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

and Applied Analysis 3 Arithmetic is any system that satisfies the whole of the ordered field axioms whose domain is a subset of R. There are infinitely many types of arithmetic, all of which are isomorphic, that is, structurally equivalent. Nevertheless, the fact that two systems are isomorphic does not exclude their separate usage. In [2], it is shown that each ordered pair of arithmetic give rise to a calculus by a sensible use of the first arithmetic or function arguments and the second arithmetic for function values. Let α and β be arbitrarily selected generators and (α-arithmetic, β-arithmetic) is the ordered pair of arithmetic. Table 1 may be useful for the notation used in α-arithmetic and β-arithmetic. Definitions for α-arithmetic are also valid for β-arithmetic. For example, β-convergence is defined by means of βintervals and their β-interiors. In the (NC), α-arithmetic is used for arguments and β-arithmetic is used for ranges; in particular, changes in arguments and ranges are measured by α-differences and β-differences, respectively. The operators of the (NC) are applied only to functions with arguments in A and values in B. The isomorphism from α-arithmetic to β-arithmetic is the unique function ι (iota) which has the following three properties: (i) ι is one to one; (ii) ι is on A and onto B; (iii) for any numbers u and V in A, ι (u +̇ V) = ι (u) +̈ ι (V) , ι (u −̇ V) = ι (u) −̈ ι (V) , ι (u ×̇ V) = ι (u) ×̈ ι (V) , ι (u ̇ /V) = ι (u) ̈ /ι (V) , V ̸ = 0̇, u ≤̇ V ⇐⇒ ι (u) ≤̈ ι (V) . (6) It turns out that ι(x) = β{α−1(x)} for all x in A and that ι( ̇ n) = ̈ n for every integer n. Since, for example, u +̇ V = ι−1{ι(u) +̈ ι(V)}, it should be clear that any statement in α-arithmetic can readily be transformed into a statement in β-arithmetic. Throughout this paper, we define the ?̇?-th ∗-exponent x?̇? and the ̇ q-th ∗-root ̇ q √x of x ∈ R∗ by x 2̇ = x ×̇ x = α {α −1 (x) × α −1 (x)} = α {[α −1 (x)] 2


Preliminaries, Background and Notations
As a popular non-Newtonian calculus, multiplicative calculus was studied by Stanley [1] in a brief overview.Bashirov et al. [2] have recently emphasized on the multiplicative calculus and gave the results with applications corresponding to the well-known properties of derivative and integral in the classical calculus.Recently, in [3], the multiplicative calculus has extended to the complex valued functions and interested in the statements of some fundamental theorems and concepts of multiplicative complex calculus and demonstrated some analogies between the multiplicative complex calculus and classical calculus by theoretical and numerical examples.Bashirov and Riza [4] have studied on the multiplicative differentiation for complex valued functions and established the multiplicative Cauchy-Riemann conditions.Bashirov et al. [5] have investigated various problems from different fields which can be modeled more efficiently using multiplicative calculus, in place of Newtonian calculus.Quite recently, C ¸akmak and Bas ¸ar [6] have showed that non-Newtonian real numbers form a field with the binary operations addition and multiplication.Further, the non-Newtonian exponent, surd, and absolute value are defined and some of their properties are given.They also proved that the spaces of all bounded, convergent, null and absolutely p-summable sequences of the non-Newtonian real numbers are the complete metric spaces.Quite recently, Tekin and Bas ¸ar have [7] proved that the corresponding classical sequence spaces are Banach spaces over the non-Newtonian complex field.Quite recently, C ¸akir [8] has defined the sets () and () of geometric complex valued bounded and continuous functions and showed that () and () form a vector space with respect to the addition and scalar multiplication in the sense of multiplicative calculus and are complete metric spaces, where  denotes the compact subset of the complex plane C. Quite recently, Uzer [9] has investigated the waves near the edge of a conducting half plane.Firstly, the series is converted into contour integrals in a complex plane and then some contour deformations are made.After that, the resultant integrals are converted back into the series forms, which are seen to be rapidly convergent near the reflection/shadow boundaries of the conducting half plane.In the second part, a multiplicative calculus is employed for evaluating the relevant integrals, approximately.By the way, he derives a simple expression, which can be used whenever the series derived in the first part of the paper is not rapidly convergent.
