A Pew survey from early 2008 asked "When your children are the age that you are now, do you think their standard of living will be much better, somewhat better, about the same, somewhat worse, or much worse than yours is now?" (The question was asked of everyone, not just parents, so in some cases the answers were about hypothetical children).
The answers, broken down by income groups:
Much Better Better Same Worse Much
Under 20,000 37% 25% 16% 12% 9%
20-40,000 33% 26% 17% 13% 11%
40-50,000 33% 27% 19% 13% 9%
50-100,000 26% 27% 23% 19% 6%
over 100,000 23% 26% 29% 17% 6%
Of course, the correct answer is unknown, but it will depend on two things: change in relative position and economic growth. The prospects of change in relative position depend on your own position: if you're at the bottom, your children aren't going to fall below you and are very likely to rise above you; if you're at the top, they can't rise and will be very likely to fall. So compared to rich people, poor people should be more likely to expect that their children will be better off than they are.
The table shows that there's some relationship of that kind, but it's not very strong. I'm pretty sure that the relationship is weaker than it "should" be given the actual correlation between parents and children's income, even once you allow for the widening gap between high and low incomes.* If that's the case, then people with low incomes are overly pessimistic about their children's chances and people with high incomes are overtly optimistic. This would be opposite to the pattern that many social scientists think exists, in which poor people accept inequality because they imagine that they or their children will make it to the top one day: see this article for an example.
*It becomes weaker and is no longer statistically significant after controlling for race and Hispanic ethnicity (blacks and Hispanics have lower incomes than whites and expect more improvement for their children).
[Data from the Roper Center for Public Opinion Research]