Norms

The choice of the Banach space to apply the Radii Polynomial Theorem (cf. Section Radii polynomial approach) is integral to the success of the computer-assisted proof. In practice, it is useful to tune the Banach space on the fly to adjust the norm estimates.

Accordingly, the spaces introduced in Section Vector spaces are not normed a priori. The norm of a Sequence or a LinearOperator is obtained via the functions norm and opnorm respectively; in both cases, one must specify a BanachSpace.

BanachSpace
├─ NormedCartesianSpace
├─ Ell1
├─ Ell2
└─ EllInf

$\ell^1$, $\ell^2$ and $\ell^\infty$

Let $\mathscr{I}$ be a set of indices such that $\mathscr{I} \subset \mathbb{Z}^d$ for some $d \in \mathbb{N}$. Consider the weighted $\ell^1, \ell^2, \ell^\infty$ spaces (cf. $\ell^p$ spaces) defined by

\[\begin{aligned} \ell^1_w &\bydef \left\{ a \in \mathbb{C}^\mathscr{I} \, : \, | a |_{\ell^1_w} \bydef \sum_{\alpha \in \mathscr{I}} |a_\alpha| w(\alpha) < \infty \right\}, \\ \ell^2_w &\bydef \left\{ a \in \mathbb{C}^\mathscr{I} \, : \, | a |_{\ell^2_w} \bydef \sqrt{\sum_{\alpha \in \mathscr{I}} |a_\alpha|^2 w(\alpha)} < \infty \right\}, \\ \ell^\infty_w &\bydef \left\{ a \in \mathbb{C}^\mathscr{I} \, : \, | a |_{\ell^\infty_w} \bydef \sup_{\alpha \in \mathscr{I}} | a_\alpha | w(\alpha) < \infty \right\}, \end{aligned}\]

where $w : \mathscr{I} \to (0, \infty)$ is a weight function.

The Banach spaces Ell1, Ell2 and EllInf wraps one or multiple Weight.

Weight
├─ AlgebraicWeight
├─ BesselWeight
├─ GeometricWeight
└─ IdentityWeight

Given a set of indices $\mathscr{I}^\prime \subset \mathbb{Z}$:

an AlgebraicWeight of rate $s \ge 0$ is defined by $w(\alpha) \bydef (1 + |\alpha|)^s$ for all $\alpha \in \mathscr{I}^\prime$.
a BesselWeight of rate $s \ge 0$ is defined by $w(\alpha) \bydef (1 + \alpha^2)^s$ for all $\alpha \in \mathscr{I}^\prime$. This weight is specific to Ell2 and Fourier as it describes the Sobolev space $H^s$.
a GeometricWeight of rate $\nu \ge 1$ is defined by $w(\alpha) \bydef \nu^{|\alpha|}$ for all $\alpha \in \mathscr{I}^\prime$.
an IdentityWeight is defined by $w(\alpha) \bydef 1$ for all $\alpha \in \mathscr{I}^\prime$. This is the default weight for Ell1, Ell2 and EllInf.

julia> a = Sequence(Taylor(2), [1.0, 1.0, 1.0]); # 1 + x + x^2
julia> norm(a, Ell1(AlgebraicWeight(1.0)))6.0
julia> b = Sequence(Fourier(1, 1.0), [0.5, 0.0, 0.5]); # cos(x)
julia> norm(b, Ell2(BesselWeight(2.0)))1.4142135623730951
julia> c = Sequence(Chebyshev(2), [1.0, 0.5, 0.5]); # 1 + 2(x/2 + (2x^2 - 1)/2)
julia> norm(c, EllInf()) # EllInf() == EllInf(IdentityWeight())1.0

Note that ℓ¹ (\ell<tab>\^1<tab>), ℓ² (\ell<tab>\^2<tab>) and ℓ∞ (\ell<tab>\infty<tab>) are the respective unicode aliases of Ell1, Ell2 and EllInf.

In the context of a $d$-dimensional TensorSpace, one prescribes weights $w_1, \dots, w_d$ for each dimension. The weight is defined by $w(\alpha) = w_1(\alpha_1) \times \ldots \times w_d(\alpha_d)$ for all $\alpha = (\alpha_1, \dots, \alpha_d) \in \mathscr{I}^{\prime\prime}$ where $\mathscr{I}^{\prime\prime} \subset \mathbb{Z}^d$ is the appropriate set of indices.

julia> a = ones(Taylor(2) ⊗ Fourier(2, 1.0) ⊗ Chebyshev(2));
julia> norm(a, Ell1((AlgebraicWeight(1.0), GeometricWeight(2.0), IdentityWeight())))390.0

However, the $d$-dimensional version of BesselWeight is defined by $w(\alpha) \bydef (1 + \alpha_1^2 + \ldots + \alpha_d^2)^s$ for all $\alpha = (\alpha_1, \dots, \alpha_d) \in \mathbb{Z}^d$. Only one BesselWeight is required for every Fourier space composing the TensorSpace.

julia> a = ones(Fourier(2, 1.0) ⊗ Fourier(3, 1.0));
julia> norm(a, Ell2(BesselWeight(2.0)))47.25462940284264

Normed cartesian space

For the norm of a CartesianSpace, one may use a NormedCartesianSpace to either:

use the same BanachSpace for each space.
use a different BanachSpace for each space.

julia> a = Sequence(Taylor(1)^2, [1.0, 2.0, 3.0, 4.0]);
julia> norm(a, NormedCartesianSpace(ℓ¹(), ℓ∞()))7.0
julia> norm(a, NormedCartesianSpace((ℓ¹(), ℓ²()), ℓ∞()))5.0

API

RadiiPolynomial.AlgebraicWeight — Type

AlgebraicWeight{T<:Real} <: Weight

Algebraic weight associated with a given rate satisfying isfinite(rate) & (rate ≥ 0).

