Convolutional Calculus

'Et moi, .... si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point alIe.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non· The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non­ linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com­ puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

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 Fréchet 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.