Non-Newtonian calculus is an alternative to the usual calculus of Newton and Leibniz.It provides differentiation and integration tools based on non-Newtonian operations instead of classical operations.Every property in classical calculus has an analogue in non-Newtonian calculus.Generally speaking, non-Newtonian calculus is a methodology that allows one to have a different look at problems which can be investigated via calculus.In some cases, for example, for wage-rate-(in dollars, euro, etc.) related problems, the use of bigeometric calculus which is a kind of non-Newtonian calculus is advocated instead of a traditional Newtonian one.
Throughout this paper, non-Newtonian calculus is denoted by (NC), and classical calculus is denoted by (CC).Also for short we use * -continuity for non-Newtonian continuity.A generator is a one-to-one function whose domain is R and whose range is a subset of R. Each generator generates exactly one type of arithmetic, and conversely each type of arithmetic is generated by exactly one generator.As a generator, we choose the function exp from R to the set R + that is to say that In the special cases  =  and  = exp,  generates the classical and geometric arithmetic, respectively, where  denotes the identity function whose inverse is itself.The set R * of non-Newtonian real numbers are defined by R * := {() :  ∈ R}.
Following Bashirov et al. [2] and Uzer [3], the main purpose of this paper is to construct the space  * (Ω) of non-Newtonian complex valued continuous functions which forms a Banach space with the norm defined on it.Finally, we give some applications to seek how (NC) can be applied to the classical Functional Analysis problems such as approximation and inner product properties.
Consider any generator  with range  ⊆ R. By arithmetic, we mean the arithmetic whose domain is  and the operations are defined as follows: for ,  ∈  and any generator , -order  <  ⇐⇒  −1 () <  −1 () . ( Particularly, if we choose , the identity function, as an -generator, then () =  and  −1 () =  for all  ∈  and therefore -arithmetic obviously turns out to the classical arithmetic.Consider If we choose exp as an -generator defined by () =   for  ∈ C, then  −1 () = ln  and -arithmetic turns out to geometric arithmetic.Consider Arithmetic is any system that satisfies the whole of the ordered field axioms whose domain is a subset of R.There are infinitely many types of arithmetic, all of which are isomorphic, that is, structurally equivalent.Nevertheless, the fact that two systems are isomorphic does not exclude their separate usage.In [2], it is shown that each ordered pair of arithmetic give rise to a calculus by a sensible use of the first arithmetic or function arguments and the second arithmetic for function values.
Let  and  be arbitrarily selected generators and (-arithmetic, -arithmetic) is the ordered pair of arithmetic.Table 1 may be useful for the notation used in -arithmetic and -arithmetic.
Definitions for -arithmetic are also valid for -arithmetic.For example, -convergence is defined by means of intervals and their -interiors.
In the (NC), -arithmetic is used for arguments and -arithmetic is used for ranges; in particular, changes in arguments and ranges are measured by -differences and -differences, respectively.The operators of the (NC) are applied only to functions with arguments in  and values in .
The isomorphism from -arithmetic to -arithmetic is the unique function  (iota) which has the following three properties: (i)  is one to one; (ii)  is on  and onto ; (iii) for any numbers  and V in , It turns out that () = { −1 ()} for all  in  and that ( ṅ ) = n for every integer .
Since, for example,  + V =  −1 {() + (V)}, it should be clear that any statement in -arithmetic can readily be transformed into a statement in -arithmetic.
Throughout this paper, we define the ṗ -th * -exponent  ṗ and the q -th * -root q √ of  ∈ R * by and q √ =  provided there exists an  ∈ R * such that  q = .
The -absolute value of a number  in  ⊂ R is defined as (| −1 ()|) and is denoted by |  | .For each -nonnegative number , the symbol ⋅ √ will be used to denote {√ −1 ()} which is the unique -nonnegative number  whose -square is equal to .For each number  in , where the absolute value |  | of  ∈ R() is defined by The * -distance between two points  1 and  2 is defined by | and has the symmetry property, since Let any  ∈ R * be given.Then,  is called a positive non-Newtonian real number if  >  (0),  is called a non-Newtonian negative real number if  <  (0), and, finally,  is called an unsigned non-Newtonian real number if  = (0).By R * + and R * − , we denote the sets of non-Newtonian positive and negative real numbers, respectively.
