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

Silvestru Sever Dragomir ()
Additional contact information
Silvestru Sever Dragomir: College of Engineering & Science Victoria University, Mathematics

A chapter in Frontiers in Functional Equations and Analytic Inequalities, 2019, pp 511-531 from Springer

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

Date: 2019
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-28950-8_26

Ordering information: This item can be ordered from
http://www.springer.com/9783030289508

DOI: 10.1007/978-3-030-28950-8_26

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().