# Library

## Domains

`ApproxFunOrthogonalPolynomials.Arc`

— Type`Arc(c,r,(θ₁,θ₂))`

represents the arc centred at `c`

with radius `r`

from angle `θ₁`

to `θ₂`

.

`ApproxFunFourier.Circle`

— Type`Circle(c,r,o)`

represents the circle centred at `c`

with radius `r`

which is positively (`o=true`

) or negatively (`o=false`

) oriented.

`ApproxFun.Curve`

— Type`Curve`

Represents a domain defined by the image of a Fun. Example usage would be

```
x=Fun(1..2)
Curve(exp(im*x)) # represents an arc
```

`ApproxFunBase.Segment`

— Type`Segment(a,b)`

represents a line segment from `a`

to `b`

. In the case where `a`

and `b`

are real and `a < b`

, then this is is equivalent to an `Interval(a,b)`

.

`IntervalSets.Interval`

— TypeAn `Interval{L,R}(left, right)`

where L,R are :open or :closed is an interval set containg `x`

such that

`left ≤ x ≤ right`

if`L == R == :closed`

`left < x ≤ right`

if`L == :open`

and`R == :closed`

`left ≤ x < right`

if`L == :closed`

and`R == :open`

, or`left < x < right`

if`L == R == :open`

`ApproxFunOrthogonalPolynomials.Line`

— Type`Line{a}(c)`

represents the line at angle `a`

in the complex plane, centred at `c`

.

`ApproxFunFourier.PeriodicSegment`

— Type`PeriodicSegment(a,b)`

represents a periodic interval from `a`

to `b`

, that is, the point `b`

is identified with `a`

.

`DomainSets.Point`

— Type`Point(x)`

represents a single point at `x`

.

`DomainSets.ProductDomain`

— Type`abstract type ProductDomain{T}`

Represents the cartesian product of other domains.

`ApproxFunOrthogonalPolynomials.Ray`

— Type`Ray{a}(c,L,o)`

represents a scaled ray (with scale factor L) at angle `a`

starting at `c`

, with orientation out to infinity (`o = true`

) or back from infinity (`o = false`

).

`DomainSets.UnionDomain`

— Type```
UnionDomain(domains...)
UnionDomain{T}(domains...)
```

The lazy union of the given domains.

See also: `uniondomain`

.

`DomainSets.uniondomain`

— Function`uniondomain(domains...)`

Return a domain that agrees with the mathematical union of the arguments.

See also: `UnionDomain`

.

`DomainSets.∂`

— FunctionReturn the boundary of the given domain as a domain.

## Accessing information about a spaces

`ApproxFunBase.canonicalspace`

— Function`canonicalspace(s::Space)`

Return a space that is used as a default to implement missing functionality, e.g., evaluation. Implement a `Conversion`

operator or override `coefficients`

to support this.

**Examples**

```
julia> ApproxFunBase.canonicalspace(NormalizedChebyshev())
Chebyshev()
```

`ApproxFunBase.itransform`

— Function`itransform(s::Space,coefficients::AbstractVector)`

Transform coefficients back to values. Defaults to using `canonicalspace`

as in `transform`

.

**Examples**

```
julia> F = Fun(x->x^2, Chebyshev())
Fun(Chebyshev(), [0.5, 0.0, 0.5])
julia> itransform(Chebyshev(), coefficients(F)) ≈ values(F)
true
julia> itransform(Chebyshev(), [0.5, 0, 0.5])
3-element Vector{Float64}:
0.75
0.0
0.75
```

`ApproxFunBase.transform`

— Function`transform(s::Space, vals)`

Transform values on the grid specified by `points(s,length(vals))`

to coefficients in the space `s`

. Defaults to `coefficients(transform(canonicalspace(space),values),canonicalspace(space),space)`

**Examples**

