1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

1.6 Mean Error and Convergence in Mean 93

This inequality is valid for anyNand therefore is also valid in the limit asN
tends to infinity. The actual fact is that, in the limit, the inequality becomes
Parseval’s equality:

1

a

∫a

−a

f^2 (x)dx= 2 a^20 +

∑∞

1

a^2 n+b^2 n. (8)

Another very important consequence of Bessel’s inequality is that the two
series

∑

a^2 nand

∑

b^2 nmust converge if the left-hand side of Eqs. (7) and (8) is
finite. Thus, the numbersanandbnmust tend to 0 asntends to infinity.
By comparing Eqs. (6) and (8), we get a different expression for the mini-
mum error:

min(EN)=a

∑∞

N+ 1

a^2 n+b^2 n.

This quantity decreases steadily to zero asNincreases. Since min(EN)is, ac-
cording to Eq. (2), a mean deviation betweenfand the truncated Fourier se-
ries off, we often say, “The Fourier series offconverges tofin the mean.”
(Another kind of convergence!)

Summary

Iff(x)has been defined in the interval−a<x<aand if
∫a

−a

f^2 (x)dx

is finite, then:

1.Among all finite series of the form

g(x)=A 0 +

∑N

1

Ancos

(nπx

a

)

+Bnsin

(nπx

a

)

the one that best approximatesfin the sense of the error described by

Eq. (2) is the truncated Fourier series off:

a 0 +

∑N

1

ancos

(

nπx

a

)

+bnsin

(

nπx

a

)

.

2.^1

a

∫a

−a

f^2 (x)dx= 2 a^20 +

∑∞

1

a^2 n+b^2 n.