Trimèche K. Transmutation Operators and Mean-Periodic Functions Associated with Differential Operators

Harwood Academic Publishers, 1988. — 282 p.When faced with a new problem, we always try to reduce it to a previously solved problem, or at least to a simpler problem. For example, to reduce a differential equation with singular coefficients to one with regular coefficients, to transform a second order differential equation into a first order one, or vice versa, to transform a Goursat problem into a Cauchy problem, or vice versa, to map certain classes of singular pseudo-differential operators onto the corresponding class of Kohn-Nirenberg pseudo-differential operators, or vice versa, and so on.Usually it is not hard to verify the properties of such a transformation, once it has been found. The problem is to
find the right transformation into a problem already studied. These transformations have been called by J. Delsarte and J .L. Lions "Transmutation operators".In this work we give the general definition of permutation and transmutation operators, the methods used for constructing these operators and we specify the relations het ween them and the classical integral transforms.Next, using the transmutation operators and the properties of the classical Harmonic Analysis, we study a Harmonic Analysis associated with differential operators (generalized translation operators, generalized convolution and Fourier transform, generalized Paley-Wiener and PIancherel type theorems).This Harmonic Analysis, the transmutation operators and the results of the classical mean-periodic functions enable us to define and study the generalized mean-periodic functions associated with differential operators.We finish this work by giving some applications of the theory of generalized mean-periodic functions.Editor's IntroductionPermutation and transmutation operators
Definition of the permutation and transmutation operators
Transmutation operators associated with semi-simple Lie groups
Transmutation operators given by the Radon transform
Delsarte's method for constructing permutation and transmutation operators
Fourier transform method for constructing permutation and transmutation operatorsGeneralized translation operators associated with differential operatorsGeneralized translation operators related to Lie groups
The method of generalized Taylor series in the sense of Delsarte
The method of spectral decomposition
The method of Cauchy problem
The method of generalized Fourier-Plancherel type theorem
Transmutation operators methodGeneralized convolution associated with differential operators
Generalized convolution of functions
Generalized convolution of functions and distributions associated with J .L. Lions's differential operator
Generalized convolution associated with differential operators on the complex domain
Generalized convolution in the multi-dimensional casePaley-wiener type theorems associated with differential operators
Classical Paley-Wiener type theorems
Paley-Wiener type theorems for lie groups
Paley-Wiener type theorems associated with differential operators on a bounded domain
Paley-Wiener type theorems associated with differential operators on an unbounded interval
Paley-Wiener type theorems associated with differential operators on the complex domain
Paley-Wiener type theorems associated with differential operators in several variablesGeneralized mean-periodic functions associated with differential operators
Spectral analysis and synthesis and mean-periodic functions
Spectral analysis and synthesis and mean-periodic functions for lie groups and symmetric spaces
Spectral analysis and synthesis; and generalized mean-periodic functions associated with differential operators
Series expansion of generalized mean-periodic functions associated with differential operatorsSome applications of the theory of mean-periodic functions
J. Delsarte's theorems
J. Delsarte's theorems for non-compact symmetric spaces of rank 1
Pompeiu's problem
Generalized Fourier series expansions
Generalized J. Delsarte' s theoremsList of symbols