```
julia> F = Fun(x -> x^2, Chebyshev());
julia> coefficients(F)
3-element Vector{Float64}:
0.5
0.0
0.5
julia> transform(Chebyshev(), values(F)) ≈ coefficients(F)
true
julia> v = map(F, points(Chebyshev(), 4)); # custom grid
julia> transform(Chebyshev(), v)
4-element Vector{Float64}:
0.5
0.0
0.5
0.0
```

`ApproxFunBase.evaluate`

— Function`evaluate(coefficients::AbstractVector, sp::Space, x)`

Evaluate the expansion at a point `x`

that lies in `domain(sp)`

. If `x`

is not in the domain, the returned value will depend on the space, and should not be relied upon. See `extrapolate`

to evaluate a function at a value outside the domain.

`DomainSets.dimension`

— Method`dimension(s::Space)`

Return the dimension of `s`

, which is the maximum number of coefficients.

## Inbuilt spaces

`ApproxFunBase.SequenceSpace`

— Type`SequenceSpace`

The space of all sequences, i.e., infinite vectors. Also denoted ℓ⁰.

`ApproxFunBase.ConstantSpace`

— Type`ConstantSpace`

The 1-dimensional scalar space.

`ApproxFunOrthogonalPolynomials.Chebyshev`

— Type`Chebyshev()`

is the space spanned by the Chebyshev polynomials

` T_0(x),T_1(x),T_2(x),…`

where `T_k(x) = cos(k*acos(x))`

. This is the default space as there exists a fast transform and general smooth functions on `[-1,1]`

can be easily resolved.

`ApproxFunOrthogonalPolynomials.Hermite`

— Type`Hermite(L)`

represents `H_k(sqrt(L) * x)`

where `H_k`

are Hermite polynomials. `Hermite()`

is equivalent to `Hermite(1.0)`

.

`ApproxFunOrthogonalPolynomials.Jacobi`

— Type`Jacobi(b,a)`

represents the space spanned by Jacobi polynomials `P_k^{(a,b)}`

, which are orthogonal with respect to the weight `(1+x)^β*(1-x)^α`

`ApproxFunOrthogonalPolynomials.Laguerre`

— Type`Laguerre(α)`

is a space spanned by generalized Laguerre polynomials `Lₙᵅ(x)`

's on `(0, Inf)`

, which satisfy the differential equations

` xy'' + (α + 1 - x)y' + ny = 0`

`Laguerre()`

is equivalent to `Laguerre(0)`

by default.

`ApproxFunOrthogonalPolynomials.Ultraspherical`

— Type`Ultraspherical(λ)`

is the space spanned by the ultraspherical polynomials

` C_0^{(λ)}(x),C_1^{(λ)}(x),C_2^{(λ)}(x),…`

Note that `λ=1`

this reduces to Chebyshev polynomials of the second kind: `C_k^{(1)}(x) = U_k(x)`

. For `λ=1/2`

this also reduces to Legendre polynomials: `C_k^{(1/2)}(x) = P_k(x)`

.

`ApproxFunFourier.Taylor`

— Type`Taylor()`

is the space spanned by `[1,z,z^2,...]`

. This is a type alias for `Hardy{true}`

.

`ApproxFunFourier.Hardy`

— Type`Hardy{false}()`

is the space spanned by `[1/z,1/z^2,...]`

. `Hardy{true}()`

is the space spanned by `[1,z,z^2,...]`

.

`ApproxFunFourier.Fourier`

— Type`Fourier()`

The space spanned by the trigonemtric polynomials

` 1, sin(θ), cos(θ), sin(2θ), cos(2θ), …`

See also `Laurent`

.

`Fourier(d::Domain)`

The space spanned by the trigonemtric polynomials

` 1, sin(2pi/L*θ), cos(2pi/L*θ), sin(2pi/L*2θ), cos(2pi/L*2θ), …`

for a domain with a period `L`

.

`ApproxFunFourier.Laurent`

— Type`Laurent()`

The space spanned by the complex exponentials

` 1,exp(-im*θ),exp(im*θ),exp(-2im*θ),…`

See also `Fourier`

.

`Laurent(d::Domain)`

The space spanned by the complex exponentials

` 1, exp(-im * (2pi/L*θ)), exp(im * (2pi/L*θ)), exp(-2im * (2pi/L*θ)), …`

for a domain with a period `L`

.

`ApproxFunFourier.CosSpace`

— Type`CosSpace()`

The space spanned by `[1, cos θ, cos 2θ, ...]`

`CosSpace(d::Domain)`

The space spanned by `[1,cos(2pi/L*θ), cos(2pi/L*2θ), ...]`

for a domain with a period `L`

`ApproxFunFourier.SinSpace`

— Type`SinSpace()`

The space spanned by `[sin θ, sin 2θ, ...]`

`SinSpace(d::Domain)`

The space spanned by `[1, sin(2pi/L*θ), sin(2pi/L*2θ), ...]`

for a domain with a period `L`

`ApproxFunSingularities.JacobiWeight`

— Type`JacobiWeight(β,α,s::Space)`

weights a space `s`

by a Jacobi weight, which on `-1..1`

is `(1+x)^β*(1-x)^α`

. For other domains, the weight is inferred by mapping to `-1..1`

.

`ApproxFunSingularities.LogWeight`

— Type`LogWeight(β,α,s::Space)`

represents a function on `-1..1`

weighted by `log((1+x)^β*(1-x)^α)`

. For other domains, the weight is inferred by mapping to `-1..1`

.

`ApproxFunBase.ArraySpace`

— Type`ArraySpace(s::Space,dims...)`

is used to represent array-valued expansions in a space `s`

. The coefficients are of each entry are interlaced.

For example,

