By R. M. Dudley, J. Feldman, B. Kostant, R. P. Langlands, E. M. Stein, C. T. Taam
Publication via Dudley, R. M., Feldman, J., Kostant, B., Langlands, R. P., Stein, E. M.
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Extra info for Lectures in Modern Analysis and Applications III
Let G(∗) be a groupoid with an identity e and H (0) be a group not necessarily commutative. 79) k(x ∗ y) = k(x)0k(y), for x, y ∈ N1 , and a and b are arbitrary constants in H. Proof. 80) f (y) = a0h(y), f (x) = g(x)0b, x, y ∈ G, respectively, where a = g(e) and b = h(e) are in H. 80) there results f (x ∗ y) = f (x)0b−10a −1 0 f (y) for x, y ∈ N1 . 79). 78) we seek. 55 there results the following theorem regarding the Pexider equations (PA) to (PE). 56. (a) The general system of solutions f, g, h : G(+) → H (+) of (PA), where G(+) is a groupoid with identity and H (+) is a group, is given by f (x) = a + A(x) + b, g(x) = a + A(x), h(x) = A(x) + b, where A is additive; that is, A satisfies (A) and a, b are arbitrary constants.
Proof. 2. 25. If a function A : R+ → R satisfies (A) and is (i) continuous at a point, (ii) monotonically increasing, (iii) nonnegative for small x, (iv) locally integrable, (v) measurable, (vi) bounded in an interval, or (vii) bounded on a bounded set of positive measure, then A(x) = cx, for x ∈ R, where c is an arbitrary constant. Proof. 2. Later we will look at it from the point of view of extension of A. 26. (Kurepa , Vrbov´a ). 41) A1 (z) = A1 (1)Re z, A2 (z) = A2 (i )Im z. Proof.
24) A1 (x n ) = x n−1 A2 (x). 24), we obtain A1 ((x + r )n ) = (x + r )n−1 [ A2 (x) + A2 (1)r ]. 26) n n−1 A1 (x 2 ) = (n − 1)x A2 (x) + A2 (1)x 2, 2 2 respectively. 25) gives n D1 (x) = D2 (x). 26) yields n A1 (x 2 ) = 2x A2(x) + (n − 2)A2 (1)x 2 = 2x[n A1(x) − (n − 1)A2 (1)x] + (n − 2)A2 (1)x 2 or A1 (x 2 ) − A1 (1)x 2 = 2x(A1(x) − A1 (1)x); that is, D1 (x 2 ) = 2x D1 (x), showing thereby that D1 is a derivation. Hence D2 is also a derivation. Case m ≥ 2 and n > m. So, n ≥ 3. 19) to have A1 ((x + r )n ) = (x + r )n−m A2 ((x + r )m ).
Lectures in Modern Analysis and Applications III by R. M. Dudley, J. Feldman, B. Kostant, R. P. Langlands, E. M. Stein, C. T. Taam