5) Fourier Series Flashcards
What is the prime period
The smallest possible a for which a periodic function holds
What is a periodic function
What are even and odd functions
When is a function piecewise continuous
A function f(x) is said to be piecewise continuous if it is continuous except for jump discontinuities, and if the number of jump discontinuities on any finite interval, x ∈ [a, b], is finite
What is a jump discontinuity
If the value of the function as we approach x = x0 from the left is different from the value as we approach x = x0 from the right
What is a periodic extension of a function
What is the inner dot product
What are the defining properties of an inner product on a vector space
What defines orthogonal and orthonormal bases in vector spaces
What conditions must a set of three vectors satisfy to be used as a basis
- Non-zero
- Linearly independent
How can any vector in an orthogonal basis of a vector space be represented using the inner product
What is an inner product for real functions
Describe the proof that the inner product for real functions is an inner product
When are two functions orthogonal
How are sine and cosine functions orthogonal on the interval [−L,L]
What is a special property about odd functions
Describe the proof that an odd function is zero on the domain [-L,L]
Give a function on [-L,L] what is its Fourier Series
How can the Fourier Series be rewritten
What is Dirichlet’s theorem
What are the implications for Fourier coefficients when a function is even or odd
How can cosine functions be expressed using (−1)^n
cos(nπ)=(−1) ^n
What are half-range Fourier series and what types can be computed when the function is known over half its range
Half-range series are employed when the function f is defined only over half of its typical range, commonly [0,L]
* Make f an odd function resulting in a Fourier sine series, because the series has only sine terms. We shall denote this function fS
* Make f an even function, resulting in a Fourier cosine series, because the series has only cosine terms. We shall denote this function fC
* Make f a periodic function with period L. We shall denote this function fP