Field:

rate :: T

Examples

julia> w = AlgebraicWeight(1.0)
AlgebraicWeight(1.0)

julia> rate(w)
1.0

source

RadiiPolynomial.BanachSpace — Type

BanachSpace

Abstract type for all Banach spaces.

source

RadiiPolynomial.BesselWeight — Type

BesselWeight{T<:Real} <: Weight

Bessel weight associated with a given rate satisfying isfinite(rate) & (rate ≥ 0).

Field:

rate :: T

Examples

julia> w = BesselWeight(1.0)
BesselWeight(1.0)

julia> rate(w)
1.0

source

RadiiPolynomial.Ell1 — Type

Ell1{T<:Union{Weight,Tuple{Vararg{Weight}}}} <: BanachSpace

Weighted $\ell^1$ space.

Field:

weight :: T

Constructors:

Ell1(::Weight)
Ell1(::Tuple{Vararg{Weight}})
Ell1(): equivalent to Ell1(IdentityWeight())
Ell1(weight::Weight...): equivalent to Ell1(weight)

Unicode alias ℓ¹ can be typed by \ell<tab>\^1<tab> in the Julia REPL and in many editors.

See also: Ell1, Ell2 and EllInf.

Examples

julia> NormedCartesianSpace(Ell1(), EllInf())
NormedCartesianSpace(ℓ¹(), ℓ∞())

julia> NormedCartesianSpace((Ell1(), Ell2()), EllInf())
NormedCartesianSpace((ℓ¹(), ℓ²()), ℓ∞())

source

RadiiPolynomial.Weight — Type

Weight

Abstract type for all weights.

source

RadiiPolynomial.ℓ² — Type

ℓ²(::Weight)
ℓ²(::Tuple{Vararg{Weight}})
ℓ²()
ℓ²(::Weight...)

Unicode alias of Ell2 representing the weighted $\ell^2$ space.

Examples

julia> ℓ²()
ℓ²()

julia> ℓ²(BesselWeight(1.0))
ℓ²(BesselWeight(1.0))

julia> ℓ²(BesselWeight(1.0), GeometricWeight(2.0))
ℓ²(BesselWeight(1.0), GeometricWeight(2.0))

source

RadiiPolynomial.ℓ¹ — Type

ℓ¹(::Weight)
ℓ¹(::Tuple{Vararg{Weight}})
ℓ¹()
ℓ¹(::Weight...)

Unicode alias of Ell1 representing the weighted $\ell^1$ space.

Examples

julia> ℓ¹()
ℓ¹()

julia> ℓ¹(GeometricWeight(1.0))
ℓ¹(GeometricWeight(1.0))

julia> ℓ¹(GeometricWeight(1.0), AlgebraicWeight(2.0))
ℓ¹(GeometricWeight(1.0), AlgebraicWeight(2.0))

source

RadiiPolynomial.ℓ∞ — Type

ℓ∞(::Weight)
ℓ∞(::Tuple{Vararg{Weight}})
ℓ∞()
ℓ∞(::Weight...)

Unicode alias of EllInf representing the weighted $\ell^\infty$ space.

Examples

julia> ℓ∞()
ℓ∞()

julia> ℓ∞(GeometricWeight(1.0))
ℓ∞(GeometricWeight(1.0))

julia> ℓ∞(GeometricWeight(1.0), AlgebraicWeight(2.0))
ℓ∞(GeometricWeight(1.0), AlgebraicWeight(2.0))

source

LinearAlgebra.norm — Function

norm(a::AbstractSequence, p::Real=Inf)

Compute the p-norm of a. Only p equal to 1, 2 or Inf is supported.

This is equivalent to:

norm(a, Ell1(IdentityWeight())) if p == 1
norm(a, Ell2(IdentityWeight())) if p == 2
norm(a, EllInf(IdentityWeight())) if p == Inf

source

LinearAlgebra.norm — Method

norm(a::Sequence, X::BanachSpace)

Compute the norm of a by interpreting space(a) as X.

source

LinearAlgebra.opnorm — Function

opnorm(A::LinearOperator, p::Real=Inf)

Compute the operator norm of A induced by the p-norm. Only p equal to 1, 2 or Inf is supported.

This is equivalent to:

opnorm(A, Ell1(IdentityWeight())) if p == 1
opnorm(A, Ell2(IdentityWeight())) if p == 2
opnorm(A, EllInf(IdentityWeight())) if p == Inf

source

LinearAlgebra.opnorm — Method

opnorm(A::LinearOperator, X::BanachSpace, Y::BanachSpace)

Compute the operator norm of A where X is the Banach space corresponding to domain(A) and Y the Banach space corresponding to codomain(A).

source

LinearAlgebra.opnorm — Method

opnorm(A::LinearOperator, X::BanachSpace)

Compute the operator norm of A where X is the Banach space corresponding to both domain(A) and codomain(A).

source

LinearAlgebra.opnorm — Method

opnorm(A::LinearOperator{<:VectorSpace,ScalarSpace}, X::BanachSpace)

Compute the operator norm of A where X is the Banach space corresponding to domain(A).

source