Let G be a locally compact group and w be a symmetric weight function on G. We define a co-product Gw on the weighted algebra L¥ (G, w-1) of essentially w-bounded Borel measurable functions on G and show that L¥ (G, w-1) becomes a Kac algebra with natural co-inverse kw and Haar weight  fw. We use the machinery of Kac algebras to introduce the weighted Fourier and Fourier-Stieltjes algebra A(G, w-1) and B(G, w-1) of G.

In this note we discuss the convergence of greedy approximants for trigonometric Fourier expansion in Lp(T), 1£p<2.

Background: In the past years, the speed of growth and economic development of countries has significantly increased the destruction of the environment, which has become a big problem for public health. In the development-environment-health triangle, economic development, environmental quality, and community health outputs can be positively and negatively related. On the other hand, this connection and the way if affects can be different in various short-term, medium-term, and long-term time horizons (frequencies). Based on this, the main purpose of this study was to empirically investigate the causal relationship between environmental sustainability, economic development, and health output in Iran in different time horizons. Methods: This was a descriptive-analytical and applied study. It investigated the causal relationship between the inverse of load capacity factor (index of environmental degradation), per capita income (index of economic growth and development), and life expectancy (index of health output) in different time horizons from 1961 to 2022 using the time series data of Iran. The data used were collected from World Development Indicators of the World Bank. Also, wavelet transformation was conducted to analyze the main time series data for short-term, medium-term, and long-term periods. Estimating the causal relationship between variables was conducted using the Fourier Toda-Yamamoto (F-TY) test, and data analysis was performed using MATLAB v24.1.0.2628055 and R v4.5.0 software. Results: In the short and medium term, there is a positive causal relationship between economic development and life expectancy, and in the long term, there is a two-way causal relationship between these two variables. The existence of a negative causal relationship from environmental destruction to life expectancy is also confirmed only in the long term; which shows that environmental degradation in the long term leads to a decrease in life expectancy through the aggravation of disease and the increase of health expenses. Moreover, in the short and medium term, there is a two-way positive causal relationship between economic development and environmental degradation, which is negative in the long term. Conclusions: Since in many cases the causal relationship between the investigated variables is not the same in the short, medium, and long term, it is necessary to design and implement applied policies in the fields of economic development, environmental sustainability, and health promotion in each time horizon separately.

In this paper we shall examine the quadratic Fourier transform which is introduced by the generalized quadratic function for one order parameter in the ordinary Fourier transform. This will be done by analyzing corresponding six subcases of the quadratic Fourier transform within a reproducing kernel Hilbert spaces framework.

In this paper, we introduce the decomposition property on locally compact groups and we give a multi-norm structure based on the Fourier and Fourier-Stieltjes algebras on locally compact groups with this property. We show that compact groups have the decomposition property. This construction generalizes the known multi-norm structure of L1-algebras on compact abelian groups. Also, we study some special multi-bounded maps on the Fourier algebras and improve some results in this theory.