Contents
Contents
Definition
For a commutative ring , the sequence algebra is the commutative -algebra , with canonical injection which sends each element in to a constant function . The commutative algebra operations are pointwise defined as:
and thus the sequence set is a commutative -algebra.
Polynomials
Given a commutative ring , a sequential polynomial is a sequence where there exists a natural number such that for all we have .
Operators
Left shift operator
Given a commutative ring and a sequence of terms in , the left shift operator
is defined as
for .
Series operator
Given a commutative ring and a sequence of terms in , the series operator
is inductively defined as
for .
For any such sequence , is called a series.
Inverse series operator
Given a commutative ring and a sequence of terms in , the inverse series operator
is inductively defined as
for .
For any such sequence , and .
Infinite product operator
Given a commutative ring and a sequence of terms in , the infinite product operator
is inductively defined as
for .
For any such sequence , is called an infinite product.
Discrete convolution
Given a commutative ring and sequence and , the discrete convolution
is defined as
for .
Given a commutative ring and sequence and , the Cauchy product of the two series is the discrete convolution of and .
Sequential derivative
Given a commutative ring and a sequence of terms in , the sequential derivative
is defined as
for .
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 and a sequence , the set of right shifted sequences of is the fiber of the left shift operator at :
A right shift operator is an element of the above set.
Every sequence in has a right shift operator for every term , defined as
for .
Sequential antiderivatives
Given a commutative ring and a sequence , the set of sequential antiderivatives of is the fiber of the sequential derivative at :
A sequential antiderivative is a element of the above set.
If is a commutative -algebra, then every sequence in has an antiderivative operator for every term , defined as
for .
Sequence functions
Given a commutative ring , a sequence function is a function .
Functional shift operator
Given a commutative ring and a sequence function , the functional shift operator
is defined as
for , and , and .
Power sequence functions
Given a commutative ring and a sequence of terms in , the power sequence function
is defined as
for and .
Functional derivative
Given a commutative ring , the functional derivative of the power sequence function
is defined as
Power series
For any such sequence , is called a formal power series or formal Maclaurin series. For any such sequence and element , is called a formal Taylor series.
Analytic functions
Suppose is a commutative ring with a partial function
making into a sequentially Hausdorff space.
Then, a partial function is an analytic function if there exists an element and sequence such that
for all elements .
See also