Some Double Generalized Weighted Fractional Integral Inequalities Associated with Monotone Chebyshev Functionals

In this manuscript, we study the unified integrals recently defined by Rahman et al. and present some new double generalized weighted type fractional integral inequalities associated with increasing, positive, monotone and measurable function F. Also, we establish some new double-weighted inequalities, which are particular cases of the main result and are represented by corollaries. These inequalities are further refinement of all other inequalities associated with increasing, positive, monotone and measurable function existing in literature. The existing inequalities associated with increasing, positive, monotone and measurable function are also restored by applying specific conditions as given in Remarks. Many other types of fractional integral inequalities can be obtained by applying certain conditions on F and Ψ given in the literature.

On the divergence of Fourier series in the general Haar system

For any countable set $D \subset [0,1]$, we construct a bounded measurable function $f$ such that the Fourier series of $f$ with respect to the regular general Haar system is divergent on $D$ and convergent on $[0,1]\backslash D$.

A systematic approach to reduced GLT

This paper concerns the spectral analysis of matrix-sequences that are generated by the discretization and numerical approximation of partial differential equations, in case the domain is a generic Peano–Jordan measurable set. It is observed that such matrix-sequences often present a spectral symbol, that is a measurable function describing the asymptotic behaviour of the eigenvalues. When the domain is a hypercube, the analysis can be conducted using the theory of generalized locally Toeplitz (GLT) sequences, but in case of generic domains, a different kind of matrix-sequences and theory has to be formalized. We thus develop in full detail the theory of reduced GLT sequences and symbols, presenting some application to finite differences and finite elements discretization for linear convection–diffusion–reaction differential equations.

On Krylov’s estimates for optional semimartingales

The estimates of N. V. Krylov for distributions of stochastic integrals by means of the L d {L_{d}} -norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W d 2 {W^{2}_{d}} . We also show how to use these estimates for the investigation of L 2 {L^{2}} -convergence of solutions of optional SDE’s.

Approximations in $$L^1$$ with convergent Fourier series

For a separable finite diffuse measure space $${\mathcal {M}}$$ M and an orthonormal basis $$\{\varphi _n\}$$ { φ n } of $$L^2({\mathcal {M}})$$ L 2 ( M ) consisting of bounded functions $$\varphi _n\in L^\infty ({\mathcal {M}})$$ φ n ∈ L ∞ ( M ) , we find a measurable subset $$E\subset {\mathcal {M}}$$ E ⊂ M of arbitrarily small complement $$|{\mathcal {M}}{\setminus } E|<\epsilon $$ | M \ E | < ϵ , such that every measurable function $$f\in L^1({\mathcal {M}})$$ f ∈ L 1 ( M ) has an approximant $$g\in L^1({\mathcal {M}})$$ g ∈ L 1 ( M ) with $$g=f$$ g = f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of $${\mathcal {M}}=G/H$$ M = G / H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

On stability of solutions of integral equations in the class of measurable functions

Consider the equation G(x)=(y,) ̃ where the mapping G acts from a metric space X into a space Y, on which a distance is defined, y ̃ ∈ Y. The metric in X and the distance in Y can take on the value ∞, the distance satisfies only one property of a metric: the distance between y,z ∈Y is zero if and only if y= z. For mappings X → Y the notions of sets of covering, Lipschitz property, and closedness are defined. In these terms, the assertion is obtained about the stability in the metric space X of solutions of the considered equation to changes of the mapping G and the element y ̃. This assertion is applied to the study of the integral equation f(t,∫_0^1▒K (t,s)x(s)ds,x(t))= y ̃(t),t ∈[0,1], with respect to an unknown Lebesgue measurable function x: [0,1] ∈R. Sufficient conditions are obtained for the stability of solutions (in the space of measurable functions with the topology of uniform convergence) to changes of the functions f,K,(y.) ̃

Regularity results of solutions to elliptic equations involving mixed local and nonlocal operators

<abstract><p>In this paper, we study the summability of solutions to the following semilinear elliptic equations involving mixed local and nonlocal operators</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{matrix} - \Delta u(x)+{{(-\Delta )}^{s}}u(x)=f(x), &amp; x\in \Omega , \\ u(x)\ge 0,~~~~~ &amp; x\in \Omega , \\ u(x)=0,~~~~~ &amp; x\in {{\mathbb{R}}^{N}}\setminus \Omega , \\ \end{matrix} \right. $\end{document} </tex-math></disp-formula></p> <p>where $ 0 &lt; s &lt; 1 $, $ \Omega\subset \mathbb{R}^N $ is a smooth bounded domain, $ (-\Delta)^s $ is the fractional Laplace operator, $ f $ is a measurable function.</p></abstract>

Lower Semi-frames, Frames, and Metric Operators

This paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.

Variable Exponent Spaces of Analytic Functions

Variable exponent spaces are a generalization of Lebesgue spaces in which the exponent is a measurable function. Most of the research done in this topic has been situated under the context of real functions. In this work, we present two examples of variable exponent spaces of analytic functions: variable exponent Hardy spaces and variable exponent Bergman spaces. We will introduce the spaces together with some basic properties and the main techniques used in the context. We will show that in both cases, the boundedness of the evaluation functionals plays a key role in the theory. We also present a section of possible directions of research in this topic.

A Variance Inequality for Meromorphic Functions Under Exterior Probability

The problem of measuring an unbounded system attribute near a singularity has been discussed. Lenses have been introduced as formal objects to study increasingly precise measurements around the singularity and a specific family of lenses called Exterior probabilities have been investigated. It has been shown that under such probabilities, measurement variance of a measurable function around a 1st order pole on a complex manifold, consists of two separable parts - one that decreases with diminishing scale of the lenses, and the other that increases. It has been discussed how this framework can lend mathematical support to ideas of non-deterministic uncertainty prevalent at a quantum scale. In fact, the aforementioned variance decomposition allows for a minimum possible variance for such a system irrespective of how close the measurements are. This inequality is structurally similar to Heisenberg uncertainty relationship if one considers energy/momentum to be a meromorphic function of a complex spacetime.