Related papers: The Taylor Integral and a Generalization of the Di…

Using the recently defined concept of Taylor measures, we propose a generalization of Taylor's theorem to measurable, non-analytic functions, that do not require differentiation. We study consequences of the generalization, including the…

We propose and study a novel collection of signed measures, which will be apply called Taylor measures. Stochastic versions of the new measures are also defined and studied. We illustrate, through examples, how the deterministic and…

In the present article, the author uses Fourier theory of tempered distributions (generalized functions) in deriving a formula for Dirichlet-like integrals. The applied method is remarkably efficient and allows a solution in a few…

In this work, we introduce a new generalized integral transform involving many potentially known or new transforms as special cases. Basic properties of the new integral transform, that investigated in this work, include the existence…

The Fourier transform is naturally defined for integrable functrions. Otherwise, it should be stipulated in which sense the Fourier transform is understood. We consider some class of radial and, generally saying, nonintegrable functions.…

In this paper we obtain new sufficient conditions for representation of a function as an absolutely convergent Fourier integral. Unlike those known earlier, these conditions are given in terms of belonging to weighted spaces. Adding weights…

Fourier transforms are ubiquitous mathematical tools in basic and applied sciences. We here report classical and quantum optical realizations of the discrete fractional Fourier transform, a generalization of the Fourier transform. In the…

The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature.…

A new derivative, called deformable derivative, is introduced here which is equivalent to ordinary derivative in the sense that one implies other. The deformable derivative is defined using limit approach like that of ordinary one but with…

In this paper, we introduce and study the Fourier transform of functions which are integrable with respect to a vector measure on a compact group (not necessarily abelian). We also study the Fourier transform of vector measures. We also…

We establish a new generalized Taylor's formula for power fractional derivatives with nonsingular and nonlocal kernels, which includes many known Taylor's formulas in the literature. Moreover, as a consequence, we obtain a general version…

The Taylor expansion is a widely used and powerful tool in all branches of Mathematics, both pure and applied. In Probability and Mathematical Statistics, however, a stronger version of Taylor's classical theorem is often needed, but only…

We analyze the conditions on the Taylor coefficients of an analytic function to admit global analytic continuation, complementing a recent paper of Breuer and Simon on general conditions for natural boundaries to form. A new summation…

In this work we define operator-valued Fourier transforms for suitable integrable elements with respect to the Plancherel weight of a (not necessarily Abelian) locally compact group. Our main result is a generalized version of the Fourier…

The convergence of DP Fourier series which are neither strongly convergent nor strongly divergent is discussed in terms of the Taylor series of the corresponding inner analytic functions. These are the cases in which the maximum disk of…

We introduce a class of integral theorems based on cyclic functions and Riemann sums approximating integrals. The Fourier integral theorem, derived as a combination of a transform and inverse transform, arises as a special case. The…

We propose a novel definition of Fourier transform, with the property that the transform of a real function is again a real function (without doubling the number of real components). We prove the inversion theorem for the novel definition,…

Fractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration and complex numbers. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has…

In this paper we present a special formula for transforming integrals to series. The resulting series involves binomial transforms with the Taylor coefficients of the integrand. Five applications are provided for evaluating challenging…

The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier transform can be uniquely characterized…