```
f = Fun(x->[exp(x),sin(x)],-1..1)
space(f) == ArraySpace(Chebyshev(),2)
```

`ApproxFunBase.TensorSpace`

— Type`TensorSpace(a::Space,b::Space)`

represents a tensor product of two 1D spaces `a`

and `b`

. The coefficients are interlaced in lexigraphical order.

For example, consider

`Fourier()*Chebyshev() # returns TensorSpace(Fourier(),Chebyshev())`

This represents functions on `[-π,π) x [-1,1]`

, using the Fourier basis for the first argument and Chebyshev basis for the second argument, that is, `φ_k(x)T_j(y)`

, where

```
φ_0(x) = 1,
φ_1(x) = sin x,
φ_2(x) = cos x,
φ_3(x) = sin 2x,
φ_4(x) = cos 2x
…
```

By Choosing `(k,j)`

appropriately, we obtain a single basis:

```
φ_0(x)T_0(y) (= 1),
φ_0(x)T_1(y) (= y),
φ_1(x)T_0(y) (= sin x),
φ_0(x)T_2(y), …
```

## Constructing a Fun

`ApproxFunBase.Fun`

— Type`Fun(s::Space, coefficients::AbstractVector)`

Return a `Fun`

with the specified `coefficients`

in the space `s`

**Examples**

```
julia> f = Fun(Fourier(), [1,1]);
julia> f(0.1) == 1 + sin(0.1)
true
julia> f = Fun(Chebyshev(), [1,1]);
julia> f(0.1) == 1 + 0.1
true
```

`Fun()`

Return `Fun(identity, Chebyshev())`

, which represents the identity function in `-1..1`

.

**Examples**

```
julia> f = Fun(Chebyshev())
Fun(Chebyshev(), [0.0, 1.0])
julia> f(0.1)
0.1
```

`Fun(s::Space)`

Return `Fun(identity, s)`

**Examples**

```
julia> x = Fun(Chebyshev())
Fun(Chebyshev(), [0.0, 1.0])
julia> x(0.1)
0.1
```

`Fun(f, d::Domain)`

Return `Fun(f, Space(d))`

, that is, it uses the default space for the specified domain.

**Examples**

```
julia> f = Fun(x->x^2, 0..1);
julia> f(0.1) ≈ (0.1)^2
true
```

`Fun(f, s::Space)`

Return a `Fun`

representing the function, number, or vector `f`

in the space `s`

. If `f`

is vector-valued, it Return a vector-valued analogue of `s`

.

**Examples**

```
julia> f = Fun(x->x^2, Chebyshev())
Fun(Chebyshev(), [0.5, 0.0, 0.5])
julia> f(0.1) == (0.1)^2
true
```

`Fun(f)`

Return `Fun(f, space)`

by choosing an appropriate `space`

for the function. For univariate functions, `space`

is chosen to be `Chebyshev()`

, whereas for multivariate functions, it is a tensor product of `Chebyshev()`

spaces.

**Examples**

```
julia> f = Fun(x -> x^2)
Fun(Chebyshev(), [0.5, 0.0, 0.5])
julia> f(0.1) == (0.1)^2
true
julia> f = Fun((x,y) -> x + y);
julia> f(0.1, 0.2) ≈ 0.3
true
```

`Base.ones`

— Method`ones(d::Space)`

Return the `Fun`

that represents the function one on the specified space.

**Examples**

