The local fractional Hilbert transform in fractal space

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1 The local fractional ilbert transform in fractal space Guang-Sheng Chen Department of Computer Engineering, Guangxi Modern Vocational Technology College, echi,guangxi, , P.. China address: Abstract: In this paper, we establish local fractional ilbert transform in fractal space, consider some properties of local fractional ilbert Transforms. Keywords: fractal space;local fractional ilbert transform; local fractional derivative Introduction In the past ten years, Local fractional calculus[-27] has been widely applied to many fields such as mathematics, image processing and signal processing etc.some authors have given many definitions of local fractional derivatives and local fractional integrals (also called fractal calculus [-6]. ereby we rewrite the following local fractional derivative which is given by [5,6] where f df ( x Δ ( x = = lim ( f ( x f( x ( 0 0 x= x0 dx δ x 0 ( x x0, for 0<, (. Δ ( f ( x f( x0 Γ ( + Δ( f( x f( x0, and local fractional integral of f ( x defined by [5-6] j= N b ( aib f( t = f(( t dt = lim f( tj( Δtj Γ ( + a Γ ( + Δ t 0 j= 0, (.2 with Δ tj = tj+ tj and Δ t = max{ Δt, Δt2,, Δt j, }, where for j =, 2,, N,[ tj, t j + ] is a partition of the interval [ ab, ] and t 0 = a, t N = b. The aim of this paper is to define local fractional ilbert transforms based on Local fractional calculus. This paper is organized as follows. In section 2, local fractional ilbert transforms is defined; Section 3 presents Properties of local fractional ilbert transforms. 2 The local fractional ilbert transform and some examples In the section, both local fractional ilbert transform and its inverse transform are defined. Definition 2. (The local fractional ilbert transform. If f ( x is defined on the real line < x <, its local fractional ilbert transform, denoted by f x, ( x is defined by ˆ f( t { f( t} = f ( x = ( dt Γ ( + (2. ( t x where x is real and the integral is treated as a Cauchy principal value, that is,

2 f( t Γ ( + ( t x ( dt x ε f( t f( t = lim[ ( dt + ( dt ] ε 0 ( ( t x ( x ε Γ + Γ + + ( t x (2.2 To obtain the inverse local fractional ilbert transform, write again (2. as ˆ f( t f ( x = ( dt Γ ( + ( t x = f ( tgx ( t( dt = f( x gx ( Γ ( + (2.3 where gx ( =, application of the local fractional Yang-Fourier transform [5,6] with x respect to x gives f F, f ( ( =, g g ( F,, F, ( = i π sgn (2.4 F, Taking the inverse Yang-Fourier transform [5,6], we find the solution for f ( x as f( x = i π sgn f ( E ( i x ( d (2 π which is, by the Convolution Theorem [5,6] F,, ˆ ( ˆ f f ( f( x = ξ ξ ( ( { ˆ dξ dξ f ( ξ} (2 ( x (2 π ξ = π ( ξ x = (2.5 Obviously, 2 { f( t} = [ { f( t}] = f( x and hence, =. Thus, the inverse local fractional ilbert transform is given by ˆ ˆ ˆ f ( ξ f( t = { f ( x} = { f ( x} = ( dξ (2 π (2.6 ( x ξ Example 2. Find the local fractional ilbert transform of f( t = t we get, by definition, t + a 2 2, a > 0 ˆ t ( 2 2 ( f x = dt Γ ( + ( t + a ( t x 2 t = [ a ( dt + x ( dt x ( dt ] ( a + x Γ ( + ( t + a ( t x ( t + a The second and third integrals as the Cauchy principal value vanish and hence, only the first integral makes a non-zero contribution. Thus, we have ˆ a t π a f ( x = arctan = a x a a x ( ( (2.7

