I don't know of any existing methods for this, so I guess I would index through each char in the String and check to see if it's a space or lowercase char. Finding a space would set a flag allowing the next character to be converted to uppercase if needed. (This should initially be set to true in order to allow the first letter to be capitalized.)
Note that chars can be treated as numeric values (although you'll need to cast back to type char after performing arithmetic on them). Uppercase letters A-Z are represented by numeric values 65-90, and lowercase (a-z) are represented by 97-122. So if you have a lowercase value, you can convert it to uppercase by subtracting 32. (Ensure first that the char you have is, in fact, within that lowercase range.) A space is represented by 32.