Functional calculus extensions on dual spaces

A $\mathcal{C}^k$-Seeley-Extension-Theorem for Bastiani's Differential Calculus

We generalize a classical extension result by Seeley in the context of Bastiani's differential calculus to infinite dimensions. The construction follows Seeley's original approach, but is significantly more involved as not only $C^k$-maps (for $k\in \mathbb{N}\cup\{\infty\}$) on (subsets of) half spaces are extended, but also continuous extensions of their differentials to some given piece of boundary of the domains under consideration. A further feature of the generalization is that we construct families of extension operators (instead of only one single extension operator) that fulfill certa…

分数阶微积分中光滑 Stieltjes 情形下 Cartwright-McMullen 定理的推广

An extension of the Cartwright-McMullen theorem in fractional calculus for the smooth Stieltjes case

本文将分数阶微积分中的 Cartwright-McMullen 定理推广至光滑积分子情形，证明了 Stieltjes 积分算子向正实数阶扩张的唯一性。

Virtual Calculus - Part II

A simultaneous extension of real numbers set and the class of real functions is discussed.

First Integrals for Problems of Calculus of Variations on Locally Convex Spaces

The fundamental problem of calculus of variations is considered when solutions are differentiable curves on locally convex spaces. Such problems admit an extension of the Euler-Lagrange equations [Orlov 2002] for continuously normally differentiable Lagrangians. Here, we formulate a Legendre condition and an extension of the classical theorem of Emmy Noether, thus obtaining first integrals for problems of the calculus of variations on locally convex spaces.

Virtual Calculus - Part I

An alternative organization for Differential and Integral Calculus, based on an extension of real numbers that include infinitesimal and infinite quantities, is presented. Only Elementary Set Theory is used, without reference to methods or results from Mathematical Logic.

Bounded $H_\infty$-calculus for cone differential operators

We prove that parameter-elliptic extensions of cone differential operators have a bounded $H_\infty$-calculus. Applications concern the Laplacian and the porous medium equation on manifolds with warped conical singularities.

Operator $k$-tone functions and analytic functional calculus

Operator $k$-tone functions on an open interval of the real line, which are higher order extensions of operator monotone and convex functions, are characterized via certain inequalities for the real and imaginary parts of analytic functional calculus by those functions.

Bounded $\mathbf{H_\infty}$-Calculus for Differential Operators on Conic Manifolds with Boundary

We derive conditions that ensure the existence of a bounded $H_\infty$-calculus in weighted $L_p$-Sobolev spaces for closed extensions $\underline{A}_T$ of a differential operator $A$ on a conic manifold with boundary, subject to differential boundary conditions $T$. In general, these conditions ask for a particular pseudodifferential structure of the resolvent $(λ-\underline{A}_T)^{-1}$ in a sector $Λ\subset\mathbf{C}$. In case of the minimal extension they reduce to parameter-ellipticity of the boundary value problem $(A,T)$. Examples concern the Dirichlet and Neumann Laplacians.

Functional Calculus for Semi-Bounded Operators

We build on the work by Davies, extending the Helffer-Sjöstrand Functional Calculus domain for semi-bounded operators on Banach spaces given a priori controlled growth of the resolvents. We employ Seeley's Extension Theorem to extend smooth functions on the half line to the whole line and thus indirectly define functions of these operators.

Functional Itô calculus and stochastic integral representation of martingales

We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by Dupire, for functionals on the space of right-continuous functions with left limits. We show that this functional derivative admits a suitable extension to the space of square-integrable martingales. This extension defines a weak derivative which is shown to be the inverse of the Ito integral and which may be viewed as a nonanticip…

Operator calculus approach to solving analytic systems

Solving analytic systems using inversion can be implemented in a variety of ways. One method is to use Lagrange inversion and variations. Here we present a different approach, based on dual vector fields. For a function analytic in a neighborhood of the origin in the complex plane, we associate a vector field and its dual, an operator version of Fourier transform. The construction extends naturally to functions of several variables. We illustrate with various examples and present an efficient algorithms readily implemented symbolically in Maple while suitable as well for numerical computations…

Who Gave you the Cauchy-Weierstrass Tale? The Dual History of Rigorous Calculus

Cauchy's contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, the seeds of the theory of rates of growth of functions as developed by Paul du Bois-…

Polyanalytic Toeplitz operators: isomorphisms, symbolic calculus and approximation of Weyl operators

We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus and present an asymptotic expansion of complex Weyl operators in terms of polyanalytic Toeplitz operators.