A New Notion of Fractional Fourier Series
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this paper, we introduce the concept of the β-orthogonality of a system of real valued functions {ϕn(x)}∞ n=0 defined on an interval (0,b), b > 0. Using this concept, we define the fractional β-cosine (or β-sine) Fourier series of a function f ∈ L2 c(0,(2π)1/β) (or f ∈L2 s(0,(2π)1/β)), β > 0. Furthermore, we show that the systems cos(nxβ) ∞ n=0 and sin(nxβ) ∞ n=1 are β-orthogonal over (0,(2π)1/β) and complete in L2 c(0,(2π)1/β) and L2 s(0,(2π)1/β), respectively. Finally, we give expansions in fractional β-cosine (β-sine) Fourier series of functions as illustrative examples.
Abstract
In this paper, we introduce the concept of the β-orthogonality of a system of real valued functions {ϕn(x)}∞ n=0 defined on an interval (0,b), b > 0. Using this concept, we define the fractional β-cosine (or β-sine) Fourier series of a function f ∈ L2 c(0,(2π)1/β) (or f ∈L2 s(0,(2π)1/β)), β > 0. Furthermore, we show that the systems cos(nxβ) ∞ n=0 and sin(nxβ) ∞ n=1 are β-orthogonal over (0,(2π)1/β) and complete in L2 c(0,(2π)1/β) and L2 s(0,(2π)1/β), respectively. Finally, we give expansions in fractional β-cosine (β-sine) Fourier series of functions as illustrative examples.