```
julia> ones(Chebyshev())
Fun(Chebyshev(), [1.0])
```

`Base.zeros`

— Method`zeros(d::Space)`

Return the `Fun`

that represents the function one on the specified space.

**Examples**

```
julia> zeros(Chebyshev())
Fun(Chebyshev(), [0.0])
```

## Accessing information about a Fun

`ApproxFunBase.domain`

— Function`domain(f::Fun)`

Return the domain that `f`

is defined on.

**Examples**

```
julia> f = Fun(x->x^2);
julia> domain(f) == ChebyshevInterval()
true
julia> f = Fun(x->x^2, 0..1);
julia> domain(f) == 0..1
true
```

`ApproxFunBase.coefficients`

— Function`coefficients(cfs::AbstractVector, fromspace::Space, tospace::Space) -> Vector`

Convert coefficients in `fromspace`

to coefficients in `tospace`

**Examples**

```
julia> f = Fun(x->(3x^2-1)/2);
julia> coefficients(f, Chebyshev(), Legendre()) ≈ [0,0,1]
true
julia> g = Fun(x->(3x^2-1)/2, Legendre());
julia> coefficients(f, Chebyshev(), Legendre()) ≈ coefficients(g)
true
```

`coefficients(f::Fun, s::Space) -> Vector`

Return the coefficients of `f`

in the space `s`

, which may not be the same as `space(f)`

.

**Examples**

```
julia> f = Fun(x->(3x^2-1)/2);
julia> coefficients(f, Legendre()) ≈ [0, 0, 1]
true
```

`coefficients(f::Fun) -> Vector`

Return the coefficients of `f`

, corresponding to the space `space(f)`

.

**Examples**

```
julia> f = Fun(x->x^2)
Fun(Chebyshev(), [0.5, 0.0, 0.5])
julia> coefficients(f)
3-element Vector{Float64}:
0.5
0.0
0.5
```

`ApproxFunBase.extrapolate`

— Function`extrapolate(f::Fun,x)`

Return an extrapolation of `f`

from its domain to `x`

.

**Examples**

```
julia> f = Fun(x->x^2)
Fun(Chebyshev(), [0.5, 0.0, 0.5])
julia> extrapolate(f, 2) # 2 lies outside the domain -1..1
4.0
```

`ApproxFunBase.ncoefficients`

— Function`ncoefficients(f::Fun) -> Integer`

Return the number of coefficients of a fun

**Examples**

```
julia> f = Fun(x->x^2)
Fun(Chebyshev(), [0.5, 0.0, 0.5])
julia> ncoefficients(f)
3
```

`ApproxFunBase.points`

— Function`points(s::Space,n::Integer)`

Return a grid of approximately `n`

points, for which a transform exists from values at the grid to coefficients in the space `s`

.

**Examples**

```
julia> chebypts(n) = [cos((2i+1)pi/2n) for i in 0:n-1];
julia> points(Chebyshev(), 4) ≈ chebypts(4)
true
```

`points(f::Fun)`

Return a grid of points that `f`

can be transformed into values and back.

**Examples**

```
julia> f = Fun(x->x^2);
julia> chebypts(n) = [cos((2i+1)pi/2n) for i in 0:n-1];
julia> points(f) ≈ chebypts(ncoefficients(f))
true
```

`ApproxFunBase.space`

— Function`space(f::Fun)`

Return the space of `f`

.

**Examples**

```
julia> f = Fun(x->x^2)
Fun(Chebyshev(), [0.5, 0.0, 0.5])
julia> space(f)
Chebyshev()
```

`Base.values`

— Method`values(iterator)`

For an iterator or collection that has keys and values, return an iterator over the values. This function simply returns its argument by default, since the elements of a general iterator are normally considered its "values".

**Examples**

```
julia> d = Dict("a"=>1, "b"=>2);
julia> values(d)
ValueIterator for a Dict{String, Int64} with 2 entries. Values:
2
1
julia> values([2])
1-element Vector{Int64}:
2
```

`values(f::Fun)`

Return `f`

evaluated at `points(f)`

.

**Examples**

