Wilson's functional equation on C |
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Authors: | Henrik Stetkaer |
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Institution: | (1) Department of Mathematics, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark |
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Abstract: | Summary We find the complete set of continuous solutionsf, g of Wilson's functional equation![rdquo](/content/b7194lnu4p148028/xxlarge8221.gif)
n = 0
N – 1
f(x + wny) = Nf(x)g(y), x, y C, given a primitiveN
th rootw of unity.Disregarding the trivial solutionf = 0 andg any complex function, it is known thatg satisfies a version of d'Alembert's functional equation and so has the formg(z) = g
(z) = N–1
n = 0
N – 1
E (wnz) for some C2. HereE
( 1, 2)(x + iy) = exp(
1x + 2).For fixedg = g
the space of solutionsf of Wilson's functional equation can be decomposed into theN isotypic subspaces for the action of Z
N
on the continuous functions on C. We prove that ther
th component, wherer {0, 1, ,N – 1}, of any solution satisfies the signed functional equation
n = 0
N – 1
f(x + wny)wnr = Ng(x)f(y), x, y C. We compute the solution spaces of each of these signed equations: They are 1-dimensional and spanned byz
n = 0
N – 1
wnr E (wnz), except forg = 1 andr 0 where they are spanned by
andz
N – r. Adding the components we get the solution of Wilson's equation. Analogous results are obtained with the action ofZ
N on C replaced by that ofSO(2).The case ofg = 0 in the signed equations is special and solved separately both for Z
N
andSO(2). |
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Keywords: | 39B32 |
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