Mathematics

We derive necessary and sufficient conditions for a continuous bounded function f:R→C to be a characteristic function of a probability measure. The Cauchy transform K f of f is used as analytic continuation of f to the upper and lower half-planes in C . The conditions depend on the behavior of K f (z) and its derivatives on the imaginary axis in C . The main results are given in terms of completely monotonic and absolutely monotonic functions.

We consider bilinear restriction estimates for wave-Schrödinger interactions and provided a sharp condition to ensure that the product belongs to L q t L r x in the full bilinear range 2 q + d+1 r <d+1 , 1⩽q,r⩽2 . Moreover, we give a counter-example which shows that the bilinear restriction estimate can fail, even in the transverse setting. This failure is closely related to the lack of curvature of the cone. Finally we mention extensions of these estimates to adapted function spaces. In particular we give a general transference type principle for U 2 type spaces that roughly implies that if an estimate holds for homogeneous solutions, then it also holds in U 2 . This transference argument can be used to obtain bilinear and multilinear estimates in U 2 from the corresponding bounds for homogeneous solutions.

In this paper we study the boundedness and compactness characterizations of the commutator of Calderón-Zygmund operators T on spaces of homogeneous type (X,d,μ) in the sense of Coifman and Weiss. More precisely, We show that the commutator [b,T] is bounded on weighted Morrey space L p,κ ω (X) ( κ∈(0,1),ω∈ A p (X),1<p<∞ ) if and only if b is in the BMO space. Moreover, the commutator [b,T] is compact on weighted Morrey space L p,κ ω (X) ( κ∈(0,1),ω∈ A p (X),1<p<∞ ) if and only if b is in the VMO space.

In this paper, we generalize fractional q -integrals by the method of q -difference equation. In addition, we deduce fractional Askey--Wilson integral, reversal type fractional Askey--Wilson integral and Ramanujan type fractional Askey--Wilson integral.

We propose a necessary and sufficient condition for a real-valued function on the real line to be a characteristic function of a probability measures. The statement is given in terms of harmonic functions and completely monotonic functions.

We derive new reduction formulas for the incomplete beta function and the Lerch transcendent in terms of elementary functions. As an application, we calculate some new integrals. Also, we use these reduction formulas to test the performance of the algorithms devoted to the numerical evaluation of the incomplete beta function.

We determine the asymptotic behaviour of certain incomplete Betafunctions.

Let f:R→C be the characteristic function of a probability measure. We study the following question: Is it true that for any closed interval I on R , which does not contain the origin, there exists a characteristic function g such that g coincides with f on I but g≡/f on R ?

In order to define the derivative f ′ ( x 0 ) of a function f:R→R we need to "know something" about f in the environment of x 0 . However, when we apply numerical routines for solving initial value problems, we do it vice versa: We use f ′ ( x 0 ) in order to "tell something" about f in the environment of x 0 . Although, it is a one-way street: If f ′ ( x 0 ) and f( x 0 ) are given, we can not determine f(x) at any point x∈R different from x 0 . That is one conceptual problem of solving differential equations in numerical mathematics. In this article, we will present a numerical algorithm to solve very simple initial value problems. However, the change of paradigm is, that we will not "leave" the point x 0 . Solving ordinary differential equations is like searching for "recipes" f . Instead of trying to find these recipes for values x∈R , we will learn them from special numbers in the "monad" of x 0 .

In the paper we propose a direct method for recovering the Sturm-Liouville potential from the Weyl-Titchmarsh m -function given on a countable set of points. We show that using the Fourier-Legendre series expansion of the transmutation operator integral kernel the problem reduces to an infinite linear system of equations, which is uniquely solvable if so is the original problem. The solution of this linear system allows one to reconstruct the characteristic determinant and hence to obtain the eigenvalues as its zeros and to compute the corresponding norming constants. As a result, the original inverse problem is transformed to an inverse problem with a given spectral density function, for which the direct method of solution from arXiv:2010.15275 is applied. The proposed method leads to an efficient numerical algorithm for solving a variety of inverse problems. In particular, the problems in which two spectra or some parts of three or more spectra are given, the problems in which the eigenvalues depend on a variable boundary parameter (including spectral parameter dependent boundary conditions), problems with a partially known potential and partial inverse problems on quantum graphs.