The Fractional Clifford-Fourier Transform

In this paper, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters. A series expansion for the kernel of the resulting integral transform is derived. In the case of even dimension, also an explicit expression for the kernel in terms of Bessel functions is obtained. Finally, the analytic properties of this new integral transform are studied in detail.     

The Clifford-Fourier Transform

A pair of Clifford-Fourier transforms is defined in the framework of Clifford analysis, as operator exponentials with a Clifford algebra-valued kernel. It is a genuine Clifford analysis construct, which is shown to be a refinement of the classical multi-dimensional Fourier transform. An adequate operational calculus is developed.     

The Class of Clifford-Fourier Transforms

Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of the classical Fourier transform. This is applied to the so-called Clifford-Fourier transform (see Brackx et al., J. Fourier Anal. Appl. 11:669–681, 2005). The integral kernel of this transform is a particular solution of a system of PDEs in a Clifford algebra, but is, contrary to the classical Fourier transform, not the unique solution. Here we determine an entire class of solutions of this system of PDEs, under certain constraints. For each solution, series expressions in terms of Gegenbauer polynomials and Bessel functions are obtained. This allows to compute explicitly the eigenvalues of the associated integral transforms. In the even-dimensional case, this also yields the inverse transform for each of the solutions. Finally, several properties of the entire class of solutions are proven.     

The Boundedness for Commutator of Fractional Integral Operator With Rough Variable Kernel

For b?∈?BMO(? ⁿ ) and 0?<?α?≤?1/2, the commutator of the fractional integral operator T _Ω,α with rough variable kernel is defined by $$ [b, T_{\Omega, \alpha}]f(x)= \int_{\mathbb{R}^n} \frac{\Omega(x,x-y)}{|x-y|^{n-\alpha}}(b(x)-b(y))f(y)dy. $$ In this paper the authors prove that the commutator [b, T _Ω,α ] is a bounded operator from $L^{\frac{2n}{n+2\alpha}}(\mathbb{R}^n)$ to L ²(? ⁿ ). The result obtained in this paper is substantial improvement and extension of some known results.     

Singular Integral Operators with Cauchy Kernel in Fractional Spaces

We single out the Besov spaces that embed into the class of continuous functions and enjoy the Fredholm theory of linear singular integral equations with Cauchy kernel. We give basic results of this theory in the class of continuous (rather than Holder continuous) functions in terms of Besov spaces. Alongside elliptic operators we consider violations of ellipticity: the degeneration of the symbol of an operator at finitely many points.     

关于粗糙核多线性分数次积分的一点注记

作者简单地证明了一类粗糙核多线性分数次积分算子及其相关的极大算子分别是关于A(p,q)权从Lp到Lq有界的以及关于幂权从Lp(1≤p＜n/α)到p/(n-a),∞有界的.     

Compactness of the Commutators of Fractional Hardy Operator with Rough Kernel

The more explicit decomposition of the operator and the kernel are utilized to investigate a characterization of the central $BMO(\mathbb{R}^{n})$-closure of $C_{c}^{\infty}(\mathbb{R}^{n})$ space via the compactness of the commutators of fractional Hardy operator with rough kernel.     

A new construction of the Clifford-Fourier kernel

In this paper, we develop a new method based on the Laplace transform to study the Clifford-Fourier transform. First, the kernel of the Clifford-Fourier transform in the Laplace domain is obtained. When the dimension is even, the inverse Laplace transform may be computed and we obtain the explicit expression for the kernel as a finite sum of Bessel functions. We equally obtain the plane wave decomposition and find new integral representations for the kernel in all dimensions. Finally we define and compute the formal generating function for the even dimensional kernels.     

A Reproducing Kernel Hilbert Space Method for Solving Integro-Differential Equations of Fractional Order

In this article, we implement a relatively new analytical technique, the reproducing kernel Hilbert space method (RKHSM), for solving integro-differential equations of fractional order. The solution obtained by using the method takes the form of a convergent series with easily computable components. Two numerical examples are studied to demonstrate the accuracy of the present method. The present work shows the validity and great potential of the reproducing kernel Hilbert space method for solving linear and nonlinear integro-differential equations of fractional order.     

