In this paper Hardy type inequalities on n-dimensional balls are studied. For the first time we establish Hardy type inequalities involving the volume distance introduced by R.S. Yulmukhametov. We obtain sharp one dimensional and multidimensional inequalities with additional terms. The weight functions in the spatial inequalities depend on the volume distance. Also, we get Hardy inequality which involves the distance to the boundary. In the case n = 2 we improve known lower estimates of the best Brezis–Marcus constants.

In this paper, we present a theoretical framework that defines and describes the fundamental aspects of the L 2 ( R d ) theory of multidimensional short-time fractional Fourier transform. We begin by defining multidimensional short-time fractional Fourier transform, before going on to describe its properties and the inner product relation. Later, we consider its d-dimensional and 2d-dimensional inverse transforms. Finally, we arrive at the sufficient conditions for the boundedness of the multidimensional short-time fractional Fourier transform acting on Lebesgue spaces.

In this work, we derive a qualitative uncertainty principle, specifically a version of Hardy's theorem, for the continuous shearlet transform.

In this study, we define Dunkl–Appell polynomials of class A ( 2 ) . Then we construct a generalization of Szász operators using these polynomials and derive some approximation properties of this generalization. Lastly, some graphical results are given to show the rate of convergence of constructed operators to a given function f.

In this paper, we apply the Laplace transform to derive closed forms for several challenging integrals that seem nearly impossible to evaluate. By utilizing the solutions to the Pythagorean equation a 2 + b 2 = c 2 , these closed forms become even more intriguing. This approach allows us to provide new integral representations for the error function. Furthermore, by leveraging an identity we derived for the inverse Laplace transform and applying a result based on Srivastava and Yürekli's identity, we provide a closed form for a nontrivial generalized integral.

Using the general solution of Bessel's differential equation, we have designed the operator B γ , c k , then using the concept of subordination a subclass of multivalent functions G p , c ( k , η , A , B ) is introduced. Inclusion properties of the above subclass based on the parameters A , B and k are derived, inequalities involving the operator which specify properties of the operator, subordination properties involving different parameter were studied.

We construct a novel class of integral operators whose eigenvalue reciprocals correspond to the roots (zeros) of two-parametric Mittag–Leffler functions E α , α ∗ ( z ) . Using the theory of μ 0 -positive operators, we demonstrate that the smallest roots (in modulus) of E α , α ∗ ( z ) must be real. Moreover, by employing the relationship between roots of Mittag–Leffler functions E α , α ∗ ( z ) and the eigenvalues of the integral spectral problem, together with new techniques involving fractional integral operators and fractional Sobolev spaces, we establish the existence, non-existence, and simplicity of the real root of E α , α ∗ ( − z ) for a large class of parameters α , α ∗ : 1 ≤ α < ∞ , 0 < α ∗ < α + ϵ . These findings extend the theory previously developed for the case 0 < α < 1 by other researchers, such as A. M. Sedletskiĭ in 2004.

The formal term-by-term differentiation with respect to parameters is demonstrated to be legitimate for the Mittag–Leffler-type functions. The justification of differentiation formulas is made by using the concept of the uniform convergence. This approach is applied to the Mittag–Leffler function depending on two parameters and, additionally, for the three-parametric Mittag–Leffler functions (namely, for the Prabhakar function and the Le Roy-type function), as well as for the four-parametric Mittag–Leffler function (and, in particular, for the Wright function). The differentiation with respect to the involved parameters is discussed also in case those special functions which are represented via the Mellin–Barnes integrals.

The main objective of this paper is to develop a new harmonic analysis related to a Fourier–Jacobi type operator Δ α , β m of the real line. We define and study the linear canonical Fourier–Jacobi transform F α , β m . We study some important properties, inversion formula, Plancherel theorem, Paley–Wiener theorem and Riemann–Lebsgue lemma. An application in solving a generalized heat equation is given.