Weak parallelogram laws on Banach spaces and applications to prediction

This paper concerns a family of weak parallelogram laws for Banach spaces. It is shown that the familiar Lebesgue spaces satisfy a range of these inequalities. Connections are made to basic geometric ideas, such as smoothness, convexity, and Pythagorean-type theorems. The results are applied to the linear prediction of random processes spanning a Banach space. In particular, the weak parallelogram laws furnish coefficient growth estimates, Baxter-type inequalities, and criteria for regularity

Weak Parallelogram Laws on Banach Spaces and Applications to Prediction

This paper concerns a family of weak parallelogram laws for Banach spaces. It is shown that the familiar Lebesgue spaces satisfy a range of these inequalities. Connections are made to basic geometric ideas, such as smoothness, convexity, and Pythagorean-type theorems. The results are applied to the linear prediction of random processes spanning a Banach space. In particular, the weak parallelogram laws furnish coefficient growth estimates, Baxter-type inequalities, and criteria for regularity

Optimal Weak Parallelogram Constants for L-p Spaces

Inspired by Clarkson\u27s inequalities for L-p and continuing work from [5], this paper computes the optimal constant C in the weak parallelogram laws parallel to f + g parallel to(r )+ C parallel to f - g parallel to(r )= 2(r-1 )(parallel to f parallel to(r) + parallel to g parallel to(r)) for the L-p spaces, 1 \u3c p \u3c infinity

Optimal Weak Parallelogram Constants for LpL^p Spaces

Inspired by Clarkson's inequalities for $L^p$ and continuing work from \cite{CR}, this paper computes the optimal constant $C$ in the weak parallelogram laws $$\|f + g \|^r + C\|f - g\|^r \leq 2^{r-1}\big( \|f\|^r + \|g\|^r \big),$$ $$\|f + g \|^r + C\|f- g \|^r \geq 2^{r-1}\big( \|f\|^r + \|g \|^r \big)$$ for the $L^p$ spaces, $1 < p < \infty$.Comment: 10 page

Duality of the Weak Parallelogram Laws on Banach Spaces

This paper explores a family of weak parallelogram laws for Banach spaces. Some basic properties of such spaces are obtained. The main result is that a Banach space satisfies a lower weak parallelogram law if and only if its dual satisfies an upper weak parallelogram law, and vice versa. Connections are established between the weak parallelogram laws and the following: subspaces, quotient spaces, Cartesian products, and the Rademacher type and co-type properties

On the Geometry of the Multiplier Space of ℓ\u3csup\u3ep\u3c/sup\u3e\u3csub\u3eA\u3c/sub\u3e

For p ∊ (1, ∞)\ {2}, some properties of the space Mp of multipliers on ℓpA are derived. In particular, the failure of the weak parallelogram laws and the Pythagorean inequalities is demonstrated for Mp. It is also shown that extremal multipliers on the ℓpA spaces are exactly the monomials, in stark contrast to the p = 2 case

On the Geometry of the Multiplier Space of ℓ\u3csup\u3ep\u3c/sup\u3e\u3csub\u3eA\u3c/sub\u3e

For p ∊ (1, ∞)\ {2}, some properties of the space Mp of multipliers on ℓpA are derived. In particular, the failure of the weak parallelogram laws and the Pythagorean inequalities is demonstrated for Mp. It is also shown that extremal multipliers on the ℓpA spaces are exactly the monomials, in stark contrast to the p = 2 case

Birkhoff–James Orthogonality and the Zeros of an Analytic Function

Bounds are obtained for the zeros of an analytic function on a disk in terms of the Taylor coefficients of the function. These results are derived using the notion of Birkhoff–James orthogonality in the sequence space ℓp with p ∈ (1,∞), along with an associated Pythagorean theorem. It is shown that these methods are able to reproduce, and in some cases sharpen, some classical bounds for the roots of a polynomial

An inner-outer factorization in ℓp with applications to ARMA processes

The following inner-outer type factorization is obtained for the sequence space ℓp: if the complex sequence F = (F0, F1,F2,...) decays geometrically, then for an p sufficiently close to 2 there exists J and G in ℓp such that F = J * G; J is orthogonal in the Birkhoff-James sense to all of its forward shifts SJ, S2J, S3J, ...; J and F generate the same S-invariant subspace of ℓp; and G is a cyclic vector for S on ℓp. These ideas are used to show that and ARMA equation with characteristic roots inside and outside of the unit circle has Symmetric-α-Stable solution,s in which the process and the given white noise are expressed as causal moving averages of a related i.i.d. SαS white noise. An autregressive representation of the process is similarly obtained