Filter design to realize Cauchy product I come from Computer Science so please pardon for my possibly wrong terminology.

I need to design a filter which has coefficients 
$$h_0, h_1, \ldots, h_n, \ldots \quad\text{such that}\quad h_0 &gt; h_1&gt; \ldots &gt; h_n &gt; \ldots$$ 

The input is $x_n$. The output is $y_n$. The relationship between $h_n$, $x_n$ and $y_n$ is a convolution:
$$y_n = h_0 x_n + h_1 x_{n-1} + \ldots + h_n x_0$$

The problem is that I have to carry out a convolution to compute the output $y_n$ upon the arrival of every new input $x_n$. So I have to keep all $x_1, x_2, \ldots, x_n, \ldots$ in memory and do $n+1$ multiplications and $n$ additions every time.

I would like to ask whether there is any way to design the coefficient $h_n$ such that I can compute $y_n$ more efficiently. I know of one solution, which is $h_n=\lambda^n$ with $0&lt;\lambda&lt;1$ but this function decays too fast.

Any idea to approximate this procedure is also welcomed.
You could implement some convolutions as recursive filters. Consider moving average as an example, to calculate the output signal, instead of summing over the averaging window, you could add the new input and subtract the part of input which exited the window from the output signal to obtain the output of next time step. Also some filters could be implemented as lattice.

But there is not a general method to implement all filters recursively or as lattice.