Convolutional Calculus

Convolutional Calculus

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Description

Presents a development of a method based on the notion of the convolution of a linear operator. This unifies approaches from operational calculus, multiplier theory, algebraic analysis and spectral theory. The most important application of the convolutional method is the extension of the Duhamel met
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Product details

  • Hardback | 208 pages
  • 165.1 x 247.65 x 19.05mm | 505g
  • Dordrecht, Netherlands
  • English
  • Revised
  • 2nd Revised edition
  • 208 p.
  • 0792306236
  • 9780792306238

Table of contents

1. Convolutions of Linear Operators. Multipliers and Multiplier Quotients.- 1.1. The Duhamel Convolution.- 1.1.1. Algebraic and functional properties of the Duhamel convolution.- 1.1.2. Multipliers of the Duhamel convolution in C (?).- 1.1.3. Duhamel representations of the commutants of the Volterra integration operator.- 1.1.4. Representations of all possible continuous convolutions of the Volterra integration operator.- 1.2. The Mikusi?ski Ring.- 1.2.1. Convolution quotients.- 1.2.2. Interpretation as multiplier quotients.- 1.2.3. The Mikusi?ski field.- 1.3. Convolutions of Linear Endomorphisms.- 1.3.1. Definition of a convolution of linear endomorphism.- 1.3.2. Divisors of zero of a convolution of a linear space endomorphism.- 1.3.3. Convolutions of similar operators.- 1.3.4. Convolutions of right inverse operators.- 1.3.5. Continuous convolutions of a Frechet space endomorphism with a cyclic element.- 1.4. The Multiplier Quotients Ring of an Annihilators-free Convolutional Algebra.- 1.4.1. Definition of the multiplier quotient ring for an annihilators-free convolutional algebra.- 1.4.2. The ring of the convolution quotients of a convolution with non-divisors of zero.- 1.4.3. Isomorphism of the multiplier quotients rings of similar operators.- 1.4.4. Convolutional approach to Taylor boundary value problems for abstract differential equations.- 1.4.5. Solvability of a Taylor boundary value problem in a resonance case.- 2. Convolutions of General Integration Operators. Applications.- 2.1. Convolutions of the Linear Right Inverses of the Differentiation Operator.- 2.1.1. A class of convolutions, depending on an arbitrary linear functional in spaces of continuous functions.- 2.1.2. Linear right inverses of the differentiation and their convolutions.- 2.1.3. Convolutional representations of the commutants of linear integration operators.- 2.1.4. The commutant of the differentiation operator in an invariant hyperplane.- 2.2. An Application of the Convolutional Approach to Dirichlet Expansions of Locally Holomorphic Functions.- 2.2.1. Delsarte-Leontiev formulas for the coefficients of Dirichlet expansions.- 2.2.2. Convolutional representation of the multipliers of the formal Leontiev expansion.- 2.2.3. Leontiev's expansions in the case of multiple zeros of the indicatrix.- 2.3. A Convolution for the General Right Inverse of the Backward Shift Operator in Spaces of Locally Holomorphic Functions.- 2.3.1. The linear right inverses of the backward shift operator in a space of locally holomorphic functions.- 2.3.2. A class of convolutions in ?($$\bar D$$)connected with the backward shift operator.- 2.3.3. The commutant of the backward shift operator in an invariant hyperplane.- 2.3.4. Convolutions of the right inverse operators of the backward shift operator in ?($$\bar D$$).- 2.3.5. Multiplier projectors on spectral subspaces of a right inverse of the backward shift operator.- 2.4. Convolutions and Commutants of the Gelfond-Leontiev Integration Operator and of Its Integer Powers.- 2.4.1. An integral representation of the Gelfond-Leontiev integration operator in a space of analytic functions in a domain star-like with respect to the origin.- 2.4.2. A convolution of the Gelfond-Leontiev integration operator in ? (?).- 2.4.3. The commutant of the Gelfond-Leontiev integration operator in ? (?).- 2.4.4. A convolutional representation of the commutant of a fixed integer power of the Gelfond-Leontiev integration operator.- 2.5. Operational Calculi for the Bernoulli Integration Operator.- 2.5.1. The ring of the convolutional quotients of the Bernoulli convolution algebra.- 2.5.2 Rings of multiplier quotients of subalgebras of the Bernoulli convolution algebra.- 3. Convolutions Connected with Second-Order Linear Differential Operators.- 3.1. Convolutions of Right Inverse Operators of the Square of the Differentiation.- 3.1.1. A convolution connected with the square of the differentiation and depending on an arbitrary linear functional.- 3.1.2. Convolutions of the first kind right inverses of d2/dt2.- 3.1.3. Convolutions of the second kind right inverses of d2/dt2.- 3.1.4. Convolutions of the third kind right inverses of d2/dt2.- 3.1.5. Operational calculi for right inverses of the square of differentiation.- 3.2. Convolutions of Initial Value Right Inverses of Linear Second-Order Differential Operators.- 3.2.1. Convolutions of the initial value right inverse of non-singular second-order linear differential operator.- 3.2.2. Convolutions of the initial value right inverses of the Bessel differential operators.- 3.3. Convolutions of Boundary Value Right Inverses of Linear Second-Order Differential Operators.- 3.3.1. Convolutions of right inverses of non-singular second-order linear differential operator, determined by a Sturm-Liouville, and a general boundary value conditions.- 3.3.2. Convolutions of the finite Sturm-Liouville integral transformations.- 3.3.3. Convolutions of the finite Bessel integral transformations.- 3.4. Applications of Convolutions to Non-Local Boundary Value Problems.- 3.4.1. Eigenexpansions for non-local spectral problems for the square of differentiation.- 3.4.2. Duhamel-type representations of solutions of non-local boundary value problems for partial differential equations of mathematical physics.- References.- Authors index.
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