The Truncated Cyclic Gradient of a Gauge Vector

In the setting of The Real Numbers: Standing Notation and Background, let $\mathbb{N}$ be the natural numbers, for $k\in\mathbb{N}$ let $[k]$ be the initial segment determined by $k$, and let $d\in\mathbb{N}$. Let $W^{\circ}_{d}$ be the set of cyclically reduced words and $W^{\circ}_{d,k}$ the set of those of length $k$, let the weights $c_{w}$ and the gauge space $E_{d}$ be as in The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge, let $D^{i}_{w}$ be the cyclic derivatives, and let word polynomials, their pointwise sums and complex multiples and their adjoints be as in Word Polynomials in Unitary Letters: the Normed *-Algebra of Finitely Supported Word Functions; finite sums of word polynomials are word polynomials as explained in Rotations and Cyclic Derivatives of Words in Unitary Letters. Let $\mathbb{C}$ be the complex numbers with conjugation $z\mapsto\overline{z}$.

(Truncated cyclic gradient)¶ Let $p\in E_{d}$ and $n\in\mathbb{N}$, and let $W^{\circ}_{\le n}$ be the union of the sets $W^{\circ}_{d,k}$, $k\in[n]$, which is finite by Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §finite and Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §finite-union. For $i\in[d]$ put

$$Z^{i}_{p,n}=\sum_{w\in W^{\circ}_{\le n}}c_{w}\,\overline{p(w)}\,D^{i}_{w},\qquad \Xi^{i}_{p,n}=\tfrac12\bigl(Z^{i}_{p,n}+(Z^{i}_{p,n})^{*}\bigr).$$

The word polynomial $\Xi^{i}_{p,n}$ is the $i$-th cyclic gradient of $p$ truncated at length $n$.