A connection between the Uncertainty Principles on the real line and on the circle
aa r X i v : . [ m a t h . F A ] J u l A CONNECTION BETWEEN THE UNCERTAINTY PRINCIPLES ON THEREAL LINE AND ON THE CIRCLE
NILS BYRIAL ANDERSEN
Abstract.
The purpose of this short note is to exhibit a new connection between the HeisenbergUncertainty Principle on the line and the Breitenberger Uncertainty Principle on the circle, byconsidering the commutator of the multiplication and difference operators on Bernstein functions. Introduction
Consider the Bernstein space B R defined as the subspace of functions f in L ( R ) whose (dis-tributive) Fourier transform F f has support in the interval [ − R, R ]. Let A δ denote the family ofnormalized backward difference operators A δ f ( z ) = f ( z ) − f ( z − δ ) δ ( f ∈ B R , z ∈ C ) , for δ ∈ (0 , B R = { f ∈ B R : xf ( x ) ∈ L ( R ) } (note that f ∈ ˙ B R ⇒ xf ( x ) ∈ B R ), and let B : ˙ B R → B R denote the multiplication operator Bf ( x ) = xf ( x ) ( f ∈ ˙ B R , x ∈ R ) . Using an operator theoretic approach, see [2, 3, 4, 5, 6, 9, 10, 11], we get the uncertainty inequalities k ( A δ − a ) f k k ( B − b ) f k ≥ |h f ( · − δ ) , f i| , for 0 = f ∈ ˙ B R , and all a, b ∈ C .At the limit δ → R arbitrary), we recover the Heisenberg Uncertainty Principle ([7])for functions on the line, and at the endpoint δ = 1 (with R = π ), we recover the BreitenbergerUncertainty Principle ([1]) for functions on the circle, thus giving a new, and easy, link betweenthe two Uncertainty Principles. Another connection between the two Uncertainty Principles wasdiscussed in [10].Finally, we show the equivalence of the Heisenberg Uncertainty Principle to another UncertaintyPrinciple on the circle ([4, § C δ f ( z ) = f ( z + δ ) − f ( z − δ )2 δ ( f ∈ B R , z ∈ C ) , for δ ∈ (0 , Date : 17 July, 2013.2010
Mathematics Subject Classification.
Key words and phrases.
Uncertainty Principles, Bernstein Spaces. Uncertainty Principles for symmetric and normal operators
Let H be a Hilbert space with inner product h· , ·i and norm k · k = h· , ·i / . For A and B linearoperators with domains D ( A ) , D ( B ) respectively, and range in H , the (normalized) expectation valueof the operator A with respect to f ∈ D ( A ) is defined as τ A ( f ) := h Af, f ih f, f i , and the standard deviation, or variance, of the operator A with respect to f ∈ D ( A ) is defined as σ A ( f ) := k Af − τ A ( f ) f k = min a ∈ C k ( A − a ) f k . We notice that τ A ( f ) f is the orthogonal projection of Af onto f . The commutator of A and B isdefined as [ A, B ] := AB − BA , with domain D ( AB ) ∩ D ( BA ).From [6, Corollary 1], [11, Theorem 3.1] and [11, Corollary 3.3], we get the following uncertaintyprinciple: Theorem 1. If A, B are symmetric or normal operators on a Hilbert space H , then k ( A − a ) f kk ( B − b ) f k ≥ σ A ( f ) σ B ( f ) ≥ |h [ A, B ] f, f i| , for all nonzero f ∈ D ( AB ) ∩ D ( BA ) , and all a, b ∈ C . Theorem 1 can be used to prove the Heisenberg Uncertainty Principle on the real line (with H = L ( R ) , Af ( x ) = xf ( x ) , Bf ( x ) = if ′ ( x )), and the Breitenberger Uncertainty Principle on thecircle (with H = L ( − π, π ) , Af ( x ) = e ix f ( x ) , Bf ( x ) = if ′ ( x )), see the references.3. The Bernstein spaces B R Let
R >
0. Recall the definition of the Bernstein spaces B R from the introduction. By theclassical Paley–Wiener theorem, B R can be identified with the space of entire functions f on C ofexponential type R , whose restriction to R belongs to L ( R ). It is well-known that B R is a Hilbertspace equipped with the L ( R ) norm k f k (of the restriction of f to the real line), which is invariantunder differentiation, and the Bernstein inequality k f ′ k ≤ R k f k holds, for all f ∈ B R . See also[8, Lecture 20] for general results concerning L p -Bernstein spaces.Let sinc ( z ) = sin( πz ) /πz . Let l ( Z ) denote the space of square-summable sequences defined onthe integers Z . Then B π and l ( Z ) are isomorphic, with proportional norms, and the isomorphismis given by the Whittaker–Kotel’nikov–Shannon Sampling Formula f ( z ) = X n ∈ Z a k sinc ( z − n ) , ( { a k } k ∈ Z ∈ l ( Z )) , which converges in L to a function f ∈ B π , given as the unique solution of the interpolation problem f ( k ) = a k , k ∈ Z . Conversely, for any function f ∈ B π , the sequence { f ( k ) } k ∈ Z belongs to l ( Z ).4. Uncertainty Principles for Bernstein spaces
Consider the family of difference operators A δ : B R → B R , with δ ∈ (0 , δ = 1, this is the usual backward difference operator ∂f ( z ) = A f ( z ) = f ( z ) − f ( z − A ∗ δ : B R → B R , is given by A ∗ δ f ( z ) = ( f ( z ) − f ( z + δ )) /δ , and since A δ A ∗ δ = A ∗ δ A δ ,we see that A is a normal operator. The multiplication operator B is obviously a symmetric operator.A small computation yields that[ A δ , B ] f ( x ) = f ( x − δ ) ( f ∈ ˙ B R , x ∈ R ) . From Theorem 1, we thus have
NCERTAINTY PRINCIPLES 3
Theorem 2.
