Corrigendum to “A Fourier type transform on translation invariant valuations on convex sets”
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ISRAEL JOURNAL OF MATHEMATICS xxx (2014), 1–2 DOI: 10.1007/s11856-014-0018-2 CORRIGENDUM TO “A FOURIER TYPE TRANSFORM ON TRANSLATION INVARIANT VALUATIONS ON CONVEX SETS” BY Semyon Alesker Department of Mathematics, Tel aviv University, Ramat Aviv, Tel Aviv 69978, Israel e-mail: [email protected] In Example 0.1.5 of the paper [1], there was given an incorrect explicit description of the Fourier transform F V on 1-homogeneous valuations on a two-dimensional vector space V (while the description of F V on 0- and 2- homogeneous valuations was correct). This description was not used anywhere in the paper and does not affect any of the other results and statements. We present now a correct description. Any valuation φ ∈ V al sm 1 (V ) can be written uniquely in the form φ(K)= S 1 h(ω)dS 1 (K, ω), where h : S 1 → C is a smooth function which is orthogonal on the unit circle S 1 to the two dimensional space of linear functionals. Let us decompose h to the even and odd parts: h = h + + h - . We decompose further the odd part h - to “holomorphic” and “anti-holomo- rphic” parts, h - = h hol - + h anti - , as follows. Let us decompose h - to the usual Fourier series on the circle S 1 : h - (ω)= k ˆ h - (k)e ikω . Received January 30, 2013 1
Transcript of Corrigendum to “A Fourier type transform on translation invariant valuations on convex sets”
ISRAEL JOURNAL OF MATHEMATICS xxx (2014), 1–2
DOI: 10.1007/s11856-014-0018-2
CORRIGENDUM TO “A FOURIER TYPE TRANSFORMON TRANSLATION INVARIANT VALUATIONS
ON CONVEX SETS”
BY
Semyon Alesker
Department of Mathematics, Tel aviv University, Ramat Aviv, Tel Aviv 69978, Israel
e-mail: [email protected]
In Example 0.1.5 of the paper [1], there was given an incorrect explicit
description of the Fourier transform FV on 1-homogeneous valuations on a
two-dimensional vector space V (while the description of FV on 0- and 2-
homogeneous valuations was correct). This description was not used anywhere
in the paper and does not affect any of the other results and statements.
We present now a correct description. Any valuation φ ∈ V alsm1 (V ) can be
written uniquely in the form
φ(K) =
∫S1
h(ω)dS1(K,ω),
where h : S1 → C is a smooth function which is orthogonal on the unit circle
S1 to the two dimensional space of linear functionals. Let us decompose h to
the even and odd parts:
h = h+ + h−.
We decompose further the odd part h− to “holomorphic” and “anti-holomo-
rphic” parts,
h− = hhol− + hanti
− ,
as follows. Let us decompose h− to the usual Fourier series on the circle S1:
h−(ω) =∑k
h−(k)eikω .
Received January 30, 2013
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2 S. ALESKER Isr. J. Math.
Then by definition
hhol− (ω) :=
∑k>0
h−(k)eikω and hanti− (ω) :=
∑k<0
h−(k)eikω .
Hence the Fourier transform of the valuation φ is equal to
(FV φ)(K) =
∫S1
(h+(Jω) + hhol− (Jω))dS1(K,ω)−
∫S1
hanti− (Jω)dS1(K,ω),
where J is the rotation of R2 by π/2, counterclockwise.
References
[1] S. Alesker, A Fourier type transform on translation invariant valuations on convex sets,
Israel Journal of Mathematics 181 (2011), 189–294.
