Re-expansions on compact Lie groups
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Re-expansions on compact Lie groups Rauan Akylzhanov1 · Elijah Liflyand2,3
· Michael Ruzhansky1,4
Received: 7 June 2019 / Revised: 25 June 2020 / Accepted: 1 July 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper we refine the re-expansion problems for the one-dimensional torus and extend them to the multidimensional tori and to compact Lie groups. First, we establish weighted versions of classical re-expansion results in the setting of multi-dimensional tori. A natural extension of the classical re-expansion problem to general compact Lie groups can be formulated as follows: given a function on the maximal torus of a compact Lie group, what conditions on its (toroidal) Fourier coefficients are sufficient in order to have that the group Fourier coefficients of its central extension are summable. We derive the necessary and sufficient conditions for the above property to hold in terms of the root system of the group. Consequently, we show how this problem leads to the re-expansions of even/odd functions on compact Lie groups, giving a necessary and sufficient condition in terms of the discrete Hilbert transform and the root system. In the model case of the group SU(2) a simple sufficient condition is given. Keywords Fourier series · Re-expansion · Compact Lie groups · Hilbert transform
The authors were supported in parts by the FWO Odysseus Project G.0H94.18N: Analysis and Partial Differential Equations, EPSRC Grant EP/R003025/1 and by the Leverhulme Grant RPG-2017-151.
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Elijah Liflyand [email protected] Rauan Akylzhanov [email protected] Michael Ruzhansky [email protected]
1
School of Mathematical Sciences, Queen Mary University of London, London, UK
2
Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel
3
Regional Mathematical Center of Southern Federal University, Rostov-on-Don, Russia
4
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium 0123456789().: V,-vol
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Mathematics Subject Classification Primary 43A30; Secondary 43A50 · 43A75 · 42A20 · 42B05
1 Introduction In the 50-s (see, e.g., [4] or in more detail [5, Chapters II and VI]), the following problem in Fourier Analysis attracted much attention: Let {ak }∞ k=0 be the sequence of the Fourier coefficients of the absolutely convergent sine (cosine) Fourier series of a function f : T = [−π, π ) → C, that is |ak | < ∞. Under which conditions on {ak } the re-expansion of f (t) ( f (t) − f (0), respectively) in the cosine (sine) Fourier series will also be absolutely convergent? In general, the answer is negative and re-expansion not always leads to an absolutely convergent series. In this paper we shall present the necessary and sufficient condition when the positive answer is the case, while in earlier works only a sufficient condition was given, sharp on the whole class. It is quite simple and is the same in both cases: ∞
|ak | ln(k + 1) < ∞.
(1.1)
k=1
In [7], a similar problem of the integrability of the
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