Kenneth Kuttler

12.2. CRITERIA FOR CONVERGENCE 281

Corollary 12.2.2 Let f be a periodic function of period 2π which is an element ofR([−π,π]). Suppose at some x, the function

y→∣∣∣∣ f (x− y)+ f (x+ y)−2s

∣∣∣∣ (12.8)

is in R((0,π]). Then limn→∞ Sn f (x) = s.

As pointed out by Apostol [3], this is a very remarkable result because even though theFourier coeficients depend on the values of the function on all of [−π,π], the convergenceproperties depend in this theorem on very local behavior of the function. There are wholebooks based on Fourier series such as Trigonometric Series by Zygmund [27] which ap-peared first in 1935 and there is a lot more in this book than the short introduction to thetopic presented here.

There is another easy to check condition which implies convergence to the midpoint ofthe jump. It was shown above that∣∣∣∣Sn f (x)− f (x+)+ f (x−)

∣∣∣∣=∣∣∣∣∣∫

1π

sin((

n+ 12

)y)

sin( y

) [f (x− y)− f (x−)+ f (x+ y)− f (x+)

]dy

∣∣∣∣∣=∫

2π

sin((

n+ 12

)y)

yy/2

sin(y/2)

[f (x− y)− f (x−)+ f (x+ y)− f (x+)

]dy+

∫π

1π

sin((

n+12

)y)[

f (x− y)− f (x−)+ f (x+ y)− f (x+)

2sin( y

) ]dy

In the second integral, the expression in [ ] is in L1 ([δ ,π]) and so, by the RiemannLebesgue lemma, this integral converges to 0.

If you know that f has finite total variation in [x−δ ,x+δ ] , then you could use Lemma10.2.7 to conclude that the first integral converges, as n→ ∞, to g(0+) where g(y) =

y/2sin(y/2)

f (x−y)− f (x−)+ f (x+y)− f (x+)2 so that g(0+) = 0. Thus, there is another corollary.

Corollary 12.2.3 Let f be a periodic function of period 2π which is an element ofR([−π,π]). Suppose at some x, f (x+) and f (x−) both exist and f is of bounded variationon [x−δ ,x+δ ] for some δ > 0. Then

limn→∞

Sn f (x) =f (x+)+ f (x−)

2. (12.9)

There are essentially two conditions which yield convergence to the mid point of thejump, the Dini conditions and the Jordan condition which is on finite total variation. Youhave to have something of this sort. If you only know you have continuity from the rightand the left, you don’t necessarily get convergence to the midpoint of the jump. At leastthis is so for the Fourier series. If you use the Ceasaro means, this kind of convergencetakes place without either of these two technical conditions. The study of Ceasaro meansis associated with Fejer whose work came much later in early 1900’s. This is consideredlater.

12.2. CRITERIA FOR CONVERGENCE 281Corollary 12.2.2 Let f be a periodic function of period 2x which is an element ofR([—2,2]). Suppose at some x, the function, f(x-y) +f (x+y) —2sy(12.8)is in R((0,2]). Then limyyoo Sn f (x) = s.As pointed out by Apostol [3], this is a very remarkable result because even though theFourier coeficients depend on the values of the function on all of [—7, 2], the convergenceproperties depend in this theorem on very local behavior of the function. There are wholebooks based on Fourier series such as Trigonometric Series by Zygmund [27] which ap-peared first in 1935 and there is a lot more in this book than the short introduction to thetopic presented here.There is another easy to check condition which implies convergence to the midpoint ofthe jump. It was shown above thatSif (x) ~~Fe ere2finer) Pen PEL g0 @ sin(3) 2 *_ fo 2sin((n+3)y) y/2 [fe-y)-fe-) +f ety) —f 4)- i 1 y sin (y/2) 2 dy+| fx=y)—f=)+f (x+y) —f (xt)[ zsin((n+5) ») | sin (3) dyIn the second integral, the expression in [ ] is in L'([6,z]) and so, by the RiemannLebesgue lemma, this integral converges to 0.If you know that f has finite total variation in [x — 6,x+ 6], then you could use Lemma10.2.7 to conclude that the first integral converges, as n — , to g(0+) where g(y) =at 5) fery)-f Ges (ety) 59 that g (0+) = 0. Thus, there is another corollary.Corollary 12.2.3 Let f be a periodic function of period 2x which is an element ofR([—2,2]). Suppose at some x, f (x+) and f (x—) both exist and f is of bounded variationon |x —6,x+ 6] for some 6 > 0. Thenlim S,f (x) = LOH + Fe)n—yoo 2(12.9)There are essentially two conditions which yield convergence to the mid point of thejump, the Dini conditions and the Jordan condition which is on finite total variation. Youhave to have something of this sort. If you only know you have continuity from the rightand the left, you don’t necessarily get convergence to the midpoint of the jump. At leastthis is so for the Fourier series. If you use the Ceasaro means, this kind of convergencetakes place without either of these two technical conditions. The study of Ceasaro meansis associated with Fejer whose work came much later in early 1900’s. This is consideredlater.