thus n0 x < ρ lim dy dx n1 n a n a n+1 na n x n 1 x 0. n + 1 a n+1 x n n0 J. Inequal. Pure Appl. Math. 51 Art

29 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 9 as well as d y dx n1 n + 1 na n+1 x n 1 n + n + 1 a n+ x n n0 n+0 or n a + 6a 3 x + n + 4 n + 3 a n+4 x n+ n0 k x y k n + 4 n + 3 a n+4 x n+ n0 Therefore from one gets the recursive relation a n x n+. k a n+4 a n n + 4 n + 3 with a a 3 0. Letting n 4m with m N 0 {0} N in this recursive relation we find the following two solutions of the equation : y 0 y 0 x m+1 kx 4 m m + 3 4m + 4 m0 1 k 3 4 x4 + y 1 y 1 x x 1 + m0 k x8 k x1 + 1 m+1 kx 4 m m + 4 4m + 5 k 4 x k 4 5 x x x13 +. We note that each one of these two power series converges by the ratio test. Thus an arbitrary solution of of a is of the form or y c 0 y 0 + c 1 y 1 k 6 f x e ax c 0 y 0 x + c 1 y 1 x where x x x m 0 π 4 k x J 1 1/4 k x y 0 Γ S { ± 1} 1 4 J. Inequal. Pure Appl. Math. 51 Art

30 30 JOHN MICHAEL ASSIAS SΓ 1 4 x J 1 1/4 k x y 1 4 k where S is defined by Besides we note that from a 0 a 3 0 the above recursive relation one gets a 6 a 10 a 14 0 as well as a 7 a 11 a 15 0 respectively. From this recursive relation n 4m with m N 0 we get the following two sequences a 4m+4 a 4m+5 such that a 4 k 3 4 a 0 a 8 k 7 8 a 4 k a 0... a 4m+4 1 m+1 k m m + 3 4m + 4 a 0 a 5 k 4 5 a 1 a 9 k 8 9 a 5 k a 1... a 4m+5 1 m+1 k m m + 4 4m + 5 a 1. Choosing a 0 c 0 1 a 1 c 1 0; a 0 c 0 0 a 1 c 1 1 one gets that y 0 y 1 are partial solutions of satisfying the initial conditions y y 0 0 0; y y Therefore the Wronskian of y 0 y 1 at x 0 is y 0 0 y 1 0 W y 0 y 1 0 y 0 0 y yielding that these p solutions y 0 y 1 are linearly independent. We note that if we divide the above power series expansion solutions y 0 y 1 we have y 1 x y 0 x y 1 x y 0 x 1 x k 0 x5 + 1 k 1 x4 + x + k 30 x5 + which obviously is nonconstant implying also that y 0 y 1 are linearly independent. Thus the above formula y a 0 y 0 + a 1 y 1 gives the general solution of also of a. Similarly we prove the following inequality H Sixth Order Moment Heisenberg Inequality. For any f L f : C any fixed but arbitrary constants x m ξ m the sixth order moment Heisenberg inequality H 3 µ 6 f µ 6 ˆf 1 56π 6 E 3f holds if µ 6 f x 6 f x dx 1 J. Inequal. Pure Appl. Math. 51 Art

31 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 31 µ 6 ˆf ξ 6 ˆf ξ dξ with x x x m ξ ξ ξ m as well as E 3f 3 ˆf ξ e iπξx f xdx f x e iπξx ˆf ξdξ 1 6π ξmx f x 3x f x 1πξ m x Im f x f x dx if E 3f < holds where Im denotes the imaginary part of. Equality holds in H 3 iff the a-differential equation f a x c 3 x 3 f a x of third order holds for a πξ m i i 1 f a e ax f a constant c 3 k 3 > 0 k 3 or equivalently iff f x e πixξm a j x j 1 + j0 1 m+1 m0 k3x 6 m j 5 + j 6 + j 6m j 6m j 6m j holds where x 0 a j j 0 1 are arbitrary constants in C. Consider the a-differential equation a 3 d 3 y dx 3 + k 3x 3 y 0 with y f a x the equivalent -differential equation 3 d 3 y dx 3 + k 3x 3 y 0 with x x x m k 3 {0} such that d 3 y/ dx 3 d 3 y/ dx 3. Employing the power series method for 3 one considers the power series expansion y n0 a nx n about x 0 0 converging absolutely in x < ρ lim n a n a n+1 x 0. Thus k 3x 3 y n0 k 3a n x n+3 J. Inequal. Pure Appl. Math. 51 Art

