In this paper, the authors introduce certain Herz type Hardy spaces with variable exponents and establish the characterizations of these spaces in terms of atomic and molecular decompositions. Using these decompositions, the authors obtain the boundedness of some singular integral operators on the Herz type Hardy spaces with variable exponents.

In this paper the author writes a simple characterization for the best copositive approximation to elements of $C(Q)$ by elements of finite dimensional strict Chebyshev subspaces of $C(Q)$ in the case when $Q$ is any compact subset of real numbers. At the end of the paper the author applies this result for different classes of $Q$.

In this paper, by using the atomic decomposition of the weighted weak Hardy space $WH_\omega^1(\mathbb{R}^n)$, the authors discuss a class of multilinear oscillatory singular integrals and obtain their boundedness from the weighted weak Hardy space $WH_\omega^1(\mathbb{R}^n)$ to the weighted weak Lebesgue space $WL_\omega^1(\mathbb{R}^n)$ for $\omega\in A_1(\mathbb{R}^n)$.

Let $Γ ⊂ \mathbb{R}^2$ be a regular anisotropic fractal. We discuss the problem of the negative spectrum for the Schrödinger operators associated with the formal expression $$H_β =id−∆+βtr^Γ_b, β∈R,$$ acting in the anisotropic Sobolev space $W^{1,α}_2(\mathbb{R}^2)$, where $∆$ is the Dirichlet Laplanian in $\mathbb{R}^2$ and $tr^Γ_b$ is a fractal potential (distribution) supported by $Γ$.

The self-affine measure $\mu_{M,D}$ associated with an iterated function system$\{\phi_{d} (x)=M^{-1}(x+d)\}_{d\in D}$ is uniquely determined. It only depends upon an expanding matrix $M$ and a finite digit set $D$. In the present paper we give some sufficient conditions for finite and infinite families of orthogonal exponentials. Such research is necessary to further understand the non-spectral and spectral of $\mu_{M,D}$. As an application, we show that the $L^2(\mu_{M, D})$ space has infinite families of orthogonal exponentials on the generalized three Sierpinski gasket. We then consider the spectra of a class of self-affine measures which extends several known conclusions in a simple manner.

In the present paper, we have considered the approximation of analytic functions represented by Laplace-Stieltjes transformations using sequence of definite integrals. We have characterized their order and type in terms of the rate of decrease of $ {E_n}( {F,\beta } )$ where $ {E_n}( {F,\beta } )$ is the error in approximating of the function $F(s)$  by definite integral polynomials in the half plane $ {{Re}} s \le \beta  < \alpha. $