Deck 11: Orthogonal Functions and Fourier Series

Question 1

The solution of the eigenvalue problem is

A)
B)
C)
D)
E)

Answer 1

E

Question 2

In order to be assured by a theorem that the Fourier Series of on converges at , to which of the following conditions need to be satisfied? Select all that apply.

A) is continuous on
B) is continuous on
C) is piecewise continuous on
D) is piecewise continuous on
E) is integrable on

Answer 2

C, D, E

Question 3

The problem is a regular Sturm-Liouville problem under certain conditions, including Select all that apply.

A) , , piecewise continuous on
B) and on
C) and on
D)
E)

Answer 3

B, E

Question 4

The square norm of the function on the interval is

A) 1
B)
C)
D)
E) 0

Question 5

The Fourier series of an even function might Select all that apply.

A) contain sine terms
B) contain cosine terms
C) contain a constant term
D) contain sine and cosine terms
E) contain sine, cosine, and constant terms

Question 6

The square norm of the function on the interval is

A) 1/2
B) 1/3
C) 1/5
D) 1
E) 0

Question 7

The differential equation is

A) Legendre's equation
B) Bessel's equation
C) the Fourier-Bessel
D) the hypergeometric
E) none of the above

Question 8

The Fourier series of the function on are Select all that apply.

A) contains only cosine terms
B) contains only sine terms
C) contains sine and cosine terms
D) contains a constant term
E) contains sine, cosine, and constant terms

Question 9

The solution of the eigenvalue problem is

A)
B)
C)
D)
E)

Question 10

The Fourier coeficients of the function on are Select all that apply.

A)
B)
C)
D)
E)

Question 11

The problem is not a regular Sturm-Liouville problem under which of the following conditions. Select all that apply.

A) on
B) on
C) on
D)
E)

Question 12

The solution of the eigenvalue problem where is bounded on , is

A)
B)
C)
D)
E)

Question 13

Which of the following differential equations are in self-adjoint form? Select all that apply.

A)
B)
C)
D)
E)

Question 14

The Fourier Series of a function defined on is where Select all that apply.

A)
B)
C)
D)
E)

Question 15

The function has a Fourier series on that converges at to

A) 7
B) 1
C) 1/2
D)
E) unknown

Question 16

Using the eigenfunctions of the previous problem, the Fourier-Legendre series for the function is , where

A)
B)
C)
D)
E)

Question 17

The function is Select all that apply.

A) odd
B) even
C) neither even nor odd
D) continuous on
E) discontinuous on

Question 18

An example of a regular Sturm-Liouville problem is with boundary conditions Select all that apply.

A)
B)
C)
D)
E) is bounded on

Question 19

The function has a Fourier series on that converges at to

A) 0
B) 1
C) 1/2
D) 2
E) unknown

Question 20

The Fourier series of the function on is

A)
B)
C)
D)
E)

Question 21

The Fourier Series of a function defined on is where Select all that apply.

A)
B)
C)
D)
E)

Question 22

Using the eigenfunctions of the previous problem, written as , the Fourier-Bessel series for the function is , where

A)
B)
C)
D)
E)

Question 23

The solution of the eigenvalue problem is bounded, is

A)
B)
C)
D)
E)

Question 24

The square norm of the function on the interval is

A) 2/3
B)
C) 1/3
D)
E) 0

Question 25

The solution of the eigenvalue problem is

A)
B)
C)
D)
E)

Question 26

Consider the differential equation . Examples of boundary conditions for this equation that make a regular Sturm-Liouville problem are Select all that apply.

A)
B)
C)
D)
E)

Question 27

The Fourier series of the function on is

A)
B)
C)
D)
E)

Question 28

The square norm of the function on the interval is

A) 1
B)
C)
D)
E) 0

Question 29

In order to be assured by a theorem that the Fourier Series of on converges to , which of the following conditions need to be satisfied? Select all that apply.

A) is continuous on
B) is continuous on
C) is piecewise continuous on
D) is piecewise continuous on
E) is integrable on

Question 30

The function is Select all that apply.

A) odd
B) even
C) neither even nor odd
D) continuous on
E) discontinuous on

Question 31

The problem is a regular Sturm-Liouville problem under certain conditions, including Select all that apply.

A) , , are continuous on
B) and on
C) and on
D)
E)

Question 32

The solution of the eigenvalue problem is

A)
B)
C)
D)
E)

Question 33

Which of the following differential equations are in self-adjoint form? Select all that apply.

A)
B)
C)
D)
E)

Question 34

The problem is a regular Sturm-Liouville problem under which of the following conditions. Select all that apply.

A) are continuous on
B) on
C) on
D)
E)

Question 35

The function has a Fourier series on that converges at to

A) 0
B) 1
C) 1/2
D)
E) unknown

Question 36

The Fourier series of an odd function might Select all that apply.

A) contain sine terms
B) contain cosine terms
C) contain a constant term
D) contain sine and cosine terms
E) contain sine, cosine, and constant terms

Question 37

The Fourier series of the function on is Select all that apply.

A) contains cosine terms
B) contains sine terms
C) contains sine and cosine terms
D) contains a constant term
E) contains sine, cosine, and constant terms

Question 38

The function has a Fourier series on that converges at to

A) 0
B) 1
C) 1/2
D)
E) unknown

Question 39

Consider the parameterized Bessel's differential equation along with the conditions is bounded, . The solution of this eigenvalue problem is

A)
B)
C)
D)
E)

Question 40

The Fourier coeficients of the function on are

A)
B)
C)
D)
E)