Fourier series Test Questions with
Answers
Name given to finite Fourier series - Answer-A trigonometric polynomial
What is related to the decay of Fourier coefficients and how? - Answer-Differentiability
of the function
Higher differentiability higher the rate of decay
Exponential decay if infinitely differentiable
3 ways to define a Fourier series for a function f(x) on domain 0 ≤ x ≤ c - Answer-1)
Translate both directions by c to get a function where 2L = P = c and use the full Fourier
series eq
2) Reflect the function in the y-axis to make it even and use the cosine series
3) Rotate it π about origin to amke it odd and use the sine series
Fourier series for an even function - Answer-F(x) = a₀/2 + ∑ (aₙ cos(nπx/L))
Called a cosine series
Fourier series for an odd function - Answer-F(x) = ∑ (bₙsin(nπx/L))
Called a sine series
The Gibb's phenomenon - Answer-The size of the overshoot of the Fourier series just
beyond a discontinuity remains non-zero for an number of terms and tends to 18%
Thus there is substantial error in the representation of the function but the size of the
region over which there is error becomes smaller with more terms
Why is the derivative of a Fourier series not always the Fourier series of the derivative
of the original function? - Answer-Because the limits don't always commute
Describe the rate of convergence of the integral of a Fourier series - Answer-Faster
convergence as integral is smoother
Describe the rate of convergence of the derivative of a Fourier series - Answer-Slower
convergence as differential is less smooth
Parseval's theorem + explain the terms - Answer-Given function f(x) with Period = 2L
1/2L ∫ [-L, L] [f(x)]² dx = ¼a₀² + ½∑(aₙ² + bₙ²)
RHS: Length/magnitude of the function
LHS: Length/Magnitude of the components
Proof of Parseval's theorem - Answer-If the mean square value of f(x) is:
Answers
Name given to finite Fourier series - Answer-A trigonometric polynomial
What is related to the decay of Fourier coefficients and how? - Answer-Differentiability
of the function
Higher differentiability higher the rate of decay
Exponential decay if infinitely differentiable
3 ways to define a Fourier series for a function f(x) on domain 0 ≤ x ≤ c - Answer-1)
Translate both directions by c to get a function where 2L = P = c and use the full Fourier
series eq
2) Reflect the function in the y-axis to make it even and use the cosine series
3) Rotate it π about origin to amke it odd and use the sine series
Fourier series for an even function - Answer-F(x) = a₀/2 + ∑ (aₙ cos(nπx/L))
Called a cosine series
Fourier series for an odd function - Answer-F(x) = ∑ (bₙsin(nπx/L))
Called a sine series
The Gibb's phenomenon - Answer-The size of the overshoot of the Fourier series just
beyond a discontinuity remains non-zero for an number of terms and tends to 18%
Thus there is substantial error in the representation of the function but the size of the
region over which there is error becomes smaller with more terms
Why is the derivative of a Fourier series not always the Fourier series of the derivative
of the original function? - Answer-Because the limits don't always commute
Describe the rate of convergence of the integral of a Fourier series - Answer-Faster
convergence as integral is smoother
Describe the rate of convergence of the derivative of a Fourier series - Answer-Slower
convergence as differential is less smooth
Parseval's theorem + explain the terms - Answer-Given function f(x) with Period = 2L
1/2L ∫ [-L, L] [f(x)]² dx = ¼a₀² + ½∑(aₙ² + bₙ²)
RHS: Length/magnitude of the function
LHS: Length/Magnitude of the components
Proof of Parseval's theorem - Answer-If the mean square value of f(x) is:
