The function applies letter substitution to the input string, maintaining the same transformation for each letter across different inputs, and leaving digits and special characters unchanged.

[CODE]: 
```python
def substitute_letter(c):
    if c.isalpha():
        substitution_map = {
            'A': 'q', 'B': 'd', 'C': 'z', 'D': 'f', 'E': 'm',
            'F': 'l', 'G': 'r', 'H': 's', 'I': 'y', 'J': 'j',
            'K': 'w', 'L': 'e', 'M': 'h', 'N': 'b', 'O': 'x',
            'P': 'i', 'Q': 'u', 'R': 'o', 'S': 'v', 'T': 'p',
            'U': 'k', 'V': 't', 'W': 'a', 'X': 'c', 'Y': 'g',
            'Z': 'n',
            'a': 'q', 'b': 'd', 'c': 'z', 'd': 'f', 'e': 'm',
            'f': 'l', 'g': 'r', 'h': 's', 'i': 'y', 'j': 'j',
            'k': 'w', 'l': 'e', 'm': 'h', 'n': 'b', 'o': 'x',
            'p': 'i', 'q': 'u', 'r': 'o', 's': 'v', 't': 'p',
            'u': 'k', 'v': 't', 'w': 'a', 'x': 'c', 'y': 'g',
            'z': 'n'
        }
        return substitution_map[c]
    else:
        return c

def approx_function(input_str):
    return ''.join([substitute_letter(c) for c in input_str])
```
This Python function approximates the behavior of the given function by applying letter substitutions to the input string based on a specific mapping.