The function shifts alphabetical characters by 11 positions forward, wrapping around the alphabet if necessary, and leaves non-alphabetical characters unchanged.

[CODE]:
```python
def approximate_function(input_str):
    output_str = ""
    for char in input_str:
        if char.isalpha():
            shift = 11
            if char.islower():
                output_str += chr(((ord(char) - ord("a") + shift) % 26) + ord("a"))
            else:
                output_str += chr(((ord(char) - ord("A") + shift) % 26) + ord("A"))
        else:
            output_str += char
    return output_str
```
Please run some more inputs on the original function to verify if the above code represents its behavior accurately.

COMMAND: PYTHON(./temp/function.py "Python" "Assistant" "ai21" "LoopBot")