Extensions of Haar Measure for Compact Connected Abelian Groups

Abstract Questions of approximative nature are considered for a space of functions 𝐿𝑝(𝐺, μ), 1 ≤ 𝑝 ≤ ∞, defined on a locally compact abelian Hausdorff group 𝐺 with Haar measure μ. The approximating subspaces which are analogs of the space of exponential type entire functions are introduced.

Download Full-text

Harmonic conjugation in L1 on compact abelian groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100075204 ◽

1992 ◽

Vol 111 (1) ◽

pp. 113-126

Author(s):

Nakhl Asmar ◽

Saleem Watson ◽

Saleem Watson

Keyword(s):

Abelian Group ◽

Haar Measure ◽

Abelian Groups ◽

Duality Theorems ◽

Character Group ◽

Compact Abelian Groups ◽

Image Position ◽

Torsion Free ◽

Harmonic Conjugation

Let G denote a compact connected abelian group with character group and normalized Haar measure . As a consequence of the duality theorems (11, theorem 2518), is torsion-free and hence can be ordered. That is, there is a sub-semigroup P of such that

Download Full-text

Power-Rich and Power-Deficient LCA Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1981-052-6 ◽

1981 ◽

Vol 33 (3) ◽

pp. 664-670 ◽

Cited By ~ 3

Author(s):

M. A. Khan

Keyword(s):

Haar Measure ◽

Algebraic Structure ◽

Abelian Groups ◽

Open Neighbourhood ◽

Locally Compact ◽

Locally Compact Abelian Groups ◽

Compact Abelian Groups ◽

Lca Group ◽

Lca Groups ◽

Dual Power

In [4], Edwin Hewitt denned a-rich LCA (i.e., locally compact abelian) groups and classified them by their algebraic structure. In this paper, we study LCA groups with some properties related to a-richness. We define an LCA group G to be power-rich if for every open neighbourhood V of the identity in G and for every integer n > 1, λ(nV) > 0, where nV = {nx ∈ G : x ∈ V} and λ is a Haar measure on G. G is power-meagre if for every integer n > 1, there is an open neighbourhood V of the identity, possibly depending on n, such that λ(nV) = 0. G is power-deficient if for every integer n > 1 and for every open neighbourhood V of the identity such that is compact, . G is dual power-rich if both G and Ĝ are power-rich. We define dual power-meagre and dual power-deficient groups similarly.

Download Full-text

Extensions of Haar measure for compact connected Abelian groups

Bulletin of the American Mathematical Society ◽

10.1090/s0002-9904-1965-11267-8 ◽

1965 ◽

Vol 71 (1) ◽

pp. 152-157 ◽

Cited By ~ 4

Author(s):

Gerald L. Itzkowitz

Keyword(s):

Haar Measure ◽

Abelian Groups

Download Full-text

Beta dynamical systems over the formal fields

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.51.2014.4.1291 ◽

2014 ◽

Vol 51 (4) ◽

pp. 454-465

Author(s):

Lu-Ming Shen ◽

Huiping Jing

Keyword(s):

Dynamical Systems ◽

Finite Field ◽

Haar Measure ◽

Laurent Series ◽

Formal Laurent Series ◽

Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

Download Full-text