arXiv subjects

Gord Sinnamon

Publications and source records attributed to Gord Sinnamon.

16 recordsLinked to original sources

arXiv2025

Best bounds for the dual Hardy operator minus identity on decreasing functions

The distance from the identity operator $I$ to $H^*$, the dual of the Hardy averaging operator, is studied on the cone of nonnegative, nonincreasing functions in Lebesgue space. The exact value is obtained. Optimal lower bounds are also given for difference, $H^*-I$, of these two operators acting on the same cone. A positive answer is given to a conjecture made in ``The norm of Hardy-type oscillation operators in the discrete and continuous settings'' by A. Ben Said, S. Boza, and J. Soria. Preprint, 2024. In addition, a direct comparison, with optimal constants, is given between the operators $H-I$ and $H^*-I$ acting on the cone.

math.FA↗

arXiv2024

Core Decreasing Functions

Given a measure space and a totally ordered ordered collection of measurable sets, called an ordered core, the notion of a core decreasing function is introduced and used to define the down space of a Banach function space. This is done using a variant of the Köthe dual restricted to core decreasing functions. To study down spaces, the least core decreasing majorant construction and the level function construction, already known for functions on the real line, are extended to this general setting. These are used to give concrete descriptions of the duals of the down spaces and, in the case of universally rearrangement invariant (u.r.i.) spaces, of the down spaces themselves. The down spaces of $L^1$ and $L^\infty$ are shown to form an exact Calderón couple with divisibility constant $1$; a complete description of the exact interpolation spaces for the couple is given in terms of level functions; and the down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property. The dual couple is also an exact Calderón couple with divisibility constant $1$; a complete description of the exact interpolation spaces for the couple is given in terms of least core decreasing majorants; and the duals of down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property.

math.FA↗

arXiv2023

A Christ-Kiselev maximal theorem in quasi-Banach function lattices

A Christ-Kiselev maximal theorem is proved for linear operators between quasi-Banach function lattices satisfying certain lattice geometrical conditions. The result is further explored for weighted Lorentz spaces, classical Lorentz spaces, and Wiener amalgams of Lebesgue function and sequence spaces. Extensions are made to Köthe dual operators and to operators on interpolation spaces of quasi-Banach function lattices. Several applications to maximal Fourier operators are presented.

math.FA↗

arXiv2023

End-point Norm Estimates for Cesàro and Copson Operators

For a large class of operators acting between weighted $\ell^\infty$ spaces, exact formulas are given for their norms and the norms of their restrictions to the cones of nonnegative sequences; nonnegative, nonincreasing sequences; and nonnegative, nondecreasing sequences. The weights involved are arbitrary nonnegative sequences and may differ in the domain and codomain spaces. The results are applied to the Cesàro and Copson operators, giving their norms and their distances to the identity operator on the whole space and on the cones. Simplifications of these formulas are derived in the case of these operators acting on power-weighted $\ell^\infty$. As an application, best constants are given for inequalities relating the weighted $\ell^\infty$ norms of the Cesàro and Copson operators both for general weights and for power weights.

math.FA↗

arXiv2023

The Fourier transform in weighted rearrangement invariant spaces

It is shown that if the Fourier transform is a bounded map on a rearrangement-invariant space of functions on $\mathbb R^n$, modified by a weight, then the weight is bounded above and below and the space is equivalent to $L^2$. Also, if it is bounded from $L^p$ to $L^q$, each modified by the same weight, then the weight is bounded above and below and $1\le p=q'\le 2$. Applications prove the non-boundedness on these spaces of an operator related to the Schrödinger equation.

math.FA↗

arXiv2023

Angular equivalence of normed spaces

Angular equivalence is introduced and shown to be an equivalence relation among the norms on a fixed real vector space. It is a finer notion than the usual (topological) notion of norm equivalence. Angularly equivalent norms share certain geometric properties: A norm that is angularly equivalent to a uniformly convex norm is itself uniformly convex. The same is true for strict convexity. Extreme points of the unit balls of angularly equivalent norms occur on the same rays, and if one unit ball is a polyhedron so is the other. Among norms arising from inner products, two norms are angularly equivalent if and only if they are topological equivalent. But, unlike topological equivalence, angular equivalence is able to distinguish between different norms on a finite-dimensional space. In particular, no two $\ell^p$ norms on $\mathbb R^n$ are angularly equivalent.

math.FA↗

arXiv2021

Hardy inequalities in normal form

A simple normal form for Hardy operators is introduced that unifies and simplifies the theory of weighted Hardy inequalities. A straightforward transition to normal form is given that applies to the various Hardy operators and their duals, whether defined on Lebesgue spaces of sequences, of functions on the half-line, or of functions on $\mathbb R^n$ or more general metric spaces. This is done by introducing an abstract formulation of Hardy operators, more general than any of these, and showing that the normal form transition applies to all operators formulated in this way. The transition to normal form is shown to preserve boundedness, compactness, and operator norm. To a large extent the transition can be carried out via well-behaved linear operators. Known results for boundedness and compactness of Hardy operators are given simple proofs and extended, via the transition, to this general setting. New estimates for the best constant in Hardy inequalities are established and a large class of Hardy inequalities is identified in which the best constants are is known precisely.

