## Approximation by polynomials of a function defined on the entire real line

Suppose is a function which can be uniformly approximated by polynomials on . Then is also a polynomial.

Note the striking difference between this result and the Weierstrass approximation theorem.

## Traian Lalescu student contest 2011 Problem 3

For a continuous function such that prove that if is uniformly continuous, then it is bounded. Prove also that the converse of the previous statement is not true.

Traian Lalescu student contest 2011

## Convex functions limit Traian Lalescu 2010

Suppose that the sequence of convex functions converges pointwise to on . Prove that it converges uniformly to on .

*Traian Lalescu student contest 2010, Iasi, Romania*

## Uniform limit of polynomials

Suppose that the sequence of polynomials converges uniformly on to a function which is not a polynomial function. Prove that the sequence is unbounded.

*J. Marsden, Elementary Classical Analysis*

It is possible that the pointwise convergence siffice for the purpose of the problem.

This means that the space is closed if we take the uniform convergence norm . Read more…

## Measurable function again

Suppose is a measurable function. Prove that there exists a sequence of polynomials such that almost uniformly, which means that for any , we can find a set with measure smaller than such that uniformly on .

**Hint:** Use Lusin’s theorem and Weierstrass’ approximation theorem, for continuous functions.

## Egorov’s Theorem

Suppose is a sequence of measurable functions defined on a measurable set with , we can find a closed set such that and uniformly on .

This theorem states that in spaces with finite measure, almost everywhere convergence is equivalent to almost uniformly convergence.

## Barbalat’s Lemma

Prove that if we have a function which is uniformly continuous on with then . Read more…