32 3 JOHN MICHAEL ASSIAS d 3 y dx 3 n3 n 3 n n 1 n a n x n 3 n + 6 n + 5 n + 4 a n+6 x n+3 6a 3 + 4a 4 x + 60a 5 x n0 P n+6 3 a n+6 x n+3 + n0 P n+6 3 a n+6 x n+3 with a 3 a 4 a 5 0 where P n+6 3 n + 6 n + 5 n + 4. Therefore from these equation 3 one gets the recursive relation 3 a n+6 k 3a n n N 0 From the null condition P3 n+6 N 3 a 3 0 a 4 0 a 5 0 the above recursive relation 3 we get a 9 a 15 a 1 0 a 10 a 16 a 0 a 11 a 17 a 3 0 respectively. From 3 n 6m with m N 0 one finds the following three sequences a 6m+6 a 6m+7 a 6m+8 such that a 6 k a 0 k 3 a a 6... k a 0 a 6m+6 1 m+1 k3 m m + 4 6m + 5 6m + 6 a 0 a 7 k a 1 k 3 a a 7... k a 1 a 6m+7 1 m+1 k3 m m + 5 6m + 6 6m + 7 a 1 J. Inequal. Pure Appl. Math. 51 Art

33 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 33 as well as a 8 k a k 3 a a 8... k a a 6m+8 1 m+1 k3 m m + 6 6m + 7 6m + 8 a. Therefore we find the following three solutions for a 0 0 for a 1 0 y 0 y 0 x m+1 k3x 6 m m + 4 6m + 5 6m + 6 m0 a6m x 6m+6 y 1 y 1 x x 1 + x + y y x x x + m0 m0 1 + m0 a 0 1 m+1 m0 a6m+7 a 1 1 m+1 m0 a6m+8 a k3x 6 m m + 5 6m + 6 6m + 7 x 6m+7 k3x 6 m m + 6 6m + 7 6m + 8 x 6m+8 for a 0 of the differential equation 3 in the form of power series converging absolutely by the ratio test. Thus an arbitrary solution of 3 of a 3 is of the form or y e ax f x a 0 y 0 + a 1 y 1 + a y f x e ax a 0 y 0 x + a 1 y 1 x + a y x x 0 with arbitrary constants a i i 0 1 in C. Choosing a 0 1 a 1 0 a 0; a 0 0 a 1 1 a 0; a 0 0 a 1 0 a 1 J. Inequal. Pure Appl. Math. 51 Art

34 34 JOHN MICHAEL ASSIAS one gets that y j j 0 1 are partial solutions of 3 satisfying the initial conditions y y y 0 0 0; y y y 1 0 0; y 0 0 y 0 0 y 0 1. Therefore the Wronskian of y j j 0 1 at x 0 is y 0 0 y 1 0 y 0 W y 0 y 1 y 0 y 0 0 y 1 0 y 0 y 0 0 y 1 0 y yielding that these p 3 partial solutions y j j 0 1 of 3 are linearly independent. Thus the above formula y j0 a jy j gives the general solution of 3 also of a 3. Analogously we establish the following inequality H 4 : 8.3. Eighth Order Moment Heisenberg Inequality. For any f L f : C any fixed but arbitrary constants x m ξ m the eighth order moment Heisenberg inequality H 4 µ 8 f µ 8 ˆf 1 104π 8 E 4f holds if µ 8 f x 8 f x dx µ 8 ˆf ξ 8 ˆfξ with x x x m ξ ξ ξ m ˆf ξ e iπξx f xdx f x as well as E 4f { 4 3 4π ξ mx + 16π 4 ξ 4 mx 4 f x dξ e iπξx ˆf ξdξ 8x 3 π ξmx f x + x 4 f x 8πξ m x 4 3 π ξmx Im f x f x } +πξ m x e f x f x x Im f x f x dx if E 4f < holds where e Im denote the real part of the imaginary part of respectively. Equality holds in H 4 iff the a-differential equation f a 4 x c 4 x 4 f a x J. Inequal. Pure Appl. Math. 51 Art

