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Certain Spaces of Functions over the Field of Non‐Newtonian Complex Numbers

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  • Ahmet Faruk Çakmak
  • Feyzi Başar

Abstract

This paper is devoted to investigate some characteristic features of complex numbers and functions in terms of non‐Newtonian calculus. Following Grossman and Katz, (Non‐Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts, 1972), we construct the field ℂ* of *‐complex numbers and the concept of *‐metric. Also, we give the definitions and the basic important properties of *‐boundedness and *‐continuity. Later, we define the space C*(Ω) of *‐continuous functions and state that it forms a vector space with respect to the non‐Newtonian addition and scalar multiplication and we prove that C*(Ω) is a Banach space. Finally, Multiplicative calculus (MC), which is one of the most popular non‐Newtonian calculus and created by the famous exp function, is applied to complex numbers and functions to investigate some advance inner product properties and give inclusion relationship between C*(Ω) and the set of C*′(Ω)*‐differentiable functions.

Suggested Citation

  • Ahmet Faruk Çakmak & Feyzi Başar, 2014. "Certain Spaces of Functions over the Field of Non‐Newtonian Complex Numbers," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:236124
    DOI: 10.1155/2014/236124
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    References listed on IDEAS

    as
    1. Sebiha Tekin & Feyzi Başar, 2013. "Certain Sequence Spaces over the Non‐Newtonian Complex Field," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    2. Sebiha Tekin & Feyzi Başar, 2013. "Certain Sequence Spaces over the Non-Newtonian Complex Field," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-11, June.
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    Cited by:

    1. Uğur Kadak & Muharrem Özlük, 2015. "Generalized Runge‐Kutta Method with respect to the Non‐Newtonian Calculus," Abstract and Applied Analysis, John Wiley & Sons, vol. 2015(1).

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