Approximation Theory and Harmonic Analysis on Spheres and Balls by Feng Dai & Yuan Xu

Author:Feng Dai & Yuan Xu
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

where the last step follows from Eq. (5.1.9). This proves Theorem 9.1.3 for p = 1.

Next, we use Hölder’s inequality and Theorem 9.1.2 to obtain

which proves Theorem 9.1.3 for .

Finally, Theorem 9.1.3 for 1 ≤ p ≤ ν follows by applying the Riesz–Thorin convexity theorem to the linear operator g↦ proj n (h κ 2; gχ c(ϖ, θ)).

For the proof of Theorem 9.1.1, we will also need the following duality result.

Lemma 9.1.4.

Assume 1 ≤ p ≤ 2 ≤ q ≤∞ and . Then the following are equivalent: (i)

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