Decimation

Abstract

Finite Riesz moments, orthogonal Laurent polynomials and scale-four projection identities.

Theorem 1.1 (parity weighted projection).

$$\forall (BC:LaurentPolynomial\mathbb{R}{\to }_l[\mathbb{R}]LaurentPolynomial\mathbb{R}{\to }_l[\mathbb{R}]\mathbb{R})(xt:\mathbb{N}),(\forall (ij:\mathbb{Z}),i\mathrm{mod}2\ne j\mathrm{mod}2\to (B(LaurentPolynomial.Ti))(LaurentPolynomial.Tj)=0)\to (\forall (e:\mathbb{Z})(q:LaurentPolynomial\mathbb{R}),(C(LaurentPolynomial.Te))q=(B(LaurentPolynomial.Te))q+((B(LaurentPolynomial.T(e-1)))q+(B(LaurentPolynomial.T(e+1)))q)/2)\to \forall (h:LaurentPolynomial\mathbb{R}),\mathrm{IsProjection}B(Ex)(LaurentPolynomial.T(-(2\cdot Nat.cast\left(t\right))))h\to \mathrm{IsProjection}C(Wx)(LaurentPolynomial.T(-(2\cdot Nat.cast\left(t\right))))h\wedge (Ch)h=(Bh)h$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Walks/Riesz/Decimation.parity_weighted_projection (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. F. Alberto Grunbaum, Luis Velazquez (2012). The quantum walk of F. Riesz. URL: https://arxiv.org/abs/1111.6630.

Commentary.

If a real bilinear form B pairs monomials of different parity to zero and C is obtained from B by adding half the pairings with the two neighbouring exponents, then for natural x and t a B projection of the monomial at negative twice t onto E x is also a C projection onto W x, with the same self-pairing. For integer arguments, mod denotes the Euclidean remainder: for nonzero d it lies between 0 inclusive and abs(d) exclusive, and e mod 0 equals e.

Theorem 1.2 (projection norm scale four).

$$\forall (xt:\mathbb{N})\{hk:LaurentPolynomial\mathbb{R}\},\mathrm{IsProjection}\mathrm{sigma}(V(2\cdot x))(LaurentPolynomial.T(-(2\cdot Nat.cast\left(t\right))))h\to \mathrm{IsProjection}\mathrm{sigma}(V(8\cdot x))(LaurentPolynomial.T(-(8\cdot Nat.cast\left(t\right))))k\to (\mathrm{sigma}h)h=(\mathrm{sigma}k)k$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Walks/Riesz/Decimation.projection_norm_scale_four (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. F. Alberto Grunbaum, Luis Velazquez (2012). The quantum walk of F. Riesz. URL: https://arxiv.org/abs/1111.6630.

Commentary.

For natural x and t, a sigma projection of the monomial at negative twice t onto V at twice x and a sigma projection of the monomial at negative eight times t onto V at eight times x have equal sigma pairings with themselves.

References

• Truth anchor: D5/S3/Quantum/Walks/Riesz/Decimation.parity_weighted_projection
• Truth anchor: D5/S3/Quantum/Walks/Riesz/Decimation.projection_norm_scale_four
• Dependency: D5/S3/Quantum/Walks/CMV/SzegoRecurrence