The function shifts each lowercase and uppercase alphabetical character in the input string 16 positions forward in the alphabet, wrapping around if needed, and leaves numbers and special characters unchanged. 
[CODE]:
```
def approximate_function(input_str):
    output_str = ""
    for char in input_str:
        if char.isalpha():
            shift = 16
            if char.islower():
                output_str += chr(((ord(char) - ord('a') + shift) % 26) + ord('a'))
            elif char.isupper():
                output_str += chr(((ord(char) - ord('A') + shift) % 26) + ord('A'))
        else:
            output_str += char

    return output_str
```