Are polynomials dense?
Are polynomials dense?
As a consequence of the Weierstrass approximation theorem, one can show that the space C[a, b] is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients; there are only countably many polynomials with rational coefficients.
Are polynomials dense in L2?
Polynomials are dense in weighted L2 space.
Can all function be approximated by polynomials?
In words, any continuous function on a closed and bounded interval can be uniformly approximated on that interval by polynomials to any degree of accuracy.
Can any function be represented as a polynomial?
Unfortunately, not all functions can be expressed as a polynomial. For example, f(x)=sinx cannot be since a polynomial has only finitely many roots and the sine function has infinitely many roots, namely {nπ|n∈Z}.
Are polynomials dense in LP?
Since, by the domi- nated convergence theorem, uniform convergence implies Lp(µ) — convergence, it follows from the Weierstrass approximation theorem (see Theorem 8.34 and Corollary 8.36 or Theorem 12.31 and Corollary 12.32) that polynomials are also dense in Lp(µ).
Are continuous functions dense in L2?
In order to show this, we will prove the equivalent statement that any function f ∈ L2([0, 1]) can be approximated by a continuous function, i.e. for every ε > 0, there exists a continuous function g such that f − gL2 = 0. The set C([0, 1], R) is dense in L2[0, 1].
Why do we approximate functions with polynomials?
This is typically done with polynomial or rational (ratio of polynomials) approximations. The objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computer’s floating point arithmetic.
Can polynomials be infinite?
The only polynomial with infinitely many roots is P(x)=0. You can prove this without appealing to the fundamental theorem of algebra. One can, by using polynomial long division, determine that there is a degree n−1 polynomial P2 such that P2(x)⋅(x−r)=P(x).
Can a polynomial have infinite terms?
A polynomial can have constants, variables and exponents, but never division by a variable. Also they can have one or more terms, but not an infinite number of terms.
Are polynomials dense in L1?
The space L1([0,1]) is a also separable. Indeed, we have shown that C([0,1]) is dense in L1([0,1]) and thus the set of polynomials with rational coefficients is dense in L1([0,1]).
Are continuous functions dense in L1?
In this supplement, we’ll show that continuous functions with compact support are dense in L1 = L1(Rn,m). The support of a complex valued function f on a metric space X is the closure of {x ∈ X : f(x) = 0}. Thus integrable functions can be approximated by continuous functions.
