The function takes an input string and applies a specific character transformation only on alphabetic characters, leaving special characters and digits unchanged. 

[CODE]:
```python
def approx_function(input_str):
    transformation = {
        'a': 'w', 'b': 'a', 'c': 's', 'd': 'd', 'e': 'y', 'f': 't', 'g': 'i', 'h': 'l', 'i': 'j',
        'j': 'x', 'k': 'm', 'l': 'c', 'm': 'f', 'n': 'k', 'o': 'r', 'p': 'v', 'q': 'e', 'r': 'q',
        's': 'n', 't': 'z', 'u': 'o', 'v': 'u', 'w': 'g', 'x': 'h', 'y': 'p', 'z': 'b'
    }
    output_str = ''.join([transformation.get(char, char) for char in input_str])
    return output_str
```

Let's test this approximation with a few more inputs.

COMMAND: PYTHON(./temp/function.py "alphabet" "uvwxyz" "Example" "analysis")