math.FA↗

arXiv2021

Norm of the discrete Cesàro operator minus identity

The norm of $C-I$ on $\ell^p$, where $C$ is the Cesàro operator, is shown to be $1/(p-1)$ when $1<p\le2$. This verifies a recent conjecture of G. J. O. Jameson. The norm of $C-I$ on $\ell^p$ is also determined when $2< p<\infty$. The two parts together answer a question raised by G. Bennett in 1996. Operator norms in the continuous case, Hardy's averaging operator minus identity, are already known. Norms in the discrete and continuous cases coincide.

math.FA↗

arXiv2020

Revisiting a sharpened version of Hadamard's determinant inequality

Hadamard's determinant inequality was refined and generalized by Zhang and Yang in [Acta Math. Appl. Sinica 20 (1997) 269-274]. Some special cases of the result were rediscovered recently by Rozanski, Witula and Hetmaniok in [Linear Algebra Appl. 532 (2017) 500-511]. We revisit the result in the case of positive semidefinite matrices, giving a new proof in terms of majorization and a complete description of the conditions for equality in the positive definite case. We also mention a block extension, which makes use of a result of Thompson in the 1960s.

math.FA↗

arXiv2017

Power majorization between the roots of two polynomials

It is shown that if two hyperbolic polynomials have a particular factorization into quadratics, then their roots satisfy a power majorization relation whenever key coefficients in their factorizations satisfy a corresponding majorization relation. In particular, a numerical observation by Klemeš is confirmed.

math.CA↗

arXiv2016

A New Algorithm for Approximating the Least Concave Majorant

The least concave majorant, $\hat F$, of a continuous function $F$ on a closed interval, $I$, is defined by \[ \hat F (x) = \inf \left\{ G(x): G \geq F, G \mbox{ concave}\right\},\; x \in I. \] We present here an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on $I$. Given any function $F \in \mathcal{C}^4(I)$, it can be well-approximated on $I$ by a clamped cubic spline $S$. We show that $\hat S$ is then a good approximation to $\hat F$. We give two examples, one to illustrate, the other to apply our algorithm.

math.CA↗

arXiv2016

Product operators on mixed norm spaces

Inequalities for product operators on mixed norm Lebesgue spaces and permuted mixed norm Lebesgue spaces are established. They depend only on inequalities for the factors and on the Lebesgue indices involved. Inequalities for the bivariate Laplace transform are given to illustrate the method. Also, an elementary proof is presented for an $n$-variable Young's inequality in mixed norm spaces.

math.FA↗

arXiv2016

Calderón-Mityagin couples of Banach spaces related to decreasing functions

A number of Calderón-Mityagin couples and relative Calderón-Mityagin pairs are identified among Banach function spaces defined in terms of the least decreasing majorant construction on the half line. The interpolation structure of such spaces is shown to closely parallel that of the rearrangement invariant spaces, and it is proved that a couple of these spaces is a Calderón-Mityagin couple if and only if the corresponding couple of rearrangement invariant spaces is a Calderón-Mityagin couple. Consequently, the class of all interpolation spaces for any couple of spaces of this type admits a complete description by the K-method if and only if the class of all interpolation spaces for the corresponding couple of rearrangement invariant spaces does. Analogous results are proved for spaces defined in terms of the level function construction. In the main, the conclusions for both types of spaces remain valid when Lebesgue measure on the half line is replaced by a general Borel measure on R. However, for certain measures the class of interpolation spaces of these new spaces may be degenerate, reducing the "if and only if" of the main results to a single implication.

math.FA↗

arXiv2015

Fourier series in weighted Lorentz spaces

The Fourier coefficient map is considered as an operator from a weighted Lorentz space on the circle to a weighted Lorentz sequence space. For a large range of Lorentz indices, necessary and sufficient conditions on the weights are given for the map to be bounded. In addition, new direct analogues are given for known weighted Lorentz space inequalities for the Fourier transform. Applications are given that involve Fourier coefficients of functions in LlogL and more general Lorentz-Zygmund spaces.

math.FA↗

arXiv2015

The D-topology for diffeological spaces

Diffeological spaces are generalizations of smooth manifolds which include singular spaces and function spaces. For each diffeological space, Iglesias-Zemmour introduced a natural topology called the $D$-topology. However, the $D$-topology has not yet been studied seriously in the existing literature. In this paper, we develop the basic theory of the $D$-topology for diffeological spaces. We explain that the topological spaces that arise as the $D$-topology of a diffeological space are exactly the $Δ$-generated spaces and give results and examples which help to determine when a space is $Δ$-generated. Our most substantial results show how the $D$-topology on the function space $C^{\infty}(M,N)$ between smooth manifolds compares to other well-known topologies.

math.DG↗

arXiv2012

The Generalized Wielandt Inequality in Inner Product Spaces

A new inequality between angles in inner product spaces is formulated and proved. It leads directly to a concise statement and proof of the generalized Wielandt inequality, including a simple description of all cases of equality. As a consequence, several recent results in matrix analysis and inner product spaces are improved.

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