Search results
Results From The WOW.Com Content Network
Cauchy–Schwarz inequality (Modified Schwarz inequality for 2-positive maps [27]) — For a 2-positive map between C*-algebras, for all , in its domain, () ‖ ‖ (), ‖ ‖ ‖ ‖ ‖ ‖. Another generalization is a refinement obtained by interpolating between both sides of the Cauchy–Schwarz inequality:
For example, to calculate the autocorrelation of the real signal sequence = (,,) (i.e. =, =, =, and = for all other values of i) by hand, we first recognize that the definition just given is the same as the "usual" multiplication, but with right shifts, where each vertical addition gives the autocorrelation for particular lag values: +
In mathematics, specifically in complex analysis, Cauchy's estimate gives local bounds for the derivatives of a holomorphic function. These bounds are optimal. These bounds are optimal. Cauchy's estimate is also called Cauchy's inequality , but must not be confused with the Cauchy–Schwarz inequality .
There are three inequalities between means to prove. There are various methods to prove the inequalities, including mathematical induction, the Cauchy–Schwarz inequality, Lagrange multipliers, and Jensen's inequality. For several proofs that GM ≤ AM, see Inequality of arithmetic and geometric means.
The Cauchy–Schwarz inequality is met with equality when the two vectors involved are collinear. In the way it is used in the above proof, this occurs when all the non-zero eigenvalues of the Gram matrix G {\displaystyle G} are equal, which happens precisely when the vectors { x 1 , … , x m } {\displaystyle \{x_{1},\ldots ,x_{m ...
In cases where the ideal linear system assumptions are insufficient, the Cauchy–Schwarz inequality guarantees a value of . If C xy is less than one but greater than zero it is an indication that either: noise is entering the measurements, that the assumed function relating x(t) and y(t) is not linear, or that y(t) is producing output due to ...
The Cauchy–Schwarz inequality shows that ... FandPLimitTool a GUI-based software to calculate the Fisher information and Cramér-Rao lower bound with application to ...
The Paley–Zygmund inequality is sometimes used instead of the Cauchy–Schwarz inequality and may occasionally give more refined results. Under the (incorrect) assumption that the events v , u in K are always independent, one has Pr ( v , u ∈ K ) = Pr ( v ∈ K ) Pr ( u ∈ K ) {\displaystyle \Pr(v,u\in K)=\Pr(v\in K)\,\Pr(u\in K)} , and ...