Chapter 15 Multiple choice questions

. Which of the following is not true about the Fourier series?

It provides a useful approximation for periodic functions.

The approximated function must be differentiable.

Symmetric functions only contain cosine harmonics and a constant term.

Antisymmetric functions only contain sine harmonics.

. Which of the following is not true about the complex Fourier series (of a real function)?

It allows the Fourier series to be written in compact notation.

The Fourier summation has to be extended to include negative harmonic order.

The Fourier coefficients are complex in general.

The Fourier coefficients of symmetric functions are imaginary.

. Which of the following is not true about Fourier transforms?

They are the limiting integral form of a Fourier series when the repeat period, or wavelength, becomes infinitely large.

The Fourier transform and its inverse come as a pair, and which one is called which is arbitrary.

The precise definition of a Fourier transform is a matter of convention but, once fixed, its inverse is uniquely defined.

The Fourier transform of a real function is real.

. If f(x) has a width w, or has structure of that characteristic length, what is the corresponding length scale for its Fourier transform F(k)?

1/w

w

w²

e^w

. Which of the following is not a property of convolutions?

The convolution of f(x) with g(x) is a blurring of f(x) with g(x).

Convolutions are not commutative: the convolution of f(x) with g(x) is not the same as that of g(x) with f(x).

The Fourier transform of a product of two functions is the convolution of their Fourier transforms.

The Fourier transform of the convolution of two functions is the product of their Fourier transforms.

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