By N. M. Chuong, P. G. Ciarlet, P. Lax

ISBN-10: 9812705503

ISBN-13: 9789812705501

ISBN-10: 9812770496

ISBN-13: 9789812770493

This quantity collects articles in natural and utilized research, partial differential equations, geometric research and stochastic and infinite-dimensional research. specifically, the participants talk about imperative and pseudo-differential operators, which play a tremendous position in partial differential equations. different tools of fixing the partial differential equations are thought of, reminiscent of the min-max method of variational difficulties and boundary worth difficulties. the rules of quantum mechanics from the viewpoints of infinite-dimensional areas and Bell's inequality and contraction also are pointed out.

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D. Sarason, Trans. Amer. Math. SOC. 207,391 (1975). 15. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, (Princeton University Press, Princeton, New Jersey, 1970). 16. L. Softova, C. R. Acad. , Paris, Ser. I, Math. 333,635 (2001). Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 43-72) @ 2007 World Scientific Publishing Co. 43 54. edu In this survey paper we give a classification of integral transforms based on their composition structure.

K. Tuan with the Bessel function of the second kind Yv(z)and the Struve function H,,(x) in the kernels' are automorphisms, but not unitary in &(O, m). Taking a composition of Fourier transforms in the form Feix3/3Ff we obtain the following pair of convolution transforms of Watson type3'l4' L 00 g ( z ) = (Aif)(z) = f (Y) = (&)(Y) = / A+ +Y)f(Y)dY, (26) A+ + Y)g(z)dzl (27) 00 --oo with the Airy function Ai(z)in the kernel1 that are unitary on L2(-m, m). Taking a composition of Fourier transforms in the form Fe-aaSinh Ff I a > 0, we obtain another pair of convolution transforms of Watson type41 with the Macdonald function K,(a) in the kernels' that is unitary on L2(-m, m).

Because of the importance of applications and relations of integral transforms with differential equations it is widely believed that new interesting integral transforms can be constructed as eigenfunction expansions of some differential operators. ~~ We will show here another method to construct new integral transforms, based on few well-known integral transforms. Many nonconvolution (index) transforms, constructed by that way, seem impossible to be obtained by eigenfunction expansions, since their kernels do not satisfy any linear differential equations.

### Advances in Deterministic and Stochastic Analysis by N. M. Chuong, P. G. Ciarlet, P. Lax

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