Abstract
Two numerical characteristics of a nonrectifiable arc \(\gamma \subset \mathbb{C}\) generalizing the notion of length are introduced. Geometrically, this notion can naturally be generalized as the least upper bound of the sums \(\sum {\Phi (a_j )}\), where \({a_j }\) are the lengths of segments of a polygonal line inscribed in the curve \(\gamma\) and \(\Phi\) is a given function. On the other hand, the length of \(\gamma\) is the norm of the functional \(f \mapsto \int_\gamma {f{\text{ }}dz{\text{ }}}\) in the space \(C{\text{(}}\gamma {\text{)}}\); its norms in other spaces can be considered as analytical generalizations of length. In this paper, we establish conditions under which the generalized geometric rectifiability of a curve \(\gamma\) implies its generalized analytic rectifiability.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