```
julia> f = Fun(x->x^2)
Fun(Chebyshev(), [0.5, 0.0, 0.5])
julia> values(f)
3-element Vector{Float64}:
0.75
0.0
0.75
julia> map(x->x^2, points(f)) ≈ values(f)
true
```

`Base.stride`

— Method`stride(f::Fun)`

Return the stride of the coefficients, checked numerically

## Modify a Fun

`ApproxFunBase.reverseorientation`

— Function`reverseorientation(f::Fun)`

Return `f`

on a reversed orientated contour.

`ApproxFunBase.setdomain`

— Function`setdomain(f::Fun, d::Domain)`

Return `f`

projected onto `domain`

.

The new function may differ from the original one, as the coefficients are left unchanged.

**Examples**

```
julia> f = Fun(x->x^2);
julia> domain(f) == ChebyshevInterval()
true
julia> g = setdomain(f, 0..1);
julia> domain(g) == 0..1
true
julia> coefficients(f) == coefficients(g)
true
```

`Base.chop`

— Method`chop(f::Fun[, tol = 10eps()]) -> Fun`

Reduce the number of coefficients by dropping the tail that is below the specified tolerance.

**Examples**

```
julia> f = Fun(Chebyshev(), [1,2,3,0,0,0])
Fun(Chebyshev(), [1, 2, 3, 0, 0, 0])
julia> chop(f)
Fun(Chebyshev(), [1, 2, 3])
```

## Bivariate Fun

`ApproxFunBase.LowRankFun`

— Type`LowRankFun(f, space::TensorSpace)`

Return an approximation to a bivariate function in a low-rank form

\[f(x,y) = \sum_i \sigma_i \phi_i(x) \psi_i(y)\]

where $\sigma_i$ represent the highest singular values, and $\phi_i(x)$ and $\psi_i(y)$ are orthogonal bases. The summation is truncated after an acceptable tolerance is reached.

**Examples**

```
julia> f = (x,y) -> x^2 * y^3;
julia> L = LowRankFun(f, Chebyshev() ⊗ Chebyshev());
julia> L(0.1, 0.2) ≈ f(0.1, 0.2)
true
```

`ApproxFunBase.ProductFun`

— Type`ProductFun(coeffs::AbstractMatrix{T}, sp::AbstractProductSpace; [tol=100eps(T)], [chopping=false]) where {T<:Number}`

Represent a bivariate function `f`

in terms of the coefficient matrix `coeffs`

, where the coefficients are obtained using a bivariate transform of the function `f`

in the basis `sp`

.

**Examples**

```
julia> P = ProductFun([0 0; 0 1], Chebyshev() ⊗ Chebyshev()) # corresponds to (x,y) -> x*y
ProductFun on Chebyshev() ⊗ Chebyshev()
julia> P(0.1, 0.2) ≈ 0.1 * 0.2
true
```

`ProductFun(M::AbstractVector{<:Fun{<:UnivariateSpace}}, sp::UnivariateSpace)`

Represent a bivariate function `f(x,y)`

in terms of the univariate coefficient functions from `M`

. The function `f`

may be reconstructed as

\[f\left(x,y\right)=\sum_{i}M_{i}\left(x\right)b_{i}\left(y\right),\]

where $b_{i}\left(y\right)$ represents the $i$-th basis function for the space `sp`

.

**Examples**

```
julia> P = ProductFun([zeros(Chebyshev()), Fun(Chebyshev())], Chebyshev()); # corresponds to (x,y)->x*y
julia> P(0.1, 0.2) ≈ 0.1 * 0.2
true
```

## Operators

`ApproxFunBase.Operator`

— Type`Operator{T}`

Abstract type to represent linear operators between spaces.

`BandedMatrices.bandwidths`

— Method`bandwidths(op::Operator)`

Return the bandwidth of `op`

in the form `(l,u)`

, where `l ≥ 0`

represents the number of subdiagonals and `u ≥ 0`

represents the number of superdiagonals.

`ApproxFunBase.domainspace`

— Function`domainspace(op::Operator)`

Return the domain space of `op`

. That is, `op*f`

will first convert `f`

to a `Fun`

in the space `domainspace(op)`

before applying the operator.

`ApproxFunBase.rangespace`

— Function`rangespace(op::Operator)`

Return the range space of `op`

. That is, `op*f`

will return a `Fun`

in the space `rangespace(op)`

, provided `f`

can be converted to a `Fun`

in `domainspace(op)`

.

`Base.getindex`

— Method`(op::Operator)[k,j]`

Return the `k`

th coefficient of `op*Fun([zeros(j-1);1],domainspace(op))`

.

`Base.getindex`

— Method`(op::Operator)[f::Fun]`

Construct the operator `op * Multiplication(f)`

, that is, it multiplies on the right by `f`

first. Note that `op * f`

is different: it applies `op`

to `f`

.

**Examples**

