By Feng Dai, Yuan Xu, Sergey Tikhonov

This e-book presents an advent to h-harmonics and Dunkl transforms. those are extensions of the standard round harmonics and Fourier transforms, during which the standard Lebesgue degree is changed via a reflection-invariant weighted degree. The authors’ concentration is at the research aspect of either h-harmonics and Dunkl transforms. Graduate scholars and researchers operating in approximation conception, harmonic research, and useful research will take advantage of this book.

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**Extra resources for Analysis on h-Harmonics and Dunkl Transforms**

**Sample text**

1, several vector-valued inequalities for self-adjoint operators on L2 (h2κ ; Sd−1 ) are established, which will be used in the proof of the main result. 2, where a Littlewood–Paley g-function deﬁned via the Poisson semi-group for h-harmonics is introduced and studied as well. 3, where two new g-functions deﬁned via the Ces`aro means play essential roles. 5. 1 Vector-valued inequalities for self-adjoint operators Here we establish several vector-valued inequalities, which will play important roles in later sections.

4. The Ces`aro (C, δ )-means of a given sequence {an }∞ n=0 of complex numbers are deﬁned by n Aδ n− j sδn := ∑ δ a j , n = 0, 1, . . 6) A n j=0 where the coefﬁcients Aδj are deﬁned by (1 − t)−1−δ = ∞ ∑ Aδn t n , t ∈ (−1, 1). n=0 For convenience, we also deﬁne Aδj = 0 for j < 0. The following useful properties follow easily from the deﬁnition: Aδj − Aδj−1 = Aδj −1 , |Aδj | ∼ δ n n j=0 j=0 ∑ Aδj = Anδ +1 , ∑ Aδn− j Aαj = Anα+δ +1 , ( j + 1) , whenever j + δ + 1 > 0. 3. 7) j=0 where Sn0 (h2κ ; f ) is the n-th partial sum.

For f ∈ L1 (Sd−1 ; h2κ ) and g ∈ L1 (wλκ ; [−1, 1]), projκn ( f ∗κ g) = gλn κ projκn f , n = 0, 1, 2 . . 5) where gλn κ is the Fourier coefﬁcient of g in the Gegenbauer polynomial, 1 gλn κ = cλκ −1 g(t) Cnλκ (t) Cnλκ (1) 1 (1 − t 2 )λκ − 2 dt. Proof. 2, 1 ωdκ projκn ( f ∗κ g)(x) = = 1 ωdκ = gλn κ Sd−1 1 ωdκ f (y) Sd−1 Sd−1 1 ωdκ ( f ∗ g)(ξ )Znκ (x, ξ )h2κ (ξ )dσ (ξ ) Sd−1 g( ξ , y )Znκ (x, ξ )h2κ (ξ )dσ (ξ ) h2κ (y)dσ (y) f (y)Znκ (x, y)h2κ (y)dσ (y) = gλn κ projκn f (x), which is what we needed to prove.