nLab sequence algebra

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Algebra

Analysis

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Definition

For a commutative ring RR, the sequence algebra is the commutative R R -algebra R\mathbb{N} \to R, with canonical injection c:R(R)c \colon R \to (\mathbb{N} \to R) which sends each element rr in RR to a constant function c(r)c(r). The commutative algebra operations are pointwise defined as:

c(r)(n)rc(r)(n) \coloneqq r
0(n)00(n) \coloneqq 0
(x+y)(n)x(n)+y(n)(x + y)(n) \coloneqq x(n) + y(n)
(x)(n)x(n)(-x)(n) \coloneqq -x(n)
(rx)(n)rx(n)(r x)(n) \coloneqq r x(n)
(1)(n)1(1)(n) \coloneqq 1
(xy)(n)x(n)y(n)(x \cdot y)(n) \coloneqq x(n) \cdot y(n)

and thus the sequence set R\mathbb{N} \to R is a commutative RR-algebra.

Polynomials

Given a commutative ring RR, a sequential polynomial is a sequence x:Rx \colon \mathbb{N} \to R where there exists a natural number NN such that for all i>Ni \gt N we have x(i)=0x(i) = 0.

Operators

Left shift operator

Given a commutative ring RR and a sequence x:Rx \colon \mathbb{N} \to R of terms in RR, the left shift operator

T:(R)(R)T \colon (\mathbb{N} \to R) \to (\mathbb{N} \to R)

is defined as

Tx(i)x(i+1)T x(i) \coloneqq x(i + 1)

for ii \in \mathbb{N}.

Series operator

Given a commutative ring RR and a sequence x:Rx:\mathbb{N} \to R of terms in RR, the series operator

Σ:(R)(R)\Sigma:(\mathbb{N} \to R) \to (\mathbb{N} \to R)

is inductively defined as

Σx(0)x(0)\Sigma x(0) \coloneqq x(0)
Σx(i+1)Σx(i)+x(i+1)\Sigma x(i + 1) \coloneqq \Sigma x(i) + x(i + 1)

for ii \in \mathbb{N}.

For any such sequence x:Rx:\mathbb{N} \to R, Σx\Sigma x is called a series.

Inverse series operator

Given a commutative ring RR and a sequence x:Rx:\mathbb{N} \to R of terms in RR, the inverse series operator

Σ 1:(R)(R)\Sigma^{-1}:(\mathbb{N} \to R) \to (\mathbb{N} \to R)

is inductively defined as

Σ 1x(0)x(0)\Sigma^{-1} x(0) \coloneqq x(0)
Σ 1x(i+1)Σ 1x(i)x(i+1)\Sigma^{-1} x(i + 1) \coloneqq \Sigma^{-1} x(i) - x(i + 1)

for ii \in \mathbb{N}.

For any such sequence x:Rx:\mathbb{N} \to R, Σ 1Σx=x\Sigma^{-1}\Sigma x = x and ΣΣ 1x=x\Sigma\Sigma^{-1}x = x.

Infinite product operator

Given a commutative ring RR and a sequence x:Rx:\mathbb{N} \to R of terms in RR, the infinite product operator

Π:(R)(R)\Pi:(\mathbb{N} \to R) \to (\mathbb{N} \to R)

is inductively defined as

Πx(0)x(0)\Pi x(0) \coloneqq x(0)
Πx(i+1)Πx(i)x(i+1)\Pi x(i + 1) \coloneqq \Pi x(i) \cdot x(i + 1)

for ii \in \mathbb{N}.

For any such sequence x:Rx:\mathbb{N} \to R, Πx\Pi x is called an infinite product.

Discrete convolution

Given a commutative ring RR and sequence x:Rx:\mathbb{N} \to R and y:Ry:\mathbb{N} \to R, the discrete convolution

()*():(R)×(R)(R)(-)*(-):(\mathbb{N} \to R) \times (\mathbb{N} \to R) \to (\mathbb{N} \to R)

is defined as

(x*y)(i) n:[0,i]x(i)y(in)(x*y)(i) \coloneqq \sum_{n:[0,i]} x(i) y(i - n)

for ii \in \mathbb{N}.

Given a commutative ring RR and sequence x:Rx:\mathbb{N} \to R and y:Ry:\mathbb{N} \to R, the Cauchy product of the two series ΣxΣx\Sigma x \cdot \Sigma x is the discrete convolution of xx and yy.

ΣxΣyx*y\Sigma x \cdot \Sigma y \coloneqq x * y

Sequential derivative

Given a commutative ring RR and a sequence x:Rx:\mathbb{N} \to R of terms in RR, the sequential derivative

𝒟:(R)(R)\mathcal{D}:(\mathbb{N} \to R) \to (\mathbb{N} \to R)

is defined as

𝒟x(i)(i+1)x(i+1)\mathcal{D} x(i) \coloneqq (i + 1) x(i + 1)

for ii \in \mathbb{N}.