35 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 35 of fourth order holds for a πξ m i i 1 f a e ax f a constant c 4 k 4 > 0 k 4 or equivalently iff 3 f x e πixξm a j x j m+1 j0 m0 k4x 8 m j 6 + j 8m j 8m j holds where x 0 a j j are arbitrary constants in C First Four Generalized Weighted Moment Inequalities. i 8.16 M 1 µ w f µ ˆf w x x x m f x dx 1 16π C 0D π C 0 A 00 I π I π w 1 1 x f x dx 1 16π E 1f because I 00 w 1 1 f x dx with w 1 x wxx x m. We note that if w 1 then E 1f C 0 D 0 I 00 w 1 1 f x dx f x dx E f 1 E ˆf ξ ξ m ˆf ξ dξ ˆf ξ dξ by the Plancherel-Parseval-ayleigh identity if E 1f < holds. Thus from 8.16 one gets the classical second order moment Heisenberg uncertainty principle which says that the product of the variance µ f of x for the probability density f the variance µ of ξ for ˆf the probability density ˆf is at least E f which is the second order moment Heisenberg 16π Inequality H 1 in our Introduction. The Heisenberg lower bound H 1 for E 4π f 1 can be different if one chooses a different formula for the Fourier transform ˆf of f. Finally the above inequality 8.16 generalizes H 1 of our Introduction there w 1. J. Inequal. Pure Appl. Math. 51 Art

36 36 JOHN MICHAEL ASSIAS ii 8.17 M µ 4 w f µ 4 ˆf w x x x m 4 f x dx ξ ξ m 4 ˆfξ dξ 1 64π C 0D C 1 D π C 4 0 A 00 I 00 + C 1 A 10 I 10 + A 11 I 11 + B 101 I I00 β I 64π I 11 βi π 4 w x f x dx w x β f + f + β Im f f x dx because e if f x Im f f x I 00 w x f x dx I 10 w x f x dx I 11 w x f x dx I 101 with w x wxx x m. It is clear that 8.17 is equivalent to M π 4 or 8.19 where π 4 E f w x e if f x dx w β w x f x dx w x f x dx 4β 4 M 1 π E f w x Im f f x dx E f C 0 D 0 + C 1 D 1 D 0 D 1 I 00 β I 10 + I 11 βi 101 w β w f w f 4βw Im f f x dx J. Inequal. Pure Appl. Math. 51 Art

37 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 37 if E f < holds. We note that if E f 1 holds then from 8.19 one gets 4 M 1 4π H while if E f 1 then 4 M 1 π 1 > H. 4π Thus we observe that the lower bound of 4 M is greater than H if E f > 1 ; the same with H if E f 1; smaller than H if 0 < E f < 1. Finally the above inequality 8.18 generalizes H of Section 8 there w 1. iii 8.1 because M 3 µ 6 w f µ 6 ˆf w x x x m 6 f x dx ξ ξ m 6 ˆfξ dξ 1 56π C 0D C 1 D π C 6 0 A 00 I 00 + C 1 A 10 I 10 + A 11 I 11 + B 101 I I00 3 β I 56π I 11 βi w 3 56π 6 3 x f x dx + 3 w 1 3 x β f + f + β Im ff x dx I 00 w 3 3 x f x dx I 10 I 11 I 101 with w 3 x wxx x m 3. It is clear that 8.1 is equivalent to M π 6 or π 6 E 3f w 1 3 x f x dx w 1 3 x e if x f x dx w 1 3 x Im w β w 1 3 f x f x dx x f x dx w 1 3 x f x dx w 1 3 f x dx + 6β w 1 3 x Im f f x dx 6 M3 1 π 3 3 E 3f J. Inequal. Pure Appl. Math. 51 Art

38 38 JOHN MICHAEL ASSIAS where E 3f w β w 1 3 f + 3w 1 3 f + 6βw 1 3 Im f f x dx if E 3f < holds. We note that if E 3f 1 then from 8.3 we find 6 M π H while if E 3f 1 then 6 M 3 1 π 3 > H. Thus we observe that the lower bound of 6 M3 is greater than H if E 3f > 1; the same with 4 H if E 3f 1; smaller than 4 H if 0 < E 3f < 1. Finally the above inequality 8. generalizes H 4 3 of our section 8 there w 1. iv 8.4 because M 4 µ 8 w f µ 8 ˆf w x x x m 8 f x dx ξ ξ m 8 ˆf ξ dξ 1 104π C 0D C 1 D 1 + C D 1 104π C 8 0 A 00 I 00 + C 1 A 10 I 10 + A 11 I 11 + B 101 I 101 +C A 0 I 0 + A 1 I 1 + A I + B 01 I 01 + B 0 I 0 + B 1 I 1 1 I00 4 β I 104π I 11 βi β 4 I 0 + 4β I 1 + I + 4β 3 I 01 + β I 0 4βI 1 1 w 4 104π 8 4 x f x dx 4 w 4 x β f + f + β Im f f x dx + w 4 x β 4 f + 4β f + f + 4β 3 Im ff β e ff + 4β Im f f x dx 1 104π 8 E 4f I 00 I 10 I 11 I 0 I 1 I w 4 4 x f x dx w 4 x f x dx w 4 x f x dx w 4 x f x dx w 4 x f x dx w 4 x f x dx J. Inequal. Pure Appl. Math. 51 Art

