1) Approximation of Functions Flashcards
(42 cards)
What does interpolation guarantee about the existence and uniqueness of a polynomial passing through given points
There is a unique polynomial of degree at most n that exactly interpolates given values at n + 1 distinct points
How can an interpolating polynomial be written explicitly
What is the error formula for an interpolating polynomial of degree at most n at n + 1 distinct points, assuming f has n + 1 continuous derivatives
What do the notations C[a,b] and C(a,b) represent
- C[a, b] denotes the set of real-valued functions f defined and continuous on the closed interval [a, b]
- C(a, b) denotes the set of real-valued functions f defined and continuous on the open interval (a, b)
What does the Weierstrass Approximation Theorem state
What is a limitation of the Weierstrass Approximation Theorem in practice
The polynomial p(x) may need to be of very high degree to achieve the desired accuracy, so the theorem is mainly of theoretical interest rather than practical use
What are Chebyshev polynomials
What recurrence relation do the Chebyshev polynomials satisfy, and what does it imply about their degree
What are the key properties of Chebyshev polynomials
What are the properties that define a norm on a vector space V over R
What is a normed linear space
The pair (V, ||·||)
What are some examples of norms on the space C[a,b]
What is the Lp-norm
What does it mean for a function g∈S to be a best approximation to f∈V from a subspace S⊂V
What does the existence theorem for best approximation state in a finite-dimensional subspace
If (V, ||·||) is a normed linear space and S is a finite dimensional subspace of V , then for every f ∈ V there exists at least one best approximation g ∈ S
What is the space Pn, and how is it used in relation to C[a,b]
Pn is the vector space consisting of all polynomials of degree at most n. For any interval [a,b], we consider Pn as a subspace of C[a,b]
What does it mean for pn ∈ Pn to be a best L∞ (minimax) approximation to f ∈ C[a,b]
What is the best L∞ approximation to a function f∈C[a,b] from P0 and how is it determined
What is the Chebyshev equioscillation theorem
The error 𝑓(𝑥) − 𝑝𝑛(𝑥) reaches its maximum absolute value at n+2
distinct points{𝑥𝑘}𝑘=0 𝑛+1 ⊂ [𝑎, 𝑏], and the sign of 𝑓(𝑥𝑘) − 𝑝𝑛(𝑥𝑘) alternates between successive points.
Why does the best L∞ approximation pn to a function f∈C[a,b] interpolate f at at least n+1 points
What can be said about the existence and uniqueness of the best L∞ approximation from Pn to f∈C[a,b]
The best L∞ approximation to f∈C[a,b] from Pn exists and is unique
What is the best L∞ approximation to x ^(n+1) on [−1,1] from polynomials of degree at most n
Which monic polynomial of degree n+1 has the smallest L∞ norm on [−1,1], and what is its norm
When are a set of functions {Φj}j=0, n considered linearly independent, and what is such a set called