The sequential derivative is a derivation, where linearity is satisfied by addition and scalar multiplication of sequences and the Leibniz rule is satisfied for the Cauchy product of sequences. As a result, every sequence algebra is a differential algebra.

Right shift operators

Given a commutative ring RR and a sequence x:Rx:\mathbb{N} \to R, the set of right shifted sequences of xx is the fiber of the left shift operator at xx:

{y:R|Ty=x}\{y:\mathbb{N} \to R \vert T y = x\}

A right shift operator is an element of the above set.

Every sequence in RR has a right shift operator T s 1:(R)(R)T^{-1}_s:(\mathbb{N} \to R) \to (\mathbb{N} \to R) for every term rRr \in R, defined as

T r 1x(0)=rT^{-1}_r x(0) = r
T r 1x(i+i)x(i)T^{-1}_r x(i + i) \coloneqq x(i)

for ii \in \mathbb{N}.

Sequential antiderivatives

Given a commutative ring RR and a sequence x:Rx:\mathbb{N} \to R, the set of sequential antiderivatives of xx is the fiber of the sequential derivative at xx:

{y:R|𝒟y=x}\{y:\mathbb{N} \to R \vert \mathcal{D} y = x\}

A sequential antiderivative is a element of the above set.

If RR is a commutative \mathbb{Q}-algebra, then every sequence in RR has an antiderivative operator 𝒟 r 1:(R)(R)\mathcal{D}^{-1}_r:(\mathbb{N} \to R) \to (\mathbb{N} \to R) for every term rRr \in R, defined as

𝒟 r 1x(0)=r\mathcal{D}^{-1}_r x(0) = r
𝒟 r 1x(i+i)1ix(i)\mathcal{D}^{-1}_r x(i + i) \coloneqq \frac{1}{i} x(i)

for ii \in \mathbb{N}.

Sequence functions

Given a commutative ring RR, a sequence function is a function f:R(R)f:R \to (\mathbb{N} \to R).

Functional shift operator

Given a commutative ring RR and a sequence function f:R(R)f:R \to (\mathbb{N} \to R), the functional shift operator

𝒯:R(R(R))(R(R))\mathcal{T}:R \to (R \to (\mathbb{N} \to R)) \to (R \to (\mathbb{N} \to R))

is defined as

𝒯 cf(r)(i)f(rc)(i)\mathcal{T}_c f(r)(i) \coloneqq f(r - c)(i)

for cRc \in R, and rRr \in R, and ii \in \mathbb{N}.

Power sequence functions

Given a commutative ring RR and a sequence x:Rx:\mathbb{N} \to R of terms in RR, the power sequence function

𝒫:(R)(R(R))\mathcal{P}:(\mathbb{N} \to R) \to (R \to (\mathbb{N} \to R))

is defined as

𝒫 x(r)(i)x(i)r i\mathcal{P}_x (r)(i) \coloneqq x(i) r^i

for ii \in \mathbb{N} and rRr \in R.

Functional derivative

Given a commutative ring RR, the functional derivative of the power sequence function f:R(R)f:R \to (\mathbb{N} \to R)

D˜:(R(R))(R(R))\tilde{D}:(R \to (\mathbb{N} \to R)) \to (R \to (\mathbb{N} \to R))

is defined as

(D˜𝒫 x)(r)(i)(𝒟x(i))r i(\tilde{D} \mathcal{P}_x)(r)(i) \coloneqq (\mathcal{D} x(i)) r^i

Power series

For any such sequence x:Rx:\mathbb{N} \to R, Σ𝒫 x(r)\Sigma\mathcal{P}_x(r) is called a formal power series or formal Maclaurin series. For any such sequence x:Rx:\mathbb{N} \to R and element cRc \in R, Σ𝒯 c𝒫 x(r)\Sigma \mathcal{T}_c \mathcal{P}_x (r) is called a formal Taylor series.

Analytic functions

Suppose RR is a commutative ring with a partial function

lim n()(n):(R)R\lim_{n \to \infty} (-)(n) \colon (\mathbb{N} \to R) \to R

making RR into a sequentially Hausdorff space.

Then, a partial function f:RRf \colon R \to R is an analytic function if there exists an element cRc \in R and sequence x:Rx \colon \mathbb{N} \to R such that

f(r)=lim nΣ𝒯 c𝒫 x(r)(n)f(r) = \lim_{n \to \infty} \Sigma \mathcal{T}_c \mathcal{P}_x (r)(n)

for all elements rdomfr \in \dom f.

See also

Last revised on May 9, 2022 at 05:16:09. See the history of this page for a list of all contributions to it.