Xiuxiu Gao,Qiang Feng,Yinyin Mei,Yi Xiang

Abstract:Fractional sine series (FRSS) and fractional cosine series (FRCS) are the discrete form of the fractional cosine transform (FRCT) and fractional sine transform (FRST).The recent studies have shown that discrete convolution is widely used in optics,signal processing and applied mathematics.In this paper,firstly,the definitions of fractional sine series (FRSS) and fractional cosine series (FRCS) are presented.Secondly,the discrete convolution operations and convolution theorems for fractional sine and cosine series are given.The relationship of two convolution operations is presented.Lastly,the discrete Young’s type inequality is established.The proposed theory plays an important role in digital filtering and the solution of differential and integral equations.

Keywords:fractional cosine series;fractional sine series;discrete convolution;discrete Young’s inequality

1 Introduction

The fractional cosine transform (FRCT) and fractional sine transform (FRST) are important applications in various fields,such as filter design[1−2],image encryption [3],radar system analysis[4],solving integral and differential equations[5−6],etc.

As a discrete form of the fractional cosine transform (FRCT) and fractional sine transform(FRST) [2,6−7],the fractional sine series (FRSS)and fractional cosine series (FRCS) are also a generalization of the Fourier cosine and sine series [8−9].Recently,the discrete Fourier cosine convolution and the discrete Fourier sine convolution associated with the Fourier cosine and sine series [8−9]are studied,and many useful theorems have been obtained.Motivated by the discretization of continuous fractional transform,we will investigate fractional sine series (FRSS) and fractional cosine series (FRCS) as well as the corresponding convolution in fractional domain.Since the convolution plays an important role in the field of mathematical and signal processing,so,it is theoretically interesting and practically useful to consider the FRSS and FRCS involving their convolution theorems.

The main contributions of this paper can be stated as follows:two novel discrete convolution structures for the fractional sine series (FRSS)and the fractional cosine series (FRCS) are proposed,and the corresponding convolution theorems are derived.The relationship of two discrete convolution operations are obtained,and the discrete Young’s type inequality for the convolution operation is discussed.The rest of this paper is organized as follows:Section 2 reviews the definition for Fourier sine and cosine series and its convolution operations.Section 3 gives two different convolution operations for FRCT and FRST,the two different kinds of convolution theorems for FRCT and FRST are derived,and the important relationship between the convolution operations for FRCS and the FRSS are also given.The discrete Young’s type inequality for FRSS is also discussed in Section 4 and conclusions are made in Section 5.

2 Preliminaries

In this section,we mainly review some basic facts on Fourier sine and cosine series,which will be needed throughout the paper.

LetLp(N0),1 ≤p<∞denote the space of sequences{x(n)}equipped with a norm

The discrete Fourier cosine convolution operations [8]and the discrete Fourier sine convolution operations [8−9]of sequencesxandyis defined by

respectively,which satisfied the following convolution theorems

respectively,whereFcDTdenotes the Fourier cosine series

andFsDTdenotes the Fourier sine series

respectively.

The classical Young’s inequality [10−12]is defined as

3 The Main Results

In this section,two novel convolution operations for the FRCS and FRSS are defined.The corresponding convolution theorems for FRCS and FRSS are derived in details.The relationships between the proposed convolution operations are also investigated.

Definition 1Letα∈R,a sequence{x(n)}∈L1(N0),the fractional cosine series and the fractional sine series are defined as

respectively,whereKφ(n,ω) is given by

Whenα=2k −1,k ∈Z,the fractional cosine series in (10) and the fractional sine series in(11) reduce to the classical Fourier cosine series[8]in (6) and Fourier sine series [9]in (7),except for some individual terms.

3.1 The Convolution Operations for FRCS and FRSS

3.2 The Convolution Theorems for FRCS and FRSS

Based on the Definition 2 and Definition 3,the convolution theorems for the FRCS and FRSS are derived respectively.

Next we prove the convolution Theorem 1,from the Definition 2,we have

The proof of Theorem 2 is achieved.

3.3 Convolution Relation of the Fractional Cosine and Sine Series

The proof is completed.

4 Discrete Young’s Type Inequality

In this section,the discrete Young’s type inequality is presented.

Theorem 4(A discrete Young’s type theorem)Letp,q,r >1,x(n)∈Lp(N0),y(n)∈Lq(N0),z(n)∈Lr(N0) and 1/p+1/q+1/r=2,then

ProofLetp1,q1,r1be the conjugate exponents ofp,q,r,respectively,and satisfied following equations

then we have 1/p1+1/q1+1/r1=1.Let

from (30),(31) and (32),we can get

Whenq >1,from the convex function inequality,we have

Therefore,we have

Similarly to (36),we get

From (36),(37) and (38),we have

With Hölder’s inequality,we have

From (39),inequality (28) is achieved,which completed the proof.

5 Conclusion

In this paper,convolution operations for the fractional cosine series (FRCS) and the fractional sine series (FRSS) are proposed,the corresponding convolution theorems are derived in detail,and the important relationship between the FRCS and the FRSS is obtained.The discrete Young’s type inequality for FRSS is also discussed.