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

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.