Differential Operators of Infinite Order with Real Arguments and Their Applications

The following sections are included:

• CONTENTS

In his infinitesimal calculus, Gottfried Wilhelm Leibniz noted that there are certain striking analogies between algebraic laws and the behavior of differential and integral operators. One of these analogies he formulated in what is now known in mathematical literature as the rule of Leibniz, which states that the form of the nth differential of a product of two functions resemble that of an nth degree binomial.

The following sections are included:

• Spaces C^k(Ω) and L_p(Ω)

• Definitions and basic properties of distributions

• Multiplication of distributions by functions

• Differentiation of distributions

• Distributions with compact support

• Convolution of distributions

• Fourier transforms of distributions

• The space of analytic functionals

• Entire functions of exponential type that are bounded on ℝⁿ

• Difference quotients and modulus of continuity

• The sampling theorems

• The de la Vallée Poussin kernel
  • The case of periodic functions in (−π,π)ⁿ
  • The case of functions given in ℝⁿ

• The Dirichlet kernels

• Markov type theorems

• The second interpolation method of Bernstein

• Orlicz classes and Orlicz spaces

• Sobolev-Orlicz spaces

• The Denjoy-Carleman classes and the quasianalyticity of functions of several real variables

• Semifinitary functions

• Weakly nonlinear equations

The following sections are included:

• The space of test functions in a neighborhood of zero
  • The Dubinskii space
  • The Bernstein - Nikolskii class 𝔐_vp(ℝⁿ)
  • The classes of all polynomials

• Differential operators of infinite order (DOIO)

• The space of generalized functions
  • The definition of
  • Examples of generalized functions in
  • The Fourier transform in

• The algebra of pseudo-differential operators with analytic symbols

• Bibliographical Notes

The following sections are included:

• Problems in the whole Euclidean space

• The Cauchy problem in the space of functions valued in

• The Cauchy problem in the space and its fundamental solution

• The Cauchy problem for ordinary pseudo-differential equations

• Boundary-value problems

• Multi-dimensional integral equations of the first kind with entire kernels

• Functional equations
  • First we consider functional equations of the type
  • Functional-differential equations
  • Cauchy problems for functional-differential equations

• A class of real analytic symbols with weights

• Bibliographical Notes

The following sections are included:

• Approximating the symbols by algebraic polynomials
  • Approximation by the Taylor series
  • Approximations by other algebraic polynomials

• Approximating the symbols by trigonometric polynomials
  • Periodic symbols
  • Non-periodic symbols

• Trigonometric interpolation

• Approximating the data by sinc functions

• Examples
  • Helmholtz's problem for Ḡ = [−π, π]
  • The Cauchy problem for the heat equation
  • The Cauchy problem for the inverse heat equation
  • The Cauchy problem for the wave equation

• Bibliographical Notes

The following sections are included:

• Introduction

• Mollification method
  • Description of the method
  • A technical lemma
  • Examples

• Numerical differentiation
  • The case of a function defined on ℝ¹
  • The case when the function f is given in (−π,π)
  • The case when the Function is given only at discrete points

• The heat equation backwards in time
  • The L₂ case
  • The L_p (1 < p ≤ ∞) case

• The Cauchy problem for the Laplace equation

• The noncharacteristic Cauchy problem for parabolic equations
  • The one-dimensional noncharacteristic Cauchy problem for a parabolic equation
  • Multi-dimensional problems
  • The case of finite domains

• Numerical case study

• Bibliographical Notes

The following sections are included:

• Monotonic limits of Banach spaces

• Nontriviality of Sobolev-Orlicz spaces of infinite order in a bounded domain

• Nontriviality of Sobolev-Orlicz spaces of infinite order on the n-dimensional torus

• Nontriviality of Sobolev-Orlicz spaces of infinite order defined on a full Euclidean space

• Nontriviality of Sobolev-Orlicz spaces of infinite order in angular domains

• Bibliographical Notes

The following sections are included:

• Traces and extensions

• Traces in LW^∞ {Φ_n, (0, a)}

• Dual spaces

• Imbedding theorems
  • Imbedding criterion for monotonic limits of Banach spaces
  • Applications to the spaces LW^∞ {Φ_α, G}
  • Sufficient algebraic conditions

• Bibliographical Notes

The following sections are included:

• Elliptic boundary value problems

• A model equation

• Periodic problems. Inhomogeneous boundary problems
  • Periodic problems
  • Inhomogeneous boundary value problems

• Degenerate nonlinear elliptic equations of infinite order
  • Solvability of Problem (8.4.1), (8.4.3)
  • Solvability of Problem (8.4.1), (8.4.4)

• Bibliographical Notes

The following sections are included:

• Bibliography