By Valery V. Volchkov, Vitaly V. Volchkov
The booklet demonstrates the improvement of essential geometry on domain names of homogeneous areas given that 1990. It covers quite a lot of themes, together with research on multidimensional Euclidean domain names and Riemannian symmetric areas of arbitrary ranks in addition to contemporary paintings on section house and the Heisenberg team. The booklet comprises many major fresh effects, a few of them hitherto unpublished, between which might be mentioned forte theorems for numerous sessions of capabilities, far-reaching generalizations of the two-radii challenge, the fashionable types of the Pompeiu challenge, and specific reconst. learn more... Offbeat imperative Geometry on Symmetric areas; Contents; Preface; half I research on Symmetric areas; bankruptcy 1 Preliminaries; 1.1 Notation; 1.2 Distributions; 1.3 a few transcendental services; 1.4 round harmonics; 1.5 The Gegenbauer polynomials; 1.6 workouts and additional effects; 1. The Titchmarsh theorem generalized (Voronin [V67]); 2. Discrete harmonics (Delsarte [D7]); three. round codes (Seidel [S10]); four. The Kelvin remodel (Axler-Bourdon-Ramey [A16]); Bibliographical notes; bankruptcy 2 The Euclidean Case; 2.1 Homeomorphisms with the generalized transmutation estate 2.2 a few completeness results2.3 platforms of convolution equations; 2.4 Abel kind necessary equations; 2.5 routines and additional effects; 1. relatives among Abel operators and Hankel transforms; 2. Fractional crucial operators (Koornwinder [K10]); three. necessary equations; four. A nonlinear Abel equation (Gorenflo and Vessella [G12]); five. aid homes of Radon transforms on curves (Quinto [Q2]); Bibliographical notes; bankruptcy three Symmetric areas of the Non-compact style; 3.1 Generalities; 3.2 The mapping; 3.3 distinctiveness theorems; 3.4 routines and extra effects; 1. The Jacobi rework bankruptcy 1 features with 0 Ball potential on Euclidean Space1.1 easiest homes of capabilities with 0 integrals over balls; 1.2 specialty effects; 1.3 Description of features within the sessions Vr(BR) and Ur(BR); 1.4 neighborhood two-radii theorems; 1.5 capabilities with 0 integrals over balls in a round annulus; 1.6 The Liouville estate; 1.7 routines and extra effects; 1. Continuation of services with vanishing integrals over balls (Zaraisky [Z9]); 2. strong point end result (Zaraisky); three. Decomposition of vector fields (Smith [S22])
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For ???? ∈ ℰ????,???? ⟨????????,???? (???? ), ????⟩ = (????????−1 )−1 2????−2 Γ2 (????/2) ∫ 0 ∞ ∫ ????????+2????−1 ℱ???????? (???? )(????) ????(????) cos(????????)????????????????. 17) It is not hard to make sure that ????????,???? (???? ) ∈ ????♮′ (ℝ1 ). Consider some basic properties of the mapping ????????,???? : ???? → ????????,???? (???? ). ′ First suppose that ???? ∈ (ℰ????,???? ∩ ???? ???? +????+????+2 )(ℝ???? ) for some ???? ∈ ℤ+ . Then ????+2????−1 ???? −???? −3 ???? ℱ???? (???? )(????) = ????(???? ) as ???? → +∞. Hence ????????,???? (???? ) ∈ ????♮???? (ℝ1 ) and ∫ ∞ (????????−1 )−1 ????????,???? (???? )(????) = ????−2 2 ????????+2????−1 ℱ???????? (???? )(????) cos(????????)????????.
I) For ???? ∈ ℰ????,???? (ℝ???? ) and ???? ∈ ℰ♮′ (ℝ???? ), one has ????????,???? (???? ∗ ???? ) = ????????,???? (???? ) ∗ Λ(???? ). 22) ′ (ii) Let ????1 , ????2 ∈ ℰ????,???? (ℝ???? ), ???? ∈ (0, +∞]. Then ????1 = ????2 in ???????? if and only if ????????,???? (????1 ) = ????????,???? (????2 ) on (−????, ????). 1. Homeomorphisms with the generalized transmutation property 49 Proof. 12). 22)). 2 makes it possible to extend ????????,???? to the space ????????,???? (???????? ), ???? ∈ (0, +∞]. 23) where ???? ∈ ????♮ (???????? ) is selected so that ???? = 1 in ????????0 (????)+???? for some ???? ∈ (0, ????− ????0 (????)).
If ???? ⩾ 1 then throughout unless otherwise stated we shall use the following basis in ℋ???? : 1 1 (????) (????) ????1 (????) = √ (????1 + ????????2 )???? , ????2 (????) = √ (????1 − ????????2 )???? . 66). 64) that ???? ????,???? (????) = ∫ ????????(2) ???? (???? −1 ????)????????????,???? (???? )????????. Assume that ???? ∈ ???? 1 (????). 64) yield ( ) ( ) ∂ ∂ ????????,1 (????) (????±1) ????,1 ′ ±???? ???? = ????????,1 (????) ∓ ???? (????), ????1 ∂????1 ∂????2 ???? ( ( ) ) ∂ ∂ ????????,2 (????) (????∓1) ′ ±???? ???? ????,2 = ????????,2 (????) ± ???? ????1 (????), ∂????1 ∂????2 ???? ????1,2 ???? ????,1 = ???????????? ????,1 , ????1,2 ???? ????,2 = −???????????? ????,2 .