Prove or disprove there are a sequnce of polynomials

vazelinahS
2021-09-04
Answered

Let $f\left(x\right)=\mathrm{sin}\left(2x\right)$

Prove or disprove there are a sequnce of polynomials${P}_{n}\left(x\right)$ which convenes to $f\left(x\right)$ uniformly on $(0,\mathrm{\infty})$ .

Prove or disprove there are a sequnce of polynomials

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StrycharzT

Answered 2021-09-05
Author has **102** answers

We have to prove or disprove there are a sequence of polynomials

We have to prove by contradiction.

Suppose there exists a sequence of polynomial

So

In particular for

We have

As

So,

This means, sequence of polynomial

That is almost all

But

And

Which leads to contradiction to fact that almost all

Hence, there doesn't exists any sequence of polynomial converges uniformly to

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