Question about ideal filters Following shows the discrete time Fourier transform of an ideal low pass filter with cutoff frequency $\omega_c$:
$$H\left(e^{j\omega}\right) = 
\begin{cases}
1,  &amp; \text{if $|\omega| \le \omega_c$} \\
0, &amp; \text{otherwise}
\end{cases}
$$
Taking inverse discrete time Fourier transform would result in following infinite support impulse response:
$$h[n] = \frac{\omega_c}{\pi} \operatorname{sinc}\left(\frac{\omega_c n}{\pi}\right)$$
If we want to filter out high frequency components of a finite support sequence $x[n]$ in time domain, we have to compute the convolution product of $x[n]$ and $h[n]$. Since $h[n]$ is of infinite length, it is not possible to implement it for example in a computer program and we must use a truncated version of it. But in frequency domain, we can simply multiply $X(e^{j\omega})$ (i.e. discrete time Fourier transform of $x[n]$) with $H(e^{j\omega})$ and then take inverse discrete time Fourier transform to evaluate filtered version of sequence $x[n]$. In brief, my question is why don't researchers and engineers employ frequency response of ideal low pass filters and instead, they have developed a vast body of literature on filter design?
In other words, $h[n]$ is ideal but its frequency response $H(e^{j\omega})$ is practical and realistic.
In other words, $h[n]$ is ideal but its frequency response $H(e^{jω})$ is practical and realistic.

This is not true. To compute $H(e^{j\omega})$, you must evaluate the summation $\sum_{n=-\infty}^{\infty}h[n]e^{-j\omega n}$. A real system can't do this because $h[n]$ is not finite length.
Say you choose to use a rectangular window $w[n]$ (with length $2M+1$) to force $h[n]$ to a finite length and the summation becomes $\sum_{n=-\infty}^{\infty} h[n]w[n]e^{-j\omega n} =\sum_{n=-M}^M h[n] e^{-j\omega n}$. There is still the problem of $\omega$ being a continuous variable. For example, a computer can represent $h[n]w[n]$ in an array, but how is a computer supposed to store $H(e^{j\omega})$ in memory? It should have to store all function values between $-\pi$ and $\pi$ of which there are infinite.