1. Fourier Series 3
N. B. Vyas
Department of Mathematics,
Atmiya Institute of Tech. and Science,
Rajkot (Guj.) - INDIA
N.B.V yas − Department of M athematics, AIT S − Rajkot

2. Partial Sum & Total Square Error
Then N th partial sum of the series is given by
N.B.V yas − Department of M athematics, AIT S − Rajkot

3. Partial Sum & Total Square Error
Then N th partial sum of the series is given by
N
a0
SN (x) = + (ar cos rx + br sinrx)
2 r=1
N.B.V yas − Department of M athematics, AIT S − Rajkot

4. Partial Sum & Total Square Error
Then N th partial sum of the series is given by
N
a0
SN (x) = + (ar cos rx + br sinrx)
2 r=1
Total Square Error
N.B.V yas − Department of M athematics, AIT S − Rajkot

5. Partial Sum & Total Square Error
Then N th partial sum of the series is given by
N
a0
SN (x) = + (ar cos rx + br sinrx)
2 r=1
Total Square Error
l
E= [f (x) − SN (x)]2 dx
−l
N.B.V yas − Department of M athematics, AIT S − Rajkot

6. Partial Sum & Total Square Error
Then N th partial sum of the series is given by
N
a0
SN (x) = + (ar cos rx + br sinrx)
2 r=1
Total Square Error
l
E= [f (x) − SN (x)]2 dx
−l
l N
2 (a0 )2
∗
E = [f (x)] − l + (a2 + b2 )
r r
−l 2 r=1
N.B.V yas − Department of M athematics, AIT S − Rajkot

7. Parseval’s Theorem
Parseval’s Formula gives the relation between Fourier
coefficients.
N.B.V yas − Department of M athematics, AIT S − Rajkot

8. Parseval’s Theorem
Parseval’s Formula gives the relation between Fourier
coefficients.
If Fourier series corresponding to f (x) converges uniformly
in (−l, l) then
l ∞
(a0 )2
[f (x)]2 dx = l + (a2 + b2 )
n n
−l 2 n=1
N.B.V yas − Department of M athematics, AIT S − Rajkot

9. Example
Ex. Find Total square error of f (x) = x2 on the
interval −π ≤ x ≤ π for N = 3.
N.B.V yas − Department of M athematics, AIT S − Rajkot

10. Example
Ex. Find Total square error of f (x) = x2 on the
interval −π ≤ x ≤ π for N = 3.
Using Parseval’s identity prove that
1 1 1 π4
1 + 4 + 4 + 4 + ... =
2 3 4 90
N.B.V yas − Department of M athematics, AIT S − Rajkot