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Functions with Bounded nth Differences

  • Donald H. Hyers
  • George Isac
  • Themistocles M. Rassias
Chapter
  • 249 Downloads
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 34)

Abstract

Let S be a commutative semigroup with a zero element and Y a real Banach space. For functions f: S → Y, we define the difference operator △ h f (x)by △ h f (x)= f (x + h) - f (x). Similarly, we define\(\Delta _{{h_1}{h_2}}^2f\left( x \right) = {\Delta _{{h_2}}}\left( {{\Delta _{{h_1}}}f\left( x \right)} \right)and\Delta _{{h_1} \cdots {h_{n + 1}}}^{n + 1}f\left( x \right) = {\Delta _{{h_{n + 1}}}}\left( {\Delta _{{h_1} \cdots {h_n}}^nf\left( x \right)} \right),n = 1,2,....\)Note that the nth difference is symmetric in the increments h1,, h n . When all the increments are equal to h we write\(\Delta _h^nf\left( x \right)\)instead of\(\Delta _{h \cdots h}^nf\left( x \right)\).

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Donald H. Hyers
  • George Isac
    • 1
  • Themistocles M. Rassias
    • 2
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece

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