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Inequalities for Symmetrized or Anti-Symmetrized Inner Products of Complex-Valued Functions Defined on an Interval

  • Silvestru Sever DragomirEmail author
Chapter

Abstract

For a function \(f:\left [ a,b\right ] \rightarrow \mathbb {C}\) we consider the symmetrical transform of f on the interval \(\left [ a,b\right ],\) denoted by f̆, and defined by
$$\displaystyle \breve {f}\left ( t\right ) :=\frac {1}{2}\left [ f\left ( t\right ) +f\left ( a+b-t\right ) \right ],t\in \left [a,b\right ] $$
and the anti-symmetrical transform of f on the interval \(\left [ a,b\right ] \) denoted by \(\tilde {f}\) and defined by
$$\displaystyle \tilde {f}:=\frac {1}{2}\left [ f\left ( t\right ) -f\left ( a+b-t\right ) \right ] ,t\in \left [ a,b\right ]. $$
We consider in this paper the inner products
$$\displaystyle \left \langle f,g\right \rangle _{\smile }:=\int _{a}^{b}\breve {f}\left ( t\right ) \overline {\breve {g}\left ( t\right ) }dt\text{ and }\left \langle f,g\right \rangle _{\sim }:=\int _{a}^{b}\tilde {f}\left ( t\right ) \overline { \tilde {g}\left ( t\right ) }dt, $$
the corresponding norms and establish their fundamental properties. Some Schwarz and Grüss’ type inequalities are also provided.

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Authors and Affiliations

  1. 1.MathematicsCollege of Engineering & Science Victoria UniversityMelbourne CityAustralia
  2. 2.DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science & Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa

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