1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Miscellaneous Exercises 131

43. & 52.f(x)=







0 , 0 <x<a

2

, x=−a,a

2

,a,

1 ,

a
2 <x<a.
53–58.For each of these exercises,

a. find the Fourier cosine integral representation of the function;

b.sketch the even extension of the function.

59–64.For each of these exercises,

a. find the Fourier sine integral representation of the function;

b.sketch the odd extension of the function.

53. & 59.f(x)=e−x, 0 <x.


54. & 60.f(x)=

{

e−x, 0 <x<a,

0 , a<x.

55. & 61.f(x)=

{

1 , 0 <x<b,

0 , b<x.

56. & 62.f(x)=

{

cos(x), 0 <x<π,

0 ,π<x.

57. & 63.f(x)=

{ 1 −x, 0 <x<1,

0 , 1 <x.

58. & 64.f(x)=

{ 1 , 0 <x<1,
2 −x, 1 <x<2,
0 , 2 <x.
65.(Cesaro summability.) Letf(x)be a periodic function with period 2π
whose Fourier coefficients area 0 ,a 1 ,b 1 ,.... Then, the partial sum

SN(x)=a 0 +

∑N

n= 1

ancos(nx)+bnsin(nx)

is an approximation tof(x)iff is sectionally smooth andNis large

enough. The average of these approximations is

σN(x)=^1

N

(

S 1 (x)+···+SN(x)

)

.

It is known thatσN(x)converges uniformly tof(x)iffis continuous.

Show that

σN(x)=a 0 +

∑N

n= 1

N+ 1 −n

N

(

ancos(nx)+bnsin(nx)

)

.