<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def resolution(file_, resolution_string):
 """ A filter to return the URL for the provided resolution of the thumbnail. """

    if sorl_settings.THUMBNAIL_DUMMY:
        dummy_source = sorl_settings.THUMBNAIL_DUMMY_SOURCE
        source = dummy_source.replace('%(width)s', '(?P<width>[0-9]+)')
        source = source.replace('%(height)s', '(?P<height>[0-9]+)')
        source = re.compile(source)
        try:
            resolution = decimal.Decimal(resolution_string.strip('x'))
            info = source.match(file_).groupdict()
            info = {dimension: int(int(size) * resolution) for (dimension, size) in info.items()}
            return dummy_source % info
        except (AttributeError, TypeError, KeyError):
            # If we can't manipulate the dummy we shouldn't change it at all
            return file_

    filename, extension = os.path.splitext(file_)
    return '%s@%s%s' % (filename, resolution_string, extension)