Herz spaces and restricted summability of Fourier transforms and Fourier series

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy-Littlewood maximal function is verified. It is proved that if the Fourier transform of θ is in a Herz space, then the restricted maximal operator of the θ-means of a distribution is of weak type (1,1), provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that over a cone-like set a.e. for all f∈L₁(Rd). Moreover, converges to f(x) over a cone-like set at each Lebesgue point of f∈L₁(Rd) if and only if the Fourier transform of θ is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the θ-summation.     

Local Hardy spaces and summability of Fourier transforms

New Wiener amalgam spaces are introduced for local Hardy spaces. A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from the amalgam space to W(L_p,ℓ_∞). This implies the almost everywhere convergence of the θ-means for all fW(L₁,ℓ_∞)L₁.     

On a Calderón–Zygmund commutator-type estimate

In this paper we present a Calderón–Zygmund commutator-type estimate. This estimate enables us to prove an embedding result concerning weighted function spaces.     

Modified Ratio Objective Approach in Mathematical Programming

A new approach to obtaining the optimality conditions for fractional mathematical programming problems involving one objective ratio in the objective function is considered. Using this approach, an equivalent optimization problem is constructed by a modification of the single-ratio objective function in the fractional programming problem. Furthermore, an η-Lagrange function is introduced for a constructed optimization problem and modified saddle-point results are presented.     

Onα-Symmetric Multivariate Characteristic Functions

Ann-dimensional random vector is said to have anα-symmetric distribution,α>0, if its characteristic function is of the form((|u₁|^α+…+|u_n|^α)^1/α). We study the classesΦ_n(α) of all admissible functions: [0, ∞)→ . It is known that members ofΦ_n(2) andΦ_n(1) are scale mixtures of certain primitivesΩ_nandω_n, respectively, and we show thatω_nis obtained fromΩ_2n−1byn−1 successive integrations. Consequently, curious relations between 1- and 2- (or spherically) symmetric distributions arise. An analogue of Askey's criterion gives a partial solution to a question of D. St. P. Richards: If(0)=1,is continuous, lim_t→∞ (t)=0, and⁽²ⁿ⁻²⁾(t) is convex, thenΦ_n(1). The paper closes with various criteria for the unimodality of anα-symmetric distribution.     

Applications of P-adic generalized functions and approximations by a system of p-adic translations of a function

Under some conditions we prove that the convergence of a sequence of functions in the space of P-adic generalized functions is equivalent to its convergence in the space of locally integrable functions. Some analogs are established of the Wiener tauberian theorem and the Wiener theorem on denseness of translations for P-adic convolutions and translations.     

Application of continuous-time programming problems to a class of variational-type inequalities

In this paper, we establish sufficiency criteria under generalized ρ−(η,θ)-invexity conditions for general continuous-time programming problems with nonlinear equality/inequality constraints. Using this we establish some existence criteria for solutions of a class of variational-type inequalities.