Let (  ) be an infinite sequence of the elements in .Then, there is at most one element  in  such that every-interval with  in its -interior contains all but finitely many terms of (  ).If there is such a number , then (  ) is said to be -convergent to , which is called the -limit of (  ).In other words, The * -limit of a function  at an element  in  is, if it exists, the unique number  in  such that, for every infinite sequence It is trivial that Since the * -integral is a weighted * -average.
Furthermore, ∫ *   () equals to the -limit of the -convergent sequence whose th term is where  1 , . . .,   is the -fold partition of -partition of [ ,  ] and   is the common value of If  is classically continuous function and  = () = , then the * -integral is a Stieltjes integral.
It is convenient to indicate the uniform relationships between the corresponding notions of the * -calculus and classical calculus.
Then, both * -lim  →  () and lim  → ā f () exist and *lim The rest of the paper is organized as follows.
In Section 2, it is shown that the set C * of non-Newtonian complex ( * -complex) numbers forms a field with the binary operations addition ( + ) and multiplication ( × ).Further, some basic properties and inequalities which play the basic role in * -convergence and * -continuity are proved.Section 3 is devoted to the space  * (Ω) of * -continuous functions of a * -complex variable.We prove that  * (Ω) is a complete metric space with the natural metric and is a Banach space with the natural norm and the space  * (; ) of all * -bounded mappings from  into  is a Banach space.As an application part, in Section 4, we try to create the * -inner product space specifically for (MC) and give an inclusion relation between  * (Ω) and the set of * -differentiable functions.
In the final section of the paper, we note the significance of the (NC) and record some further suggestions.

* -Complex Field and * -Inequalities
In this section, following Tekin and Bas,ar [7], we give some knowledge on the * -complex field and some concerning inequalities.
Let ȧ ∈ (, + , − , × , / , ≤ ) and b ∈ (, + , − , × , / , ≤ ) be arbitrarily chosen elements from corresponding arithmetic.Then, the ordered pair ( ȧ , b ) is called as a * -point.The set of all * -points is called the set * -complex numbers and is denoted by C * , that is, Define the binary operations addition (⊕) and multiplication 2 ) as follows: ⊙ : where Then, Here and after, we know that C * is a field and the distance between two points in C * is computed by the relation  * .Now, we will see whether this relation  * is metric over C * or not, define * -norm, and try to obtain some required inequalities in the sense of non-Newtonian complex calculus.
where  * = ( ȧ , b ) and  * = ( 0 , 0 ).In this paper, we mainly focus on the sup metric on the * -complex numbers because * -continuity always required to use that metric relation.Therefore, we present the completeness of the set C * with respect to the sup metric.Theorem 8. C * is a Banach space with the norm ‖ ⋅ ‖ defined by where  * = ( ȧ , b ) and  * = ( 0 , 0 ).

Continuous Function Space over the Field C *
In this section, we construct the space of continuous functions over the field C * and show that this space is a complete metric space with max metric such that It would not be too hard to find out that the space of *continuous functions creates a normed space with the norm reduced from the sup metric.Finally, we investigate the completeness property of the spaces of * -bounded and *continuous functions.
Let Ω ⊂ C * be compact.Then, by  * (Ω), we denote the space of * -continuous functions defined on the set Ω.One can easily see that the set  * (Ω) forms a vector space over C * with respect to the algebraic operations addition (+) and scalar multiplication (×) defined on  * (Ω) as follows: ( In order to show that  * (Ω) is a metric space with the metric  * defined by (32), we give the following lemma.Proof.Let  and  be the generators on the sets of arguments and values, respectively.
(ii) One can easily see for every ,  ∈  * (Ω) that which shows that the symmetry axiom (M2) also holds.
(iii) By a routine verification for every , , ℎ ∈  * (Ω), if we apply then we obtain that ( This means that the triangle inequality (M3) also holds.Therefore, since (i)-(iii) are satisfied,  * is a metric on  * (Ω).This completes the proof.The * -norm on  * (Ω) defines a metric  * on  * (Ω) given by and is called the induced * -metric by the * -norm.
