Operator triplets in applied mathematics. Integral transforms and fractional integrals (Q2847905)

The book is devoted to operator triplets. An operator triplet \({\mathcal K},{\mathcal L},{\mathcal M}\) due to the author's definition is the triple of operators \({\mathcal K}\), \({\mathcal L}\), \({\mathcal M}\) together with their inverses \({\mathcal K}^{-1},{\mathcal L}^{-1},{\mathcal M}^{-1}\) for which the following relations hold: NEWLINE\[NEWLINE{\mathcal M} = {\mathcal K}{\mathcal L}, \qquad {\mathcal M}^{-1} = {\mathcal K}^{-1}{\mathcal L}^{-1},NEWLINE\]NEWLINE NEWLINE\[NEWLINE{\mathcal K} = {\mathcal M}{\mathcal L}^{-1}, \qquad {\mathcal K}^{-1} = {\mathcal M}^{-1}{\mathcal L},\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE{\mathcal L} = {\mathcal M}{\mathcal K}^{-1}, \qquad {\mathcal L}^{-1} = {\mathcal M}^{-1}{\mathcal K}.NEWLINE\]NEWLINE All operators are considered as integral ones. The fact that the operators \({\mathcal K}\) and \({\mathcal K}^{-1}\) with kernels \(K(x,\lambda)\) and \(K^{-1}(x,\lambda)\) are inverse to each other is described in the following way: NEWLINE\[NEWLINE\int\limits_\Gamma K(\hat{x},\lambda)K^{-1}(x,\lambda) \, d\lambda = \delta(x - \hat{x}), \qquad \int\limits_\omega K(x,\lambda)K^{-1}(x,\hat{\lambda}) \, d\lambda = \delta(\lambda - \hat{\lambda});\tag{2}NEWLINE\]NEWLINE these relations are called by the author orthogonality ones. The formulas for the kernels of operators in (1) are understood in the same manner. As the author writes, ``The dominant purpose of this work is to find basic regularities of creation of different combinations of orthogonal structures, to construct a sequential theory of operator triplets. These and similar structures form a power tool for the investigation of problems of applied mathematics and signal processing.'' The book contains the following chapters: 0. Introduction; 1. Integral transforms; 2. Fractional integrals; 3. Operator triplets; 4. Triplets of integral transforms; 5. Fourier transforms and exponential kernels; 6. Fourier transforms with trigonometrical kernels; 7. Havelock transforms; 8. Mellin transform; 9. Hankel transform; 10. Titchmarsh transform; 11. Weber transform; 12. Fourier-Bessel transform; 13. Lebedev transforms; 14. Mehler/Fock and Lebedev/Skalskaya-2 transforms; 15. Legendre transform; 16. Laggere transform; 17. Jacobi transform; 18. Integral equations; 19. Dual integral equations; two applications: A. Generalized functions; B. Abel integral equation; and Bibliography (35 items).NEWLINENEWLINE Really, the book is a list of operator triplets without any explanations. Since the formulas (1)--(2) and similar to them are senseless without the description of domains for the corresponding operators and the sense of integrals in these operators and the orthogonality relations, the book will hardly be useful for mathematicians and specialists in close fields.