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Hermite functions and uncertainty principles for the\n Fourier and the windowed Fourier transforms

Aline Bonami, Bruno Demange, Philippe Jaming

Published Jan 1, 2003• 119 citations
OpenAlex

Abstract

We extend an uncertainty principle due to Beurling into a characterization of\nHermite functions. More precisely, all functions $f$ on $\\mathbb{R}^d$ which may\nbe written as $P(x)\\exp (-\\langle Ax, x\\rangle)$, with $A$ a real symmetric\ndefinite positive matrix, are characterized by integrability conditions on the\nproduct $f(x) \\widehat{f}(y)$. We then obtain similar results for the windowed\nFourier transform (also known, up to elementary changes of functions, as the\nradar ambiguity function or the Wigner transform). We complete the paper with a\nsharp version of Heisenberg's inequality for this transform.

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Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms

Aline Bonami, Bruno Demange, Philippe Jaming

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on \mathbb{R}^d which may be written as P(x)\exp (-\langle Ax, x\rangle) , with A a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x) \widehat{f}(y) . We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.

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Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms

Aline Bonami, Bruno Demange, Philippe Jaming

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions $f$ on $\R^d$ which may be written as $P(x)\exp (Ax,x)$, with $A$ a real symmetric definite positive matrix, are characterized by integrability conditions on the product $f(x)\hat{f}(y)$. We also give the best constant in uncertainty principles of Gelf'and Shilov type. We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.

arXiv:math/0102111•Feb 14, 2001•0 citations

The uncertainty principle and quantum wave functions that are their own Fourier transforms

Keith Zengel, Nick DeVitto, Nathanael Hillyer +2 more

We present several variations of a proof of the position-momentum uncertainty principle that are based on the calculus of variations and does not rely on the Cauchy–Schwartz inequality. We show that the stationary uncertainty wave functions are the Hermite–Gaussian solutions to the quantum harmonic oscillator problem, that the minimum uncertainty wave function is the Gaussian, and that stationary uncertainty wave functions must be their own Fourier transforms. We also provide a calculus of variations proof of the Cauchy–Schwartz inequality. Finally, we discuss the properties of wave functions that are their own Fourier transforms and provide examples of such functions that may be of interest to teachers of undergraduate and graduate level quantum mechanics courses.

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Uncertainty principles invariant under the fractional Fourier transform

David Mustard

Abstract Uncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.

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Fourier uncertainty principles

A mathematical uncertainty principle is an inequality or uniqueness theorem concerning the joint localization of a function or system and its spectrum. A Fourier uncertainty principle is thus a statement that a function and its Fourier transform cannot both decay too rapidly. The most familiar form is the Wiener-Heisenberg inequality, 5.1 $$ \left\| {(x - x_0 )f(x)} \right\|_2 \left\| {(\xi - \xi _0 )\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} (\xi )} \right\|_2 \geqslant \frac{{\mathop {\left\| f \right\|}\nolimits_2^2 }} {{4\pi }},$$ which becomes an equality only for multiples of Gaussians of the form exp(−πα(x − x0)2 + 2πiξ0x). In this chapter, D will denote the normalized differentiation or momentum operator D = (1/2πi)d/dx. The inequality (5.1) is often written 5.2 $$ \sigma _X (f)\sigma _D (f) \equiv \left\| {Xf - \left\langle {Xf,f} \right\rangle f} \right\|_2 \left\| {Df - \left\langle {Df,f} \right\rangle f} \right\|_2 \geqslant 1,$$ where X is the multiplication operator f(x) ↦ xf(x). Barnes [22] attributes the first proof of (5.1) to a lecture delivered by Wiener in Hilbert’s mathematical physics seminar in Göttingen in 1925. In fact, Wiener discusses this lecture, though not the explicit proof of (5.1), at length in his autobiography (see [365], p. 105). What distinguished Wiener’s approach, however, was his macroscopic interpretation of (5.1), a point of view that led to further important achievements, including the Landau, Pollack and Slepian “Bell Labs” uncertainty principles in Chapter 3

Birkhäuser Boston eBooks•Dec 20, 2007•0 citations

THE HEISENBERG INEQUALITY ON ABSTRACT WIENER SPACES

Yuh-Jia Lee

【The Heisenberg inequality associated with the uncertainty principle is extended to an infinite dimensional abstract Wiener space (H, B) with an abstract Wiener measure p $_1$ . For $\phi$ $\in$ L $^2$ (p $_1$ ) and T $\in$ L(B, H), it is shown that (※Equations, See Full-text), where F(sub) $\phi$ is the Fourier-Wiener transform of $\phi$ . The conditions when the equality holds also discussed.】

