Get Convolution type functional equations on topological Abelian PDF

By Laszlo Szekelyhidi

ISBN-10: 9810206585

ISBN-13: 9789810206581

This publication is dedicated to the potential purposes of spectral research and spectral synthesis for convolution kind useful equations on topological abelian teams. the answer area of convolution kind equations has been synthesized within the experience that the final suggestions are equipped up from exponential monomial strategies. particularly, equivalence of platforms of practical equations might be validated. This ends up in a unified remedy of classical equations and to attention-grabbing new effects.

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Extra info for Convolution type functional equations on topological Abelian groups

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2. Let G, S be commutative semigroups and n a nonnegative integer. KA:G —*Sis n-additive and symmetric, then we have for ail x, y in G n A-{x + y) = Y A(\x} M^ ). j k k it=0 P R O O F : Let n > 1 , then we have for all x, y in G A'(x + y) = A(x +y,[z + t/J^O = A{x, [x + ;/]„_,) + A(y, [x + As the function (xj,x ,... -* A(z, xi, x ,... , £ _ i ) is evidently n — 1additive and symmetric for any fixed z in G, and its diagonalization is the function x H* A(z, [z]„_i), the above equation gives the statement by induction as follows 2 A*{x + y) = n-l 2 (" .

N . /)U^) = 0 A i=l for all x , y in G and for j = 0 , l , . . , n , as multiplication by n! is bijective in S. To prove the converse, we multiply our last equation by (*), and add the equations for k =j,j + 1 , . . , t i , and for j = 0 , 1 , . . ,n. Now we introduce a translation invariant subclass of V{G). I f m is a positive integer, P : R —• C is a (classical) polynomial in m variables and at : G —» R (k = 1,2,... ,Wl) are continuous additive functions, then the function x w P(ai(x),a (x),...

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Convolution type functional equations on topological Abelian groups by Laszlo Szekelyhidi


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