The definition of space of continuous functions makes it possible to give a much more intuitive meaning to the classical notion of uniform convergence.Convergence in the space of continuous functions space turns into the uniform convergence.One of the most important results of the concept of the space of continuous functions is the famous Stone-Weierstrass approximation theorem which is a very powerful tool for proof of general results on continuous functions.Using this theorem, we can prove some results fits for functions of special type and later extend them to all continuous functions by a density argument.In this paper, we show, with the rules of non-Newtonian calculus, its advantages to Stone-Weierstrass theorem in the space of  * (Ω), or not.The answer of this question is affirmative in some cases, but not every time when we want.In [3], Uzer showed by using multiplicative calculus which is a kind of non-Newtonian calculus that it is more flexible than the classical calculus for Bessel functions in a special domain.We can reproduce more examples for the same situation but we mainly focused on the theoretical properties of the space  * (Ω).
(i) One can easily show that That is to say that the axiom (N1) holds.
(ii) From the property of vector space axioms of the space  * (Ω), it is immediate that Hence, the absolute homogeneity axiom (N2) also holds.
(iii) It is obtained by the similar way used in the proof of Lemma 9 that This means that the triangle inequality axiom (N3) is satisfied.Since (i)-(iii) are fulfilled, ‖ ⋅ ‖ * , defined by (41), is a norm for the space  * (Ω).Definition 12. Let  be any set and let  ⊂ C * be a complex normed space.A mapping  from  into  is bounded if () is bounded in , or equivalently if max ∈ ‖()‖ * is finite.The set of all bounded mappings from  into  is denoted by  * (; ).which plays an important role in decision making in Neutral Networks.
As a second application of (MC), we mention the * -inner product property.A * -inner product space  is a vector space with an inner product ⟨, ⟩ * defined on it.A * -norm ‖ ⋅ ‖ * is defined by ‖‖ * = * √⟨, ⟩ * and if ⟨, ⟩ * = 0 holds, then  and  are called * -orthogonal vectors.A * -Hilbert space  is a complete * -inner product space.The spaces to be considered are defined as follows.
Definition 16.Let  be a vector space over the field C * or R * .A * -inner product on  is a mapping from  ×  into the scalar field  = C * (or R * ) of ; that is, with every pair of vectors  and , there is a scalar ⟨, ⟩ * called the * -inner product of  and , such that for all vectors , ,  and any scalar , the following axioms hold: then we say that  is multiplicative inner product space.
If  = (, ), then, using again the equalities (24) and [10, p. 88], we conclude that Thus, in (MC), we have  ⊙ z = exp( 2 +  2 ).In a (real or complex) * -inner product space , two vectors ,  ∈  are called orthogonal and we write  ⊥  provided ⟨, ⟩ * = 0 .For a subset  ⊆ , the set  ⊥ is defined by Corollary 19.A multiplicative inner product space satisfies the parallelogram equality.Let  = ( ẋ , ẏ ) and  = ( u , V ) such that If we apply  −1 to (64) and consider the iota function  =  ∘  −1 , we obtain that Since (  ) is -convergent to , the difference of  −1 (  ) −  −1 () converges to 0. Therefore, we conclude that Now, we have by applying  to (66) that which means that * -lim  → ∞ (  ) = () and this step concludes the proof.Following Wen [11], we give a counterexample such that there is a nowhere differentiable continuous function constructed by infinite products.Suppose 0 <   < 1 and   is an even integer for each , and ∑ ∞ =1   is convergent and set is a continuous nowhere differentiable function.
Now, let us consider that the Non-Newtonian * -calculus is multiplicative calculus, which means that the generator functions  and  are equal to () =  and exp() =   , respectively.Then, the function  defined by Wen [11] as in (68) is (1 +   sin   ) . (69) The * -continuity of  given by ( 69) is obtained from uniform convergence of the function   .Besides, as we already know from [10], and so forth, multiplicative differentiation has a relationship between the classical differentiation such as  * () = exp [ln  ()]  =    ()/() . (70) Therefore, this formula does not allow * -differentiability to the function (69), too.

