eu THEORY OF DISCRETE FOURIER SERIES GENERATED BY GENERALIZED DIFFERENCE OPERATOR

Abstract: Constant amplitude transforms like Discrete Fourier Transform (DFT), Walsh transform, nonlinear phase Walsh-like transforms and Gold codes have been successfully used in many wire-line and wireless communication technologies including code division multiple access (CDMA), discrete multi-tone (DMT) and orthogonal frequency division multiplexing (OFDM) types. In this paper, we derive the discrete Fourier Series using orthonormal functions and generalized difference operator with its inverse. Suitable examples are provided to illustrate the main results.


Introduction
In 1807, Fourier astounded some of his contemporaries by asserting that an "arbitrary" function could be expressed as a linear combination of sine and cosine functions.For a brief but excellent account of the history of this subject and its impact on the development of mathematics, one can refer [1,2,3,4,5].These linear combinations, now called Fourier series, have become an indispensable tool in the analysis of certain periodic phenomena (such as vibrations, planetary and wave motion) which are applied in physics and engineering [6,7,8,9].
In 1989, Miller and Rose [10] introduced the discrete analogue of the Riemann-Liouville fractional derivative and proved some properties of the fractional difference operator.The general fractional h-difference Riemann-Liouville operator and its inverse ∆ −ν h f (t) were mentioned in [11,12].As application of ∆ −ν h , by taking ν = m (positive integer) and h = ℓ, the sum of the m th partial sums on n th powers of arithmetic, arithmetic-geometric progressions and products of n consecutive terms of an arithmetic progression have been derived using ∆ −m ℓ u(k), where ∆ ℓ u(k) = u(k + ℓ) − u(k) [13].
The basic problems in the theory of discrete Fourier series are described in the setting of discrete orthogonal functions.Therefore, first we present some terminology concerning discrete orthogonal function and then we develop the theory of discrete Fourier series.Throughout this paper, we assume that the and N is a positive integer.

Preliminaries
An n th root of unity is a complex number satisfying the equation If z holds equation (1) but z m = 1; 0 < m < N − 1, then z is defined as a primitive N th root of unity.The complex number z 0 = e j(2π/N ) , where j 2 = −1, is the primitive N th root of unity with the smallest positive argument.The other primitive N th roots of unity are expressed as where n and N are co-prime.All N th roots of unity satisfy the unique summation property of a geometric series expressed as A periodic with period of N, constant modulus, complex discrete-time sequence e r (k) is defined as This complex sequence over a finite discrete time interval in a geometric series is expressed according to equation (3) as follows.
This mathematical property is utilized with the factorization into two orthogonal exponential functions, where one defines the Discrete Fourier Transform(DFT) {e n (k)} satisfying where m, n are integers and the notation ( * ) represents the complex conjugate of a function.The equation ( 6) motivates us to define the generalized discrete orthonormal system and Fourier series by replacing ∆ by ∆ ℓ and e n (k) by u n (k).Definition 2.1.[14] Let u(k), k ∈ [0, ∞), be a real or complex valued function and ℓ ∈ (0, ∞).Then, the generalized difference operator ∆ ℓ on u(k) is defined as and the inverse of ∆ ℓ denoted by ∆ −1 ℓ is defined as where c is a constant.

Discrete Orthogonal Systems of Functions
The function u , where k = ∞.Hence, we consider only bounded functions on .., φ m } be a collection of bounded complex valued functions defined on I.If (φ n , φ m ) ℓ = 0 whenever m = n, the collection S ℓ is said to be a discrete orthogonal system on I with respect to ℓ.If in addition, each u n has norm 1, then S ℓ is said to be an orthonormal system.
and N is a positive integer.
Proof.First, we shall prove that From the linearity of ∆ −1 ℓ and (17), we have Using the linearity of ∆ −1 ℓ , the orthonormality of φ n and (17), we obtain where ĉq and bq are complex conjugates of c q and b q respectively.Now (29) is derived from (30) and the following relation Taking b q = c q in (29), we get Now, (28) follows from ( 29), (31) and will mean that the numbers c 0 , c 1 , c 2 , ... are given by the formula The series in (32) is called the Discrete Fourier Series of u relative to S ℓ and the numbers c 0 , c 1 , c 2 , ... are called the Discrete Fourier Coefficients of u relative to and S ℓ is the orthonormal system of trigonometric functions described in (18), then the series obtained by (32) is called discrete Fourier series generated by u.In this case, we can write (32) in the form the coefficients being given below. and The coefficients described in ( 35)-( 37) can be obtained either by closed form or summation form of ∆ −1 ℓ u(k) depending on u(k) and when N → ∞ the Discrete Fourier Series converges to Fourier Series.To obtain orthonormal system and Discrete Fourier Series we develop certain results of ∆ −1 ℓ on trigonometric functions with u(k).
Proof.Since 1 − cos pk k = 0 at k = 0 and it is bounded for all k, By Theorem (4.6), as u is bounded, we have Now, we have From (43) we get the required result.c q φ q (k).
Proof.We take b q = c q in (29) and observe that the left member is nonnegative.Therefore Theorem 4.9.Assume that {φ 0 , φ 1 , ..., φ n , ..., φ M } is a system of discrete orthonormal functions on I, N is very large and ℓ = b − a 2N is very small.Let {c n } be any sequence of complex numbers such that |c q | 2 converges.Then there is a function u bounded on I such that: (a)(u, φ q ) (ℓ) = c q for each q ≥ 0, and Proof.Since {φ q } is discrete orthonormal, we have Take u(k) = M q=0 c q φ q (k), k ∈ I.
Proof of Part (b) follows from (45)

Conclusion
When ℓ → 0, the Discrete Fourier Series and Discrete Fourier Transforms become usual Fourier Series and the Fourier Transforms.If (•)dx is not exist, then we can replace (•) by ℓ∆ −1 ℓ (•) and we can get several applications using discrete Fourier transform and its series using summation form of ∆ −1 ℓ .

4 .
Discrete Fourier Series of a Function Relative to an Orthonormal System Definition 4.1.Let S ℓ = {φ 0 , φ 1 , φ 2 , ..., φ M } be an orthonormal system on I, ℓ = b − a 2N and assume that u is a complex valued bounded function on I.The notation

Theorem 4 . 8 .
Let s ℓ = {φ 0 , φ 1 , φ 2 , ..., φ M } be a system of discrete orthonormal functions defined on I, assume that, f is bounded complex-valued function defined on I and suppose that u(k) ℓ ≈ M n=0 c n φ n (k).Then for n ≤ M , (a) The series M n=0 |c n | 2 converges and satisfies the inequality holds if and only if u − s M (ℓ) = 0, where {s n } is the sequence of partial sums defined by s n (k) = M q=0

n q=0 |c q | 2 − s n 2 (
≤ u 2 (ℓ) .This establishes (a).To prove (b), we again put b k = c k in (29) to obtain u Part (b) follows at once from this equation.