3 Example 2.2 Find the local fractional ilbert transform of (a f ( t = cos t and (b f ( t = sin t ˆ cos ( t x cos x sin ( t x sin x f ( x = ( dt ( Γ + ( t x cos ( t x sin ( t x = cos x ( dt sin x ( dt Γ ( + ( t x Γ ( + ( t x which is, the new variable T = t x, ˆ cos ( cos T f x = x ( dt Γ ( + T sin T sin x ( dt Γ ( + T (2.8 Obviously, the first integral vanishes because its integrand is an odd function of T. On the other hand, the second integral makes a non-zero contribution so that (2.8 gives Similarly, it can be shown that {cos t } = π sin x (2.9 {sin t } = π cos x (2.0 3 Basic Properties of the local fractional ilbert Transforms Theorem 3. If { ( } ˆ f t = f ( x, then the following properties hold: (a (b (c (d (e { ( } ˆ f t+ a = f ( x+ a, (3. { ( } ˆ f at = f ( ax, (3.2 { ( } ˆ f at = f ( ax, (3.3 ( { ( } f t d fˆ ( x =, (3.4 dx ˆ { tf( t} = x f ( x + f( t( dt Γ ( +, (3.5 (f F { { f( t}} i = sgn F { f( x}, (3.6 (g { f( t} = f( t (3.7 where f ( t = f, f denotes the norm in L2, (, (h [ ]( ˆ f x = f ( x, [ ˆ f ]( x = f (eciprocity relations, (3.8

4 (i (Parseval s formulas. f, g = f, g and f, g = f, g, (3.9 Proof. (a We obtain, by definition,set u = t+ a f( t a { f( t + a} = + ( dt ( Γ + ( t x f( u ( du fˆ = = ( x + a ( Γ + ( u ( x+ a (b We find, by definition,set u = at f( at f( u { f ( at} = ( ( ˆ dt du f ( ax ( ( t x ( Γ + = Γ + ( u ax = Similarly, result (c can be proved. (d We have, by definition ( ( f ( t { f ( t} = ( dt Γ ( + ( t x f( t ( = f( t[ ] ( dt ( ( t x Γ + ( t x Γ ( f ( t d fˆ ( x = ( = Γ( 2 Γ ( + ( dt 2 t x dx Proofs of (e (i are similar and consequently,we omit it Theorem 3.2 If f ( t is an even function of t, then, an alternative form of the local fractional ilbert transform is ˆ x f( t f( x f ( x = ( dt Γ ( + t x Proof. As the Cauchy principal value, we obtain consequently, Γ ( + ( t x ( ( dt = 0 ˆ f( t f( x f ( x = ( dt ( Γ + t x t ( f( t f( x x ( f( t f( x = ( dt + ( dt ( t x ( Γ + Γ + t x (3. Since f ( t is an even function, the integrand of the first integral of (3. is an odd function; hence, the first integral vanishes, and (3. gives (3.0. Since result (2.3 discovers that the local fractional ilbert transform can be written as a convolution transform, we state the following. Theorem 3.3 If f and g, ( are such that their local fractional ilbert transforms are L

5 also in L, (,Then and ( f g( x = ( f g( x = ( f g( x (3.2 ( f g( x = ( f g( x. (3.3 Proof. We obtain, by definition, ξ = t y ( f g( x = [ f( y g( t y( dy ]( dt Γ ( + ( t x Γ ( + = f( y( g( x y( dy = ( f g( x Γ ( + Likewise, we can obtain the second result in (3.2. To prove (3.3, we replace g by g in (3.2 and then use 2 g = g. 4 discussions In present paper we give local fractional ilbert transforms as follows: and its inverse transform ˆ f( t { f( t} = f ( x = ( dt ( for 0< (4. Γ + ( t x ˆ ˆ ˆ f ( ξ f( t = { f ( x} = { f ( x} = ( dξ (2 π for 0< (4.2 ( x ξ The transforming functions are local fractional continuous. That is to say (in other words, it means that, it is fractal function defined on fractal sets. ilbert transforms in integer space are the special case of fractal dimension =. It is a tool to deal with differential equation with local fractional derivative. eferences [] J. Tenreiro Machado, Virginia Kiryakova,Francesco Mainardi, ecent istory of Fractional Calculus,Communications in Nonlinear Science and Numerical Simulations,Elsevier, vol. 6, Issue 3, pp , March 20. [2] J. A. Tenreiro Machado, Manuel F. Silva, amiro S. Barbosa, Isabel S. Jesus, Cecília M. eis, Maria G. Marcos, Alexandra F. Galhano, Some Applications of Fractional Calculus in Engineering, Mathematical Problems in Engineering, indawi, vol. 200, Article ID 63980, doi:0.55/200/ [3] S.G. Samko, A.A. Kilbas, O.I. Marichev: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, 993. [4] Kolwankar K.M., Gangal A.D., Local Fractional Fokker Planck Equation. Phys. ev. Lett., 80, 998, [5] Babakhani A., Daftardar-Gejji V., On Calculus of Local Fractional Derivatives. J. Math. Anal. Appl., 270, 2002, [6] Chen Y., Yan Y., Zhang K., On the local fractional derivative. J. Math. Anal. Appl..

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