On sequential greedy-type bases

It is known that a basis is almost greedy if and only if the thresholding greedy algorithm gives essentially the smallest error term compared to errors from projections onto intervals or in other words, consecutive terms of $\mathbb{N}$. In this paper, we fix a sequence $(a_n)_{n=1}^\infty$ and compare the TGA against projections onto consecutive terms of the sequence and its shifts. We call the corresponding greedy-type condition the $\mathcal{F}_{(a_n)}$-almost greedy property. Our first result shows that the $\mathcal{F}_{(a_n)}$-almost greedy property is equivalent to the classical almost greedy property if and only if $(a_n)_{n=1}^\infty$ is bounded. Then we establish an analog of the result for the strong partially greedy property. Finally, we show that under a certain projection rule and conditions on the sequence $(a_n)_{n=1}^\infty$, we obtain a greedy-type condition that lies strictly between the almost greedy and strong partially greedy properties.