| Abstract: |
| The first Heisenberg group $\mathbb{H}=\mathbb{R}^3$ is a homogeneous group equipped with the non-Euclidean metric $d_{\mathbb{H}}((x,y,t),(x`,y`,t`))=[((x`-x)^2+(y`-y)^2)^2+(t`-t+2(xy`-x`y))^2]^{\frac{1}{4}}$. Corresponding to the notorious fractal Sierpi\`nski gasket $(SG)$, one can get a fractal in $\mathbb{H}$ which is entitled as lifted Sierpi\`nski gasket $(\widetilde{SG})$.
Motivated from the previous work on Heisenberg group and $\alpha$-fractal functions on $SG$ (Publ. Mat. 47 (2003), no. 1, 237-259, Proc. London Math. Soc. (3) 91 (2005), no. 1, 153-183, J. Math. Anal. Appl. 487 (2020), no. 2, 124036, 16 pp.), we first construct the class of $\alpha$-fractal interpolation functions (FIFs) in numerous spaces on $\widetilde{SG}$. Next, we define the $\alpha$-fractal operator with respect to the class of $\alpha$-FIFs constructed in numerous spaces, and also discuss the several properties of the same operators. We conclude the talk by providing the glimpse of the dimension of the graphs of constructed $\alpha$-FIFs. |
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