```
julia> x = Fun()
Fun(Chebyshev(), [0.0, 1.0])
julia> D = Derivative()
ConcreteDerivative : ApproxFunBase.UnsetSpace() → ApproxFunBase.UnsetSpace()
julia> Dx = D[x] # construct the operator y -> d/dx * (x * y)
TimesOperator : ApproxFunBase.UnsetSpace() → ApproxFunBase.UnsetSpace()
julia> twox = Dx * x # Evaluate d/dx * (x * x)
Fun(Ultraspherical(1), [0.0, 1.0])
julia> twox(0.1) ≈ 2 * 0.1
true
```

`Base.:\`

— Method`\(A::Operator,b;tolerance=tol,maxlength=n)`

solves a linear equation, usually differential equation, where `A`

is an operator or array of operators and `b`

is a `Fun`

or array of funs. The result `u`

will approximately satisfy `A*u = b`

.

`LinearAlgebra.qr`

— Method`qr(A::Operator)`

returns a cached QR factorization of the Operator `A`

. The result `QR`

enables solving of linear equations: if `u=QR`

, then `u`

approximately satisfies `A*u = b`

.

`LazyArrays.cache`

— Method`cache(op::Operator)`

Caches the entries of an operator, to speed up multiplying a Fun by the operator.

## Inbuilt operators

`ApproxFunBase.Conversion`

— Type`Conversion(fromspace::Space, tospace::Space)`

Represent a conversion operator between `fromspace`

and `tospace`

, when available.

See also `PartialInverseOperator`

that might be able to represent the inverse, even if this isn't banded.

`ApproxFunBase.Derivative`

— Type`Derivative(sp::Space, k::Int)`

Return the `k`

-th derivative operator on the space `sp`

.

**Examples**

```
julia> Derivative(Chebyshev(), 2) * Fun(x->x^4) ≈ Fun(x->12x^2)
true
```

`Derivative(sp::Space, k::AbstractVector{Int})`

Return a partial derivative operator on a multivariate space. For example,

```
Dx = Derivative(Chebyshev()^2,[1,0]) # ∂/∂x
Dy = Derivative(Chebyshev()^2,[0,1]) # ∂/∂y
```

Using a static vector as the second argument would help with type-stability.

**Examples**

```
julia> ∂y = Derivative(Chebyshev()^2, [0,1]);
julia> ∂y * Fun((x,y)->x^2 + y^2) ≈ Fun((x,y)->2y)
true
```

`Derivative(sp::Space)`

Return the first derivative operator, equivalent to `Derivative(sp,1)`

.

**Examples**

```
julia> Derivative(Chebyshev()) * Fun(x->x^2) ≈ Fun(x->2x)
true
```

`Derivative(k)`

Return the `k`

-th derivative, acting on an unset space. Spaces will be inferred when applying or manipulating the operator. If `k`

is an `Int`

, this returns a derivative in an univariate space. If `k`

is an `AbstractVector{Int}`

, this returns a partial derivative in a multivariate space.

**Examples**

```
julia> Derivative(1) * Fun(x->x^2) ≈ Fun(x->2x)
true
julia> Derivative([0,1]) * Fun((x,y)->x^2+y^2) ≈ Fun((x,y)->2y)
true
```

`Derivative()`

Return the first derivative on an unset space. Spaces will be inferred when applying or manipulating the operator.

**Examples**

