By Feng Dai, Yuan Xu, Sergey Tikhonov

ISBN-10: 3034808860

ISBN-13: 9783034808866

ISBN-10: 3034808879

ISBN-13: 9783034808873

This publication presents an creation to h-harmonics and Dunkl transforms. those are extensions of the normal round harmonics and Fourier transforms, during which the standard Lebesgue degree is changed by means of a reflection-invariant weighted degree. The authors’ concentration is at the research facet of either h-harmonics and Dunkl transforms. Graduate scholars and researchers operating in approximation idea, harmonic research, and sensible research will take advantage of this book.

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**Example text**

If {μ j } is a sequence of complex numbers satisfying the condition (Ak ) for some positive integer k, then, as can be easily veriﬁed from the deﬁnition, {μ j } satisﬁes the condition (Ai ) for all 1 ≤ i ≤ k, with a possibly different absolute constant M. 3]. 46 Chapter 4. 4. Let {μ j }∞j=0 be a bounded sequence of complex numbers satisfying the condition (Aδ +1 ) for some nonnegative integer δ . Assume that {a j }∞j=0 is another sequence of complex numbers. Let sδn and σnδ denote the Ces`aro (C, δ )-means of the sequences {a j }∞j=0 and {a j μ j }∞j=0 , respectively.

Put T t = Prκ with r = e−t . Using the above lemma, it is easy to see that T t is a diffusion semi-group. We will need another semi-group, which is the discrete analog of the heat operator: Htκ f := f ∗κ qtκ , qtκ (s) := ∞ ∑ e−n(n+2λκ )t n=0 n + λκ λκ Cn (s). 6. The family of operators {Htκ } is a symmetric diffusion semi-group. 4. Maximal functions 29 Proof. The kernel qtκ is known to be nonnegative, from which it immediately follows that Htκ are positive and that qtκ λκ ,1 = 1, by the orthogonality of the Gegenbauer polynomials.

By Minkowski’s inequality, it sufﬁces to show that G(x, ·) κ,r ≤ g λκ ,r , where G(x, y) = Vκ [g( x, · )](y). 6). 6) that G(x, ·) κ,1 ≤ 1 ωdκ Sd−1 Vκ [|g( x, · )|] (y)h2κ (y)dσ = cλκ The log-convexity of the L p -norm implies then G(x, ·) κ,r 1 −1 |g(t)|wλ (t)dt = g ≤ g λκ ,r . λκ ,1 . 24 Chapter 3. 3), the projection projκn is a convolution operator projκn f = f ∗κ Znκ , Znκ (t) := n + λκ λκ Cn (t). 3. For f ∈ L1 (Sd−1 ; h2κ ) and g ∈ L1 (wλκ ; [−1, 1]), projκn ( f ∗κ g) = gλn κ projκn f , n = 0, 1, 2 .

### Analysis on h-Harmonics and Dunkl Transforms by Feng Dai, Yuan Xu, Sergey Tikhonov

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