Kenneth Kuttler

328 CHAPTER 12. INNER PRODUCT SPACES, LEAST SQUARES

the following assertions and eventually conclude that under these very reasonableconditions (more general ones are possible.)

limn→∞

Sn f (x) = ( f (x+)+ f (x−))/2

the mid point of the jump. In words, the Fourier series converges to the midpoint ofthe jump of the function.

Sn f (x) =∫

−π

f (y)Dn (x− y)dy =∫

−π

f (x− y)Dn (y)dy

You just change variables and then use 2π periodicity to get this.∣∣∣∣Sn f (x)− f (x+)+ f (x−)2

∣∣∣∣=

∣∣∣∣∫ π

−π

(f (x− y)− f (x+)+ f (x−)

)Dn (y)dy

∣∣∣∣=

∣∣∣∣∫ π

0f (x− y)Dn (y)dy+

∫π

0f (x+ y)Dn (y)dy

−∫

0( f (x+)+ f (x−))Dn (y)dy

∣∣∣∣≤∣∣∣∣∫ π

0( f (x− y)− f (x−))Dn (y)dy

∣∣∣∣+ ∣∣∣∣∫ π

0( f (x+ y)− f (x+))Dn (y)dy

∣∣∣∣Now apply some trig. identities and use the result of Problem 37 to conclude thatboth of these terms must converge to 0.

328 CHAPTER 12. INNER PRODUCT SPACES, LEAST SQUARESthe following assertions and eventually conclude that under these very reasonableconditions (more general ones are possible.)lim S,f (x) = (FH) +f (e-)) 2the mid point of the jump. In words, the Fourier series converges to the midpoint ofthe jump of the function.Tl 1S£() = [| FO)Dae-y)dy= | F—-y) Duly) dyYou just change variables and then use 27 periodicity to get this.sp) Fett fe)= f(r x-y)- Lt) + fe FEC) p, (yar2= [Pre y)D no)dy+ [fF f (x+y) Dn (y) dy-[ sios-spoeyvia0<| [ (ro) Fo) Ds Oday] 4 [Fety)- £64) Pn yayNow apply some trig. identities and use the result of Problem 37 to conclude thatboth of these terms must converge to 0.