By V. P. Havin, N. K. Nikol’skij (auth.), V. P. Havin, N. K. Nikol’skij (eds.)

ISBN-10: 3642578543

ISBN-13: 9783642578540

ISBN-10: 3642633803

ISBN-13: 9783642633805

This EMS quantity indicates the nice energy supplied via sleek harmonic research, not just in arithmetic, but in addition in mathematical physics and engineering. geared toward a reader who has discovered the rules of harmonic research, this e-book is meant to supply numerous views in this very important classical topic. The authors have written a good e-book which distinguishes itself through the authors' first-class expository style.
it may be invaluable for the professional in a single zone of harmonic research who needs to acquire broader wisdom of alternative facets of the topic and likewise via graduate scholars in different components of arithmetic who want a normal yet rigorous advent to the subject.

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Extra resources for Commutative Harmonic Analysis III: Generalized Functions. Application

Example text

Q, Theorem 2. For every generalized function u defined in a convex domain the Fourier transform F' (u) can be written as an integral F'(u)(1jJ) == u(F(1jJ)) = J 1jJhdf. Q. Example 1. , the functional on Z(JRn) acting according to the formula 88(1jJ) = 1jJ(O). 2) one can proceed as follows: using the fact that the function 1jJ is harmonic, we represent 1jJ(O) as the average of 1jJ over a sphere of radius r and then average this representation with respect to r with some smooth finite density p( r) such that J= P 1.

In lR. n we consider the quadratic form q(x) = xI - x~ - ... - x;; let C be a convex cone in the half-space Xl ~ 0, where q ~ 0 (the future cone). M. Riesz (1949) studied in detail the family of distributions defined for Re>. > 0 by the formula q~('(x). = -1, -2, ... and>' = -2"' -2" - 1, ... (cf. also §2 of Chapt. -1 . )r ( >. 3) becomes an entire function of the parameter>' with values in S'(JRn), and supp Z>.

X) they follow from Theorem 2. 3. Restriction of Generalized Functions to Submanifolds. Suppose a closed submanifold Z C X is a fiber of some foliation F, and U is a generalized function on X that is smooth across F. In this case one can define the restriction of 11. to Z, which we shall denote ulZ E K,I(Z). To give a precise meaning to this statement we define an adequate convergence in the space K,IF (X). By definition Uk --+ 11. in K"F (X) if this sequence converges to U in K,I(X) and in addition for every chart (Xa , tPa) of the foliation F, any element T E K,(Xa/lRn), and any continuous functional hE &'(Rn) the numerical sequence h( tPa.

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Commutative Harmonic Analysis III: Generalized Functions. Application by V. P. Havin, N. K. Nikol’skij (auth.), V. P. Havin, N. K. Nikol’skij (eds.)

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