39 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 39 I 101 w 4 x e if x f x dx I 01 w 4 x e if x f x dx I 0 w 4 x e f x f x dx I 1 w 4 x e if x f x dx w 4 x Im with w 4 x wxx x m 4. It is clear that 8.4 is equivalent to M4 1 π 4 where E 4f w 4 4 4β w 4 + β 4 w 4 8β 4 E 4f w 4 x Im f x f x dx w 4 x Im f x f x dx w 4 x e f x f x dx f x f x dx f + 4 w 4 + β w 4 f + w 4 f w 4 β w 4 Im ff 4β w 4 e ff + 8βw 4 Im f f x dx if E 4f < holds.we note that if E 4f 1 then from 8.5 we find 8 M π H while if E 4f 1 then 8 M 4 1 π 4 > H. Thus we observe that the lower bound of 8 M4 is greater than H if E 4f > 1; the same with 8 H if E 4f 1; smaller than 8 H if 0 < E 4f < 1. Finally the above inequality 8.4 generalizes H 8 4 of our Section 8 there w First form of 8.3 if w 1 x m 0 ξ m 0. We note that β πξ m 0 w p x x p w p p x p! p Therefore the above-mentioned four special cases i iv yield the four formulas: 8.6 E 1f f x dx E f 8.7 E f f x x f x dx 8.8 E 3f 3 f x 3x f x dx 8.9 E 4f 1 f x 4x f x + x 4 f x dx respectively if E pf < holds for p It is clear that in general q q A ql β q 0 1 if l q A ql β l q l if l q for 0 l q. J. Inequal. Pure Appl. Math. 51 Art

40 40 JOHN MICHAEL ASSIAS Thus if β 0 one gets 1 if l q 8.30 A ql 0 if l q It is obvious if β 0 that 8.31 B qkj 1 q k q k lq the Kronecker delta 0 l q. q j β q j k 0 0 k < j q such that j + k < q for 0 k < j q; that is β q j k β 0 1 for 0 k < j q. Therefore from we obtain 8.3 D q A qq I qq I qq 1 p q w p p q x f q x dx if D q < holds for 0 q p. We note that if w 1 x m 0 ξ m 0 or β 0 then w p x x p p w p q p x x p p q p p 1 p p q + 1 x p p q or 8.33 w p q p x From we get the formula p! p p q! xq p! q! xq 8.34 D q 1 p q p! q! if D q < holds for 0 q p. Therefore from 8.34 one finds that p/ E pf C q D q q0 p/ q0 1 q or the formula 8.35 E pf q0 p p q p q q p/ 1 p q p p q x q f q x dx 0 q p. 1 p q p! x q f q x dx q! p! p q q! q x q f q x dx if E pf < holds for 0 q p when w 1 xm ξ m 0. Let 8.36 m p f x p f x dx be the p th moment of x for f about the origin x m m p ˆf ξ p ˆf ξ dξ J. Inequal. Pure Appl. Math. 51 Art

41 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 41 the p th moment of ξ for ˆf about the origin ξ m 0. Denote 8.38 ε pq 1 p q p p! p q p if p N 0 q. p q q! q Thus from we find 8.39 E pf p/ ε pq x q f q x dx q0 if E pf < holds for 0 q p. If w 1 x m ξ m 0 one gets from the following Corollary 8.. Corollary 8.. Assume that f : C is a complex valued function of a real variable x w 1 x m ξ m 0 ˆf is the Fourier transform of f described in Theorem 8.1. Denote m p f or ˆf ε pq as in 8.36 or respectively for all p N. If f L all the above assumptions hold then p 8.40 m p f p mp ˆf 1 p/ π p p ε pq m q f q holds for any fixed but arbitrary p N 0 q p where 8.41 m q f q x q f q x dx. Equality in 8.40 holds iff the differential equation f p x c p x p f x of p th order holds for some c p > 0 any fixed but arbitrary p N. If q 0 then we note that 8.41 yields m 0 f q0 f x dx E f. We also note that if p 5 then p / ; q 0 1. Thus from 8.39 we get ε ! ! 0 10 ε ! ! 1 ε ! 5! Therefore 8.4 E 5f f x 60x f x + 5x 4 f x dx if E 5f < holds. Similarly if p 6 then p 3; q Thus from 8.39 one finds ε ε ε 6 70 ε 63. J. Inequal. Pure Appl. Math. 51 Art

