292 CHAPTER 12. SERIES AND TRANSFORMS

18. In the formula for the Bernstein polynomials, suppose f (0) = f (1) = 0. Show thatpn (0) = pn (1) = 0. Now if f is continuous on R, 2π periodic, and f (−π) = f (π) ,show there is a sequence of periodic continuous functions fn such that fn is a polyno-mial on (−π,π) and fn (−π) = f (π) such that limn→∞ ∥ f − fn∥∞

= 0. Explain whyfn satisfies an appropriate Dini condition at every point and hence limm→∞ Sm fn (x) =fn (x) where Sm fn is the mth partial sum for the Fourier series of fn.

12.8 The Fourier TransformThe Fourier transform is very useful in applications. It is essentially a characteristic func-tion in probability for example. These completely characterize probability measures. It isused in many other places also. To do it right, you really ought to be using the Lebesgueintegral, but this has not been discussed yet so the presentation ends up being a little fussierthan it would be if it were based on a better integral.

Definition 12.8.1 For f Riemann integrable on finite intervals, the Fourier trans-form is defined by

F f (t)≡ limR→∞

1√2π

∫ R

−Re−itx f (x)dx

whenever this limit exists. Of course this happens if f ∈ L1 (R) thanks to Lemma 10.0.4.The inverse Fourier transform is defined the same way except you delete the minus sign inthe complex exponential.

F−1 f (t)≡ limR→∞

1√2π

∫ R

−Reitx f (x)dx

Does it deserve to be called the “inverse” Fourier transform? This question will beexplored somewhat below.

The next theorem justifies the terminology above which defines F−1 and calls it theinverse Fourier transform. Roughly it says that the inverse Fourier transform of the Fouriertransform equals the mid point of the jump. Thus if the original function is continuous, itrestores the original value of this function. Surely this is what you would want by callingsomething the inverse Fourier transform.

Now for certain special kinds of functions, the Fourier transform is indeed in L1 andone can show that it maps this special kind of function to another function of the samesort. This can be used as the basis for a general theory of Fourier transforms. However, thefollowing does indeed give adequate justification for the terminology that F−1 is called theinverse Fourier transform.

Theorem 12.8.2 Let g ∈ L1 (R) and suppose g is locally Holder continuous fromthe right and from the left at x as in 10.8 and 10.9, or the Jordan condition which says thatg is of finite total variation on [x−δ ,x+δ ] for some δ > 0. Then

limR→∞

12π

∫ R

−Reixt∫

∞

−∞

e−ityg(y)dydt =g(x+)+g(x−)

2.

Proof: Note that∫ R

−Reixt∫

∞

−∞

e−ityg(y)dydt =∫

∞

−∞

e−ityg(y)dy∫ R

−Reixtdt

=∫

∞

−∞

e−ityg(y)∫ R

−Reixtdydt