The function takes an input string and counts the occurrences of uppercase and lowercase letters of the alphabet, returning a string with the count of each in the order they appear.

[CODE]:
```python
def approximate_function(input_str):
    result = []
    for char in input_str:
        if char.isalpha():
            count = 0
            for c in input_str:
                if c == char:
                    count += 1
            result.append(str(count))
    return ' '.join(result)
```

Let's test this approximation with some more inputs:

COMMAND: PYTHON(./temp/function.py "Test123" "!Vz?_CC" "12=AA4*8_!B")