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We obtain new convolutions for quadratic-phase Fourier integral operators (which include, as subcases, e.g., the fractional Fourier transform and the linear canonical transform). The structure of these convolutions is based on properties of the mentioned integral operators and takes profit of weight-functions associated with some amplitude and Gaussian functions. Therefore, the fundamental properties of that quadratic-phase Fourier integral operators are also studied (including a Riemann–Lebesgue type lemma, invertibility results, a Plancherel type theorem and a Parseval type identity). As applications, we obtain new Young type inequalities, the asymptotic behaviour of some oscillatory integrals, and the solvability of convolution integral equations.

Working in the frame of variable bounded variation spaces in the sense of Wiener, introduced by Castillo, Merentes, and Rafeiro, we prove convergence in variable variation by means of the classical convolution integral operators. In the proposed approach, a crucial step is the convergence of the variable modulus of smoothness for absolutely continuous functions. Several preliminary properties of the variable p ( · ) -variation are also presented.

We consider the -linear problem (also known as the Markushevich problem and the generalized Riemann boundary value problem) and the convolution integral equation of the second kind on a finite interval (also known as the truncated Wiener–Hopf equation). We find new conditions for correct solvability of the -linear problem and the truncated Wiener–Hopf equation.

We establish some algebraic, metric, and spectral properties of convolution integral operators in L2(ℝN).

ABSTRACT This article uses both operational and symbolic methods to establish and treat a hybrid form of the Mittag-Leffler function and the Konhauser polynomials. Some well-known polynomials are special instances of this class of hybrid form. To the best of our knowledge, the literature has not yet tackled the problem of treating a hybrid class of Mittag-Leffler functions with Kornhauser polynomials, which is what makes the results in this work novel. Identities such as operational rules, expansions, quasi-monomials, differential equations, generating functions, and integral representations for the Mittag-Leffler-Konhauser polynomials are obtained. We then solve a convolution integral equation whose kernel contains the Mittag-Leffler-Konhauser polynomials. Resolving this equation requires the addition of a new family of double Mittag-Leffler functions. We also develop formulas for fractional calculus for our newly defined functions and polynomials. Finally, we show the importance of the Mittag-Leffler function and the Mittag-Leffler-Konhauser polynomials in physical applications by developing and solving two generalized fractional kinetic equations.

In this article, we study the existence and uniqueness of solutions to some kinds of singular integral equations with Cauchy kernel and convolution kernel. In order to transform our equations into Riemann-Hilbert problems (R-H problems), we establish the relation between Fourier integral transform and Cauchy type integral, and we generalize Sokhotski–Plemelj formula. By means of the classical R-H problems and of regularity theory of Fredholm integral equations, we present the necessary and sufficient conditions of Noether solvability and the explicit solutions for such equations. Especially, we discuss the property and asymptotic behavior of solutions at nodes. This paper will be of great significance for the study of improving and developing complex analysis, integral equations, and R-H problems.

The commutation relation K L = L K between finite convolution integral operator K and differential operator L has implications for spectral properties of K . We characterize all operators K admitting this commutation relation. Our analysis places no symmetry constraints on the kernel of K extending the well-known results of Morrison for real self-adjoint finite convolution integral operators.

This review discusses the development of studies that evaluated the essentiality and requirements of iron from the ancient to the present. The therapeutic effects of iron compounds were recognized by the ancient Greeks and Romans. The earliest recognition of the essentiality of iron was stated by Paracelsus, a distinguished physician alchemist, in the sixteenth century. Iron was included in the earliest nutritional standard prepared for the Royal Army by E. A. Parkes, the first professor of hygiene. The League of Nations Health Organisation determined average iron requirements based on literature review. In the first US Recommended Dietary Allowances (RDA), the RDA of iron was determined from the results of iron balance studies. In the current Dietary Reference Intakes, iron requirements were determined based on the factorial method with the aid of Monte Carlo simulation for combining basal and menstrual iron losses. Population data analysis is a recently developed alternative that does not use the pre-estimated iron absorption rate and requires the prevalence of inadequacy instead. Population data analysis uses the convolution integral for combining basal and menstrual iron losses to ensure the required accuracy. This review also provides new estimates of hair and nail iron losses.

We introduce and fully analyze a new commutation relation K ¯ L 1 = L 2 K between finite convolution integral operator K and differential operators L 1 and L 2 , that has implications for spectral properties of K . This work complements our explicit characterization of commuting pairs K L = L K and provides an exhaustive list of kernels admitting commuting or sesquicommuting differential operators.

The asymptotic behavior at large times of the solution of the Cauchy problem for the Airy equation—a third-order evolutionary equation—is established. We assume that the initial function is locally Lebesgue integrable and has a power-law asymptotics at infinity. For the solution in the form of a convolution integral with the Airy function, we use the auxiliary parameter method and the regularization of singularities to obtain an asymptotic Erdélyi series in inverse powers of the cubic root of the time variable with coefficients depending on the self-similar variable and the logarithm of time.