Sparse representations of approximation to identity via time-space fractional heat equations

In this paper, we investigate sparse representations of approximation to identity via time-space fractional heat equations: \[ \left\{ \begin{aligned} & \partial<^>{\beta}_{t}u(t,x)=-\nu(-\Delta)<^>{\alpha/2}u(t,x),\quad (t,x)\in\mathbb R<^>{1+n}_{+};\cr & u(0,x)=f(x),\ x\in\mathbb R<^>{n}. \end{aligned}\right. \]{partial derivative t beta u(t,x)=-nu(-Delta)alpha/2u(t,x),(t,x)is an element of R+1+n;u(0,x)=f(x), x is an element of Rn. Due to the time-fractional derivative, the semigroup property is invalid for the solutions $ u(\cdot,\cdot ) $ u(& sdot;,& sdot;) to the above problem. This deficiency makes it difficult to verify the boundary vanishing condition of $ u(\cdot,\cdot ) $ u(& sdot;,& sdot;), which is essential for getting the sparse representations. We develop a new method to avoid using the semigroup property. The analogous results are obtained for stochastic time-space fractional heat equations. As an application, we apply the adaptive Fourier decomposition to establish sparse representations of the solutions to the concerned equations.