The Littlewood-Paley Theorem for ℝ, $$\mathbb{T}$$ and ℤ: Dyadic Intervals

1977 ◽  
pp. 134-147
Author(s):  
R. E. Edwards ◽  
G. I. Gaudry
Keyword(s):  
Dyadic Intervals

Duality for dyadic triangles

2019 ◽  
Vol 29 (01) ◽  
pp. 61-83 ◽  
Author(s):  
K. Matczak ◽  
A. Mućka ◽  
A. B. Romanowska
Keyword(s):  
Binary Operation ◽  
Arithmetic Mean ◽  
Unit Cube ◽  
Unit Interval ◽  
Real Plane ◽  
Finitely Generated ◽  
One Dimensional ◽  
Additional Constant ◽  
The One ◽  
Dyadic Intervals

This paper is a direct continuation of the paper “Duality for dyadic intervals” by the same authors, and can be considered as its second part. Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are, respectively, defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent groupoid under the binary operation of arithmetic mean. The first paper dealt with the structure of finitely generated subgroupoids of the dyadic line, which were shown to be isomorphic to dyadic intervals. Then a duality between the class of dyadic intervals and the class of certain subgroupoids of the dyadic unit square was described. The present paper extends the results of the first paper, provides some characterizations of dyadic triangles, and describes a duality for the class of dyadic triangles. As in the case of intervals, the duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the dyadic unit cube, considered as (commutative, idempotent and entropic) groupoids with additional constant operations.

scholarly journals Scheduling dyadic intervals

1995 ◽  
Vol 63 (2) ◽  
pp. 101-116 ◽  
Author(s):  
James R. Driscoll ◽  
Dennis M. Healy ◽  
Garth T. Isaak
Keyword(s):  
Dyadic Intervals

Duality for dyadic intervals

2019 ◽  
Vol 29 (01) ◽  
pp. 41-60 ◽  
Author(s):  
K. Matczak ◽  
A. Mućka ◽  
A. B. Romanowska
Keyword(s):  
Convex Sets ◽  
Convex Polytopes ◽  
Arithmetic Mean ◽  
Unit Interval ◽  
Finitely Generated ◽  
One Dimensional ◽  
Additional Constant ◽  
Dyadic Space ◽  
The One ◽  
Dyadic Intervals

In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of (real) polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. This paper is a first step in finding a duality for dyadic polytopes, analogues of real convex polytopes, but defined over the ring [Formula: see text] of dyadic rational numbers instead of the ring of reals. A dyadic [Formula: see text]-dimensional polytope is the intersection with the dyadic space [Formula: see text] of an [Formula: see text]-dimensional real polytope whose vertices lie in the dyadic space. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polytopes carry the structure of a commutative, entropic and idempotent groupoid under the operation of arithmetic mean. Such dyadic polytopes do not preserve all properties of real polytopes. In particular, there are infinitely many (pairwise non-isomorphic) dyadic intervals. We first show that finitely generated subgroupoids of the groupoid [Formula: see text] are all isomorphic to dyadic intervals. Then, we describe a duality for the class of dyadic intervals. The duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the square of the dyadic unit interval with additional constant operations. A second paper deals with a duality for dyadic triangles.

scholarly journals Two remarks on primary spaces

2012 ◽  
Vol 153 (3) ◽  
pp. 505-523 ◽  
Author(s):  
PAUL F. X. MÜLLER
Keyword(s):  
Dyadic Intervals

AbstractWe prove that for an operator T on ℓ∞(H1 ()), respectively ℓ∞(L1 ()), the identity factors through T or Id - T. Hence ℓ∞(H1 ()) and ℓ∞(L1 ()) are primary spaces. We re-prove analogous results of H.M. Wark for the spaces ℓ∞(Hp()), 1 < p < ∞. In the present paper direct combinatorics of colored dyadic intervals replaces the dependence on Szemerédi's theorem in [11].

DYADIC POLYGONS

2011 ◽  
Vol 21 (03) ◽  
pp. 387-408 ◽  
Author(s):  
K. MATCZAK ◽  
A. B. ROMANOWSKA ◽  
J. D. H. SMITH
Keyword(s):  
Binary Operation ◽  
Arithmetic Mean ◽  
Real Plane ◽  
Finitely Generated ◽  
Idempotent Algebra ◽  
One Dimensional ◽  
The Real ◽  
Pythagoras Theorem ◽  
The One ◽  
Dyadic Intervals

Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are, respectively, defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The one-dimensional analogs are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent algebra under the binary operation of arithmetic mean. In this paper, dyadic intervals and triangles are classified to within affine or algebraic isomorphism, and dyadic polygons are shown to be finitely generated as algebras. The auxiliary results include a form of Pythagoras' theorem for dyadic affine geometry.

scholarly journals A new structure for scaling functions system with Dyadic intervals

2017 ◽  
Author(s):  
Raghad S. Shamsah ◽  
Anvarjon A. Ahmedov ◽  
Hishamuddin Zainuddin ◽  
Adem Kilicman ◽  
Fudziah Ismail
Keyword(s):  
Scaling Functions ◽  
Dyadic Intervals

scholarly journals Combinatorics of Dyadic Intervals: Consistent Colourings

2014 ◽  
Vol 62 (2) ◽  
pp. 101-115
Author(s):  
Anna Kamont ◽  
Paul F. X. Müller
Keyword(s):  
Dyadic Intervals
Sign in / Sign up

Export Citation Format

Share Document