Journal of the Korean Mathematical Society•Jan 1, 2001•1 citation
We extend an uncertainty principle due to Beurling into a characterization of\nHermite functions. More precisely, all functions fff on mathbbRd\\mathbb{R}^dmathbbRd which may\nbe written as P(x)exp(−langleAx,xrangle)P(x)\\exp (-\\langle Ax, x\\rangle)P(x)exp(−langleAx,xrangle), with AAA a real symmetric\ndefinite positive matrix, are characterized by integrability conditions on the\nproduct f(x)widehatf(y)f(x) \\widehat{f}(y)f(x)widehatf(y). We then obtain similar results for the windowed\nFourier transform (also known, up to elementary changes of functions, as the\nradar ambiguity function or the Wigner transform). We complete the paper with a\nsharp version of Heisenberg's inequality for this transform.
We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on \mathbb{R}^d which may be written as P(x)\exp (-\langle Ax, x\rangle) , with A a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x) \widehat{f}(y) . We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.
We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions fff on Rd\R^dRd which may be written as P(x)exp⁡(Ax,x)P(x)\exp (Ax,x)P(x)exp(Ax,x), with AAA a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x)f^(y)f(x)\hat{f}(y)f(x)f^​(y). We also give the best constant in uncertainty principles of Gelf'and Shilov type. We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.
We present several variations of a proof of the position-momentum uncertainty principle that are based on the calculus of variations and does not rely on the Cauchy–Schwartz inequality. We show that the stationary uncertainty wave functions are the Hermite–Gaussian solutions to the quantum harmonic oscillator problem, that the minimum uncertainty wave function is the Gaussian, and that stationary uncertainty wave functions must be their own Fourier transforms. We also provide a calculus of variations proof of the Cauchy–Schwartz inequality. Finally, we discuss the properties of wave functions that are their own Fourier transforms and provide examples of such functions that may be of interest to teachers of undergraduate and graduate level quantum mechanics courses.
Abstract Uncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.
A mathematical uncertainty principle is an inequality or uniqueness theorem concerning the joint localization of a function or system and its spectrum. A Fourier uncertainty principle is thus a statement that a function and its Fourier transform cannot both decay too rapidly. The most familiar form is the Wiener-Heisenberg inequality, 5.1 $$ \left\| {(x - x_0 )f(x)} \right\|_2 \left\| {(\xi - \xi _0 )\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} (\xi )} \right\|_2 \geqslant \frac{{\mathop {\left\| f \right\|}\nolimits_2^2 }} {{4\pi }},$$ which becomes an equality only for multiples of Gaussians of the form exp(−πα(x − x0)2 + 2πiξ0x). In this chapter, D will denote the normalized differentiation or momentum operator D = (1/2πi)d/dx. The inequality (5.1) is often written 5.2 σX(f)σD(f)≡∥Xf−⟨Xf,f⟩f∥2∥Df−⟨Df,f⟩f∥2⩾1, \sigma _X (f)\sigma _D (f) \equiv \left\| {Xf - \left\langle {Xf,f} \right\rangle f} \right\|_2 \left\| {Df - \left\langle {Df,f} \right\rangle f} \right\|_2 \geqslant 1,σX​(f)σD​(f)≡∥Xf−⟨Xf,f⟩f∥2​∥Df−⟨Df,f⟩f∥2​⩾1, where X is the multiplication operator f(x) ↦ xf(x). Barnes [22] attributes the first proof of (5.1) to a lecture delivered by Wiener in Hilbert’s mathematical physics seminar in Göttingen in 1925. In fact, Wiener discusses this lecture, though not the explicit proof of (5.1), at length in his autobiography (see [365], p. 105). What distinguished Wiener’s approach, however, was his macroscopic interpretation of (5.1), a point of view that led to further important achievements, including the Landau, Pollack and Slepian “Bell Labs” uncertainty principles in Chapter 3
【The Heisenberg inequality associated with the uncertainty principle is extended to an infinite dimensional abstract Wiener space (H, B) with an abstract Wiener measure p 1_11​ . For ϕ\phiϕ ∈\in∈ L 2^22 (p 1_11​ ) and T ∈\in∈ L(B, H), it is shown that (※Equations, See Full-text), where F(sub) ϕ\phiϕ is the Fourier-Wiener transform of ϕ\phiϕ . The conditions when the equality holds also discussed.】