粗糙核多线性分数次积分算子在加权Herz空间的有界性

研究两类带粗糙核的多线性分数次积分算子T_(Ω,α)~A, T_(Ω,α)~Af(x)=∫R_m(A;x,y)/R~n|x- y|~(n+m-α-1)Ω(x-y)f(y)dy及其相关的极大算子M_(Ω,α)~A在加权Herz空间的有界性,其中Ω∈L~s(S~(n-1))(s>1)是R~n中的零次齐次函数,m∈N,A有m=1阶导数且D~γA∈BMO(R~n)或D~γA∈L~r(R~n)(|γ|=m -1,1相似文献   

再生核移位勒让德基函数法求解分数阶微分方程

该文以再生核理论为基础,用移位Legendre多项式作为基函数构造了一个新的再生核空间,并给出了该空间下的再生核函数.与经典的再生核函数有所不同的是该空间下的再生核函数不再是分段函数,因此可以减小分数阶算子作用在核函数上时的计算量,使近似解更为精确.数值算例表明该方法的有效性.     

The Clifford-Fourier Transform {mathcal {F}_o} and Monogenic Extensions

In the last decade several versions of the Fourier transform have been formulated in the framework of Clifford algebra. We present a (Clifford-Fourier) transform, constructed using the geometric properties of Clifford algebra. We show the corresponding results of operational calculus, and a connection between the Fourier transform and this new transform. We obtain a technique to construct monogenic extensions of a certain type of continuous functions, and versions of the Paley-Wiener theorems are formulated.     

粗糙核分数次积分算子的多线性算子在Hardy空间上的有界性

该文证明带有粗糙核的分数次积分算子的多线性算子\[T_{\Omega,\alpha}^{A}(f)(x)={\rm {\rm p.v.}}\int_{R^{n}}P_{m}(A;x,y)\frac{\Omega(x-y)}{|x-y|^{n-\alpha+m-1}}f(y){\rm d}y\]的$(H^{1}(\rr^{n}),L^{\frac{n}{n-\alpha},\infty}(\rr^{n}))$有界性.     

A Characterization of Boundedness of Fractional Maximal Operator with Variable Kernel on Herz-Morrey Spaces

A significant number of studies have been carried out on the generalized Lebesgue spaces L~(p(x)), Sobolev spaces W~(1,p(x)) and Herz spaces. In this paper, we demonstrated a characterization of boundedness of the fractional maximal operator with variable kernel on Herz-Morrey spaces.     

On the Reproducing Kernel Hilbert Spaces Associated with the Fractional and Bi-Fractional Brownian Motions

We present decompositions of various positive kernels as integrals or sums of positive kernels. Within this framework we study the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions. As a tool, we define a new function of two complex variables, which is a natural generalization of the classical Gamma function for the setting we consider. D. Alpay thanks the Earl Katz family for endowing the chair which supports his research.     

Boundedness of Fractional Integrals with a Rough Kernel on the Pro duct Triebel-Lizorkin Spaces

By using the Littlewood-Paley decomposition and the interpolation the-ory, we prove the boundedness of fractional integral on the product Triebel-Lizorkin spaces with a rough kernel related to the product block spaces.     

Conservatory of Kaup-Kupershmidt Equation to the Concept of Fractional Derivative with and without Singular Kernel

Making use of the traditional Caputo derivative and the newly introduced Caputo-Fabrizio derivative with fractional order and no singular kernel, we extent the nonlinear Kaup-Kupershmidt to the span of fractional calculus. In the analysis, different methods of fixed-point theorem together with the concept of piccard L-stability are used, allowing us to present the existence and uniqueness of the exact solution to models with both versions of derivatives. Finally, we present techniques to perform some numerical simulations for both non-linear models and graphical simulations are provided for values of the order α = 1.00; 0.90. Solutions are shown to behave similarly to the standard well-known traveling wave solution of Kaup-Kupershmidt equation.     

Boundedness for the Singular Integral with Variable Kernel and Fractional Differentiation on Weighted Morrey Spaces

Let T be the singular integral operator with variable kernel, T*be the adjoint of T and T~#be the pseudo-adjoint of T. Let T_1T_2 be the product of T_1 and T_2, T_1? T_2 be the pseudo product of T_1 and T_2. In this paper, we establish the boundedness for commutators of these operators and the fractional differentiation operator Dγon the weighted Morrey spaces.