Let δ ∈ (0 , . Let = f ∈ ˙ B R . Then k ( A δ − a ) f k k ( B − b ) f k ≥ σ A δ ( f ) σ B ( f ) ≥ |h f ( · − δ ) , f i| , for all a, b ∈ C . Let { f ( n ) } n ∈ Z be a sequence in l ( Z ), and denote by ˇ f ∈ L ( − π, π ) the Fourier inverse of f . TheFourier series corresponding to the function e iθ ˇ f is the sequence { f ( n − } . The Fourier inverse of xf ( x ) ∈ B π , or { nf ( n ) } ∈ l ( Z ), is i ddθ ˇ f . Let δ = 1 and R = π , then Theorem 2 yields Corollary 3.
The Breitenberger Uncertainty Principle for functions on the circle. Let f ∈ ˙ B π .Then (cid:13)(cid:13) ( e iθ − a ) ˇ f (cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ddθ − b (cid:19) ˇ f (cid:13)(cid:13)(cid:13)(cid:13) ≥ (cid:12)(cid:12)(cid:10) e iθ ˇ f , ˇ f (cid:11)(cid:12)(cid:12) , for all a, b ∈ C . We can rewrite this in terms of the Fourier coefficients { f ( n ) } , X n ∈ Z | f ( n − − af ( n ) | ! X n ∈ Z | ( n − b ) f ( n ) | ! ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ∈ Z f ( n − f ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , which holds for all square-summable sequences { f ( n ) } n ∈ Z ∈ l ( Z ).Let δ →
0, then Theorem 2 yields
Corollary 4.
The Heisenberg Uncertainty Principle for functions on the line. Let f ∈ B R , for some R > . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ddx − a (cid:19) f (cid:13)(cid:13)(cid:13)(cid:13) k ( x − b ) f k ≥ k f k , and all a, b ∈ C . The inequality holds for f ∈ ˙ B R by Theorem 2, and easily extends to all f ∈ B R . A standarddensity argument furthermore extends the inequality to all functions f ∈ L ( R ).Finally, let us consider the central difference operators C δ . Since[ C δ , B ] f ( x ) = f ( x + δ ) + f ( x − δ )2 ( f ∈ ˙ B R , x ∈ R ) , Theorem 1 gives
Theorem 5.
Let δ ∈ (0 , . Let = f ∈ ˙ B R . Then k ( C δ − a ) f k k ( B − b ) f k ≥ σ C δ ( f ) σ B ( f ) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) f ( · + δ ) + f ( · − δ )2 , f (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) , for all a, b ∈ C . In the limit δ →
0, Theorem 5 yields the Heisenberg Uncertainty Principle as before. So let δ = 1and R = π , then, Corollary 6.
Let f ∈ ˙ B π . Then (cid:13)(cid:13) (sin( θ ) − a ) ˇ f (cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ddθ − b (cid:19) ˇ f (cid:13)(cid:13)(cid:13)(cid:13) ≥ (cid:12)(cid:12)(cid:10) cos( θ ) ˇ f , ˇ f (cid:11)(cid:12)(cid:12) , for all a, b ∈ C . NILS BYRIAL ANDERSEN Final remarks
Normally, when we discuss Uncertainty Principles mathematically, we say that f or F f cannot belocalized at the same time. Here, we assume that F f is localized as supp F f ⊂ [ − R, R ], or f ∈ B R ,which is another reason why it may be interesting to look at B R .The Bernstein inequality k f ′ k ≤ R k f k , together with the Heisenberg Uncertainty Principle,also yields the following inequality, for 0 = f ∈ B R , and a ∈ C ,12 k f k ≤ k f ′ k k ( x − a ) f k ≤ R k f k k ( x − a ) f k , or k ( x − a ) f k ≥ k f k R , which indeed supports the claim that localization of the frequency, i.e., R small, implies indetermi-nacy of the position. References [1] E. Breitenberger,
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J. Fourier Anal. Appl. (1997),207–233.[4] T.N.T. Goodman, S.S. Goh, Uncertainty principles and optimality on circles and spheres, Advances in construc-tive approximation: Vanderbilt 2003, 207218, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2004.[5] S.S. Goh, T.N.T. Goodman, Uncertainty principles and asymptotic behavior, Appl. Comput. Harmon. Anal. (2004), 19–43.[6] S.S. Goh, C.A. Micchelli, Uncertainty principles in Hilbert spaces, J. Fourier Anal. Appl. (2002), 335–373.[7] W. Heisenberg, ¨Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. f. Physik. (1927), 172–198.[8] B.Ya. Levin, Lectures on entire functions , Translations of Mathematical Monographs, Vol. 150, Amer. Math.Soc., Providence, RI.[9] J. Prestin, E. Quak, Optimal functions for a periodic uncertainty principle and multiresolution analysis,
Proc.Edinburgh Math. Soc. (2) (1999), 225–242.[10] J. Prestin, E. Quak, H. Rauhut, K. Selig, On the connection of uncertainty principles for functions on the circleand on the real line, J. Fourier Anal. Appl. (2003), 387-409.[11] K. Selig, Uncertainty Principles revisited, Electron. Trans. Numer. Anal. (2002), 165–177. Department of Mathematics, Aarhus University, Ny Munkegade 118, Building 1530, DK-8000 AarhusC, Denmark
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