FRACTIONAL BROWNIAN VECTOR FIELDS 1. Introduction. A one ...

24 P. D. TAFTI AND M. UNSER
with
e ζ 2γ+d−1
irr = e ξ irr
2γ −
d−1
2γ eξ sol
and e ζ sol
= − 1
2γ eξ 2γ+1
irr +
2γ eξ sol .
−γ
−γ
Appendix B. Conjugacy of (− ´∆) and (− `∆) .
−ξ −ξ
We proceed to show that for all test functions f and g ∈ d ,
−γ
−γ
〈(− ´∆) f , g〉 = 〈f ,(− `∆) g〉.
−ξ −ξ
−γ
Using Parseval’s identity and the definition of (− `∆) in (2.14) we can write
−ξ
∫
−γ
〈f ,(− `∆) g〉 = −ξ (2π)−d [ ˆf (ω)] H ˆΦ−γ
d −ξ (ω)[Rγ ĝ](ω) dω
⎡
⎤
∫
= (2π) −d [ ˆf (ω)] H ˆΦ−γ (ω) ⎢
d −ξ ⎣ĝ(ω) −
∑
T k [ĝ]ω k ⎥
⎦ dω.
|k|≤⌊2γ− d 2 ⌋
(B.1)
Moreover,
T k [ĝ] = ĝ (k) (0)
k!
(−j)
=
∫ k x k
g(x) dx.
d k!
By combining this and (B.1) we get
∫
−γ
〈f ,(− `∆) g〉 = −ξ (2π)−d [ ˆf (ω)] H ˆΦ−γ (ω)
d −ξ
⎡
⎤
· ⎢
⎣
e
∫ −j〈x,ω〉 −
∑ (−j) k x k ω k
g(x) dx⎥
d k!
⎦ dω.
|k|≤⌊2γ− d 2 ⌋
∫
−γ
= (− ´∆) f
d −ξ (x)H g(x) dx;
where the last step follows from exchanging the order of integration and using the definition
of the right inverse given in (2.15), together with the identity ˆΦ−γ
−γ
−ξ (ω)H = ˆΦ (ω). The
−ξ
−γ
last integral is by definition equal to the scalar product 〈(− ´∆) f , g〉.
−ξ
REFERENCES
[1] MUTHUVEL ARIGOVINDAN, Variational Reconstruction of Vector and Scalar Images from non-Uniform
Samples, PhD thesis, EPFL, Biomedical Imaging Group, 2005. Available from: http://library.epfl.
ch/theses/nr=3329.
[2] MARCO AVELLANEDA AND ANDREW J. MAJDA, Simple examples with features of renormalization for
turbulent transport, Philos. Trans. Roy. Soc. London Ser. A, 346 (1994), pp. 205–233.
[3] ALBERT BENASSI, STÉPHANE JAFFARD, AND DANIEL ROUX, Elliptic gaussian random processes, Rev. Mat.
Iberoamericana, 13 (1997), pp. 19–90.
[4] THIERRY BLU AND MICHAEL UNSER, Self-similarity: Part II—Optimal estimation of fractal processes, IEEE
Trans. Signal Process., 55 (2007), pp. 1364–1378.