Conclusion
One of the purposes of this work is to extend the classical calculus to the non-Newtonian real calculus for dealing with complex valued functions.Some of the analogies between (CC) and the (NC) are demonstrated by theoretical examples.We derive classical continuous function space in the sense of non-Newtonian calculus and try to understand their structure of being non-Newtonian vector space.Generally, we work on the vector spaces which concern physics and computing.There are lots of techniques that have been developed in the sense of (CC).If (NC) is employed together with (CC) in the formulations, then many of the complicated phenomena in physics or engineering may be analyzed more easily.Even some biological and finance problems can be solved by exponential calculus, which is just a sort of non-Newtonian calculus.Quite recently, Talo and Bas,ar have studied the certain sets of sequences of fuzzy numbers and introduced the classical sets ℓ ∞ (), (),  0 (), and ℓ  () consisting of bounded, convergent, null, and absolutely -summable sequences of fuzzy numbers in [12].Next, they have defined the -, -, and -duals of a set of sequences of fuzzy numbers and gave the duals of the classical sets of sequences of fuzzy numbers together with the characterization of the classes of infinite matrices of fuzzy numbers transforming one of the classical set into another one.Following Bashirov et al. [2] and Uzer [3], we have given the corresponding results for non-Newtonian calculus to the results obtained for fuzzy valued sequences in Talo and Bas ¸ar [12], as a beginning.As a natural continuation of this paper, we should record that it is meaningful to define the -, -, and -duals of a space of sequences of non-Newtonian real elements and to determine the duals of classical spaces ℓ * ∞ ,  * ,  * 0 , and ℓ * p together with the characterization of matrix transformations between the classical sequence spaces over the non-Newtonian complex field C * .Further, one can obtain the similar results by using another type of calculus instead of non-Newtonian calculus.
Non-Newtonian calculus is a new area in mathematics and has very pristine subjects to discuss.We just begin with the space of continuous and bounded functions which would step us to investigate more complicated theoretical structures and properties of (NC).We are trying to develop something valuable about non-Newtonian Functional Analysis, but only the mathematical authorities can decide that.

A
function  is * -continuous at a point  in  if and only if  is an argument of  and * -lim  →  () = ().When  and  are the identity function , the concepts of * -limit and * -continuity are identical with those of classical limit and classical continuity.The -change of a function  over an interval [ ,  ] is the number () − ().A * -uniform function is a function in , is * -continuous, and has the same -change over any two -interval of equal -extent.The * -uniform functions are those expressible in the form {( × ) + }, where  and  are constants in  and  is unrestricted in .By choosing  = 1 and  = 0 , we see that  is * -uniform.It is characteristic of a * -uniform function that, for each -progression of arguments, the corresponding sequence of values is a progression.The * -slope of a * -uniform function is its change over any  interval of -extent 1 .For example, the * -slope of the function {( × ) + } turns out to be ().In particular, the * -slope of  equals 1 , and the * -slope of a constant function on  equals 0 .The * -gradient of a function  over [ ,  ] is the * -slope of the * -uniform function containing (, ()) and (, ()) showed as  *   and turns out to be  *   = [ () −  ()] / [ () −  ()] .(14) If the following * -limit exists, the * -derivative of  [ * ]() at , and say that  is * -differentiable at , [ * ] () = *lim  →  {[ () −  ()] / [ () −  ()]} .(15) If it exists, [ * ]() is necessarily in .The * -derivative  *  of  is the function that assigns to each number in  the number [ * ](), if it exists.

Theorem 3 .
The * -derivative and * -integral are inversely related in the sense indicated by the following two statements.
The classical derivatives []() and [ * ]() do not necessarily coexist and are seldom equal; however, if the following exist, is denoted by  *    and defined to be the -limit of the -convergent sequence whose th term is the -average of ( 1 ), . . ., (  ), where  1 , . . .,   is the -fold -partition of [ ,  ] .The * -average of a * -uniform function on [ ,  ] is equal to the -average of its values at  and  and is equal to its value at the -average of  and .
Furthermore,  is * -continuous at  if and only if f is classically continuous at ā .