42 4 JOHN MICHAEL ASSIAS Therefore 8.43 E 6f 360 f x 1080x f x + 135x 4 f x x 6 f x dx if E 6f < holds. In the same way one gets 8.44 E 7f 7 70 f x 50x f x + 40x 4 f x 7x 6 f x dx 8.45 E 8f 0160 f x 80640x f x x 4 f x 8.46 E 9f 9 448x 6 f x + x 8 f 4 x dx 4030 f x x f x x 4 f x 1680x 6 f x + 9x 8 f 4 x dx if E pf < holds for p We note that the cases E pf : p are given above via the four formulas Second form of 8.3 if ξ m 0. In general for w p x wxx p with x x x m where x m is any fixed arbitrary real w : a real valued weight function as well as ξ m 0 we get from that D q I qq 1 p q w x x p p q f q x dx or 1 p q 1 p q p q m0 p q m D q 1 p q p q m p q m p q m0 w m x x p p q m f q x dx w m x p! p q q + m! m p! p p q m! xp p q m f q x dx w m x x m x q f q x dx if D q < holds for 0 q p. If m 0 then one finds from 8.47 the formula Therefore from 8.47 one gets that 8.48 E pf p/ 1 p q p p q p q q0 m0 p q q p! p q q + m! m w m x x m x q f q x dx if E pf < holds for 0 q p when w : is a real valued weight function xm any fixed but arbitrary real constant ξ m 0. If m 0 x m 0 then we find from 8.48 the formula J. Inequal. Pure Appl. Math. 51 Art

43 ON THE HEISENBEG-PAULI-WEYL INEQUALITY 43 If we denote 8.49 ε pqw x 1 p q p p q then one gets from 8.48 that 8.50 E pf p/ q0 ε pqw xx q if E pf < holds for 0 q p. It is clear that the formula p/ 8.51 E pf 1 p q p p q q0 p q p q q p q m0 f q x dx p q q m0 p! p q q + m! m p/ q0 p! p q q + m! m ε pqw x x q w m x x m f q x dx w m x x m x q f q x dx if E pf < holds for 0 q p. Therefore from we get the following Corollary 8.3. Corollary 8.3. Assume that f : C is a complex valued function of a real variable x w : a real valued weight function x m any fixed but arbitrary real number ξ m 0 ˆf is the Fourier transform of f described in our above theorem. Denote µ p w f m p ˆf ε pqw x as in the preliminaries of the above theorem respectively for all p N. If f L all the above assumptions hold then p 8.5 µ p w f p mp ˆf 1 π p p/ q0 holds for any fixed but arbitrary p N 0 q p. Equality in 8.5 holds iff the differential equation f p x c p x p f x ε pqw x x q of p th order holds for some c p > 0 any fixed but arbitrary p N. f q x dx We note that for p ; q 0 1 w x w x x with x x x m ; ξ m 0 we get from 8.49 that ε 0w x w x + 4w x x + w x x ε 1w x w x. Therefore from 8.48 one obtains w 8.53 E f x + 4w x x + w x x f x w x x f x dx 1 p if E f <. J. Inequal. Pure Appl. Math. 51 Art

44 44 JOHN MICHAEL ASSIAS This result 8.53 can be found also from 8.0 where β πξ m 0 thus 8.54 E f w x f x w x f x dx if E f < holds with w x w x x x m w x x w 1 x w x x + w x x w x w x + 4w x x + w x x Third form of 8.3 if w 1. In general with any fixed but arbitrary real numbers x m ξ m x x x m ξ ξ ξ m one finds w p x p w r p x p r p! p r! xp r w p p q p! q! xq. Therefore the integrals I ql I qkj of the above theorem take the form 8.55 I ql 1 p q p! q! µ q f l 0 l q p 8.56 I qkj 1 p q p! q! µ q fkj 0 k < j q where f kj : are real valued functions of x such that 8.57 f kj x e r qkj f k x f j x 0 k < j q k+j q with r qkj µ q f l x q f l x dx is the q th moment of x for f l 8.59 µ q fkj x q f kj xdx is the q th moment of x for f kj. We note that if 0 k j l q p then 8.60 f ll x s ql f l x thus 0 l q 0 k < j q 8.61 µ q fll s ql µ q f l 0 l q p where s ql 1 q l. p p p p J. Inequal. Pure Appl. Math. 51 Art