```
julia> Derivative() * Fun(x->x^2) ≈ Fun(x->2x)
true
```

`ApproxFunBase.Dirichlet`

— Type`Dirichlet(sp,k)`

is the operator associated with restricting the `k`

-th derivative on the boundary for the space `sp`

.

`Dirichlet(sp)`

is the operator associated with restricting the the boundary for the space `sp`

.

`Dirichlet()`

is the operator associated with restricting on the the boundary.

`ApproxFunBase.Evaluation`

— Type`Evaluation(sp,x,k)`

is the functional associated with evaluating the `k`

-th derivative at a point `x`

for the space `sp`

.

`Evaluation(sp,x)`

is the functional associated with evaluating at a point `x`

for the space `sp`

.

`Evaluation(x)`

is the functional associated with evaluating at a point `x`

.

`ApproxFunBase.Integral`

— Type`Integral(sp::Space, k::Int)`

Return the `k`

-th integral operator on `sp`

. There is no guarantee on normalization.

`Integral(sp::Space)`

Return the first integral operator, equivalent to `Integral(sp,1)`

.

`Integral(k::Int)`

Return the `k`

-th integral operator, acting on an unset space. Spaces will be inferred when applying or manipulating the operator.

`Intergral()`

Return the first integral operator on an unset space. Spaces will be inferred when applying or manipulating the operator.

`ApproxFunBase.Laplacian`

— Type`Laplacian(sp::Space)`

Return the laplacian operator on space `sp`

.

`Laplacian()`

Return the laplacian operator on an unset space. Spaces will be inferred when applying or manipulating the operator.

`ApproxFunBase.Multiplication`

— Type`Multiplication(f::Fun,sp::Space)`

is the operator representing multiplication by `f`

on functions in the space `sp`

.

`Multiplication(f::Fun)`

is the operator representing multiplication by `f`

on an unset space of functions. Spaces will be inferred when applying or manipulating the operator.

`ApproxFunBase.Neumann`

— Function`Neumann(sp)`

is the operator associated with restricting the normal derivative on the boundary for the space `sp`

. At the moment it is implemented as `Dirichlet(sp,1)`

.

`Neumann()`

is the operator associated with restricting the normal derivative on the boundary.

`ApproxFunBase.PartialInverseOperator`

— Type`PartialInverseOperator(O::Operator, bandwidths = (0, Infinities.ℵ₀))`

Return an approximate estimate for `inv(O)`

, such that `PartialInverseOperator(O) * O`

is banded, and is approximately `I`

up to a bandwidth that is one less than the sum of the bandwidths of `O`

and `PartialInverseOperator(O)`

.

Only upper-triangular operators are supported as of now.

**Examples**

```
julia> C = Conversion(Chebyshev(), Ultraspherical(1));
julia> P = PartialInverseOperator(C); # default bandwidth
julia> P * C
TimesOperator : Chebyshev() → Chebyshev()
1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ⋯
⋅ 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ⋱
⋅ ⋅ 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ⋱
⋅ ⋅ ⋅ 1.0 0.0 0.0 0.0 0.0 0.0 0.0 ⋱
⋅ ⋅ ⋅ ⋅ 1.0 0.0 0.0 0.0 0.0 0.0 ⋱
⋅ ⋅ ⋅ ⋅ ⋅ 1.0 0.0 0.0 0.0 0.0 ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 0.0 0.0 0.0 ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 0.0 0.0 ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 0.0 ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 ⋱
⋮ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱
julia> P = PartialInverseOperator(C, (0, 4)); # specify an upper bandwidth
julia> P * C
TimesOperator : Chebyshev() → Chebyshev()
1.0 0.0 0.0 0.0 0.0 0.0 -0.5 ⋅ ⋅ ⋅ ⋅
⋅ 1.0 0.0 0.0 0.0 0.0 0.0 -1.0 ⋅ ⋅ ⋅
⋅ ⋅ 1.0 0.0 0.0 0.0 0.0 0.0 -1.0 ⋅ ⋅
⋅ ⋅ ⋅ 1.0 0.0 0.0 0.0 0.0 0.0 -1.0 ⋅
⋅ ⋅ ⋅ ⋅ 1.0 0.0 0.0 0.0 0.0 0.0 ⋱
⋅ ⋅ ⋅ ⋅ ⋅ 1.0 0.0 0.0 0.0 0.0 ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 0.0 0.0 0.0 ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 0.0 0.0 ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 0.0 ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.0 ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋱
```