By Feng Dai, Yuan Xu, Sergey Tikhonov
This ebook presents an advent to h-harmonics and Dunkl transforms. those are extensions of the standard round harmonics and Fourier transforms, within which the standard Lebesgue degree is changed by means of a reflection-invariant weighted degree. The authors’ concentration is at the research part of either h-harmonics and Dunkl transforms. Graduate scholars and researchers operating in approximation idea, harmonic research, and useful research will reap the benefits of this book.
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Additional resources for Analysis on h-Harmonics and Dunkl Transforms
If t ∈ (0, π2 ) and r > 0, then ∼ Kr ( f ,t)κ,p . 5 can be found in [16, Chapter 10]. 6 yields the following direct Jackson inequality and the inverse inequality. 6. Let f ∈ L p (h2κ ; Sd−1 ) if 1 ≤ p < ∞, and f ∈ C(Sd−1 ) if p = ∞. Then for any r > 0 and n ∈ N, En ( f )κ,p ≤ c Kr ( f ; n−1 )κ,p , and Kr ( f , n−1 )κ,p ≤ c n−r n ∑ (k + 1)r−1 Ek ( f )κ,p . 4 The sharp Marchaud inequality In this section, we will prove the following sharp Marchaud inequality. 1.
Deﬁne θ Λ(θ , x) = 0 Tφκ | f |(x)(sin φ )2λ dφ . 4) implies that Λ(θ , x) ≤ Mκ f (x) θ 0 (sin φ )2λ dφ for all x ∈ Sd−1 . 2, | f ∗κ g(x)| =cλ π Tφκ f (x)g(cos φ )(sin φ )2λ dφ 0 π ≤ cλ 0 Tφκ | f |(x)k(φ )(sin φ )2λ dφ . Integrating by parts, we obtain π | f ∗κ g(x)| ≤ cλ Λ(π, x)k(π) − ≤ cλ Mκ f (x) k(π) 0 π 0 Λ(θ , x)k (θ )dθ (sin φ )2λ dφ − π 0 k (θ ) θ 0 (sin φ )2λ dφ dθ , since k (cos θ ) ≤ 0. Integrating by parts again, we conclude that | f ∗κ g(x)| ≤ Mκ f (x)cλ π 0 k(θ )(sin θ )2λ dθ ≤ cMκ f (x).
4 The sharp Marchaud inequality In this section, we will prove the following sharp Marchaud inequality. 1.
Analysis on h-Harmonics and Dunkl Transforms by Feng Dai, Yuan Xu, Sergey Tikhonov