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Ulam’s Type Stability and Generalized Norms
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A symmetric functional equation is one whose form is the same regardless of the order of the arguments. A remarkable example is the Cauchy functional equation: f ( x + y ) = f ( x ) + f ( y ) . Interesting results in the study of the rigidity of quasi-isometries for symmetric spaces were obtained by B. Kleiner and B. Leeb, using the Hyers-Ulam stability of a Cauchy equation. In this paper, some results on the Ulam’s type stability of the Cauchy functional equation are provided by extending the traditional norm estimations to ther measurements called generalized norm of convex type (v-norm) and generalized norm of subadditive type (s-norm).
Title: Ulam’s Type Stability and Generalized Norms
Description:
A symmetric functional equation is one whose form is the same regardless of the order of the arguments.
A remarkable example is the Cauchy functional equation: f ( x + y ) = f ( x ) + f ( y ) .
Interesting results in the study of the rigidity of quasi-isometries for symmetric spaces were obtained by B.
Kleiner and B.
Leeb, using the Hyers-Ulam stability of a Cauchy equation.
In this paper, some results on the Ulam’s type stability of the Cauchy functional equation are provided by extending the traditional norm estimations to ther measurements called generalized norm of convex type (v-norm) and generalized norm of subadditive type (s-norm).
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