The phenomenon in the essence of classical uncertainty principles is well known since the thirties of the last century. We introduce a new phenomenon which is in the essence of a new notion that we introduce: "Generalized Uncertainty Principles". We show the relation between classical uncertainty principles and generalized uncertainty principles. We generalized "Landau-Pollak-Slepian" uncertainty principle. Our generalization relates the following two quantities and two scaling parameters: 1) The weighted time spreading $\int_{-\infty}^\infty |f(x)|^2w_1(x)dx$ , ( $w_1(x)$ is a non-negative function). 2) The weighted frequency spreading $\int_{-\infty}^\infty |\hat{f}(ω)|^2w_2(ω)dω$ . 3) The time weight scale $a$ , ${w_1}_a(x)=w_1(xa^{-1})$ and 4) The frequency weight scale $b$ , ${w_2}_b(ω)=w_2(ωb^{-1})$ . "Generalized Uncertainty Principle" is an inequality that summarizes the constraints on the relations between the two spreading quantities and two scaling parameters. For any two reasonable weights $w_1(x)$ and $w_2(ω)$ , we introduced a three dimensional set in $R^3$ that is in the essence of many uncertainty principles. The set is called "possibility body". We showed that classical uncertainty principles (such as the Heiseneberg-Pauli-Weyl uncertainty principle) stem from lower bounds for different functions defined on the possibility body. We investigated qualitative properties of general uncertainty principles and possibility bodies. Using this approach we derived new (quantitative) uncertainty principles for Landau-Pollak-Slepian weights. We found the general uncertainty principles related to homogeneous weights, $w_1(x)=w_2(x)=x^k$ , $k\in N$ , up to a constant.The phenomenon in the essence of classical uncertainty principles is well known since the thirties of the last century. We introduce a new phenomenon which is in the essence of a new notion that we introduce: "Generalized Uncertainty Principles". We show the relation between classical uncertainty principles and generalized uncertainty principles. We generalized "Landau-Pollak-Slepian" uncertainty principle. Our generalization relates the following two quantities and two scaling parameters: 1) The weighted time spreading $\int_{-\infty}^\infty |f(x)|^2w_1(x)dx$, ($w_1(x)$ is a non-negative function). 2) The weighted frequency spreading $\int_{-\infty}^\infty |\hat{f}(ω)|^2w_2(ω)dω$. 3) The time weight scale $a$, ${w_1}_a(x)=w_1(xa^{-1})$ and 4) The frequency weight scale $b$, ${w_2}_b(ω)=w_2(ωb^{-1})$. "Generalized Uncertainty Principle" is an inequality that summarizes the constraints on the relations between the two spreading quantities and two scaling parameters. For any two reasonable weights $w_1(x)$ and $w_2(ω)$, we introduced a three dimensional set in $R^3$ that is in the essence of many uncertainty principles. The set is called "possibility body". We showed that classical uncertainty principles (such as the Heiseneberg-Pauli-Weyl uncertainty principle) stem from lower bounds for different functions defined on the possibility body. We investigated qualitative properties of general uncertainty principles and possibility bodies. Using this approach we derived new (quantitative) uncertainty principles for Landau-Pollak-Slepian weights. We found the general uncertainty principles related to homogeneous weights, $w_1(x)=w_2(x)=x^k$, $k\in N$, up to a constant.

arXiv:0807.2190Jul 14, 20084 citations

Uncertainty Principle

James M. Feagin

Springer eBooksJan 1, 19942 citations

Uncertainty Principles for Time-Frequency Representations

Karlheinz Gröchenig

We present a machine to produce new uncertainty principles from old ones. Traditionally an uncertainty principle is an inequality involving both a functionfand its Fourier transformf.To generate new uncertainty principles, we interpret the pair(f,f)as a time-frequency representation, replace it by a different time-frequency representation, and formulate a corresponding inequality. We discuss a few recent uncertainty principles for the short-time Fourier transform and the Wigner distribution that can be obtained in this way and suggest further problems.We present a machine to produce new uncertainty principles from old ones. Traditionally an uncertainty principle is an inequality involving both a functionfand its Fourier transformf.To generate new uncertainty principles, we interpret the pair(f,f)as a time-frequency representation, replace it by a different time-frequency representation, and formulate a corresponding inequality. We discuss a few recent uncertainty principles for the short-time Fourier transform and the Wigner distribution that can be obtained in this way and suggest further problems.

Birkhäuser Boston eBooksJan 1, 200376 citations

UNCERTAINTY PRINCIPLE

WORLD SCIENTIFIC eBooksJun 1, 20000 citations

\int_{-\infty}^\infty |f(x)|^2w_1(x)dx

, (

w_1(x)

is a non-negative function). 2) The weighted frequency spreading

\int_{-\infty}^\infty |\hat{f}(ω)|^2w_2(ω)dω

. 3) The time weight scale

a

{w_1}_a(x)=w_1(xa^{-1})

and 4) The frequency weight scale

b

{w_2}_b(ω)=w_2(ωb^{-1})

. "Generalized Uncertainty Principle" is an inequality that summarizes the constraints on the relations between the two spreading quantities and two scaling parameters. For any two reasonable weights

w_1(x)

and

w_2(ω)

, we introduced a three dimensional set in

R^3

that is in the essence of many uncertainty principles. The set is called "possibility body". We showed that classical uncertainty principles (such as the Heiseneberg-Pauli-Weyl uncertainty principle) stem from lower bounds for different functions defined on the possibility body. We investigated qualitative properties of general uncertainty principles and possibility bodies. Using this approach we derived new (quantitative) uncertainty principles for Landau-Pollak-Slepian weights. We found the general uncertainty principles related to homogeneous weights,

w_1(x)=w_2(x)=x^k

k\in N

, up to a constant.