By Feng Dai, Yuan Xu
1 round Harmonics.- 2 Convolution and round Harmonic Expansion.- three Littlewood-Paley thought and Multiplier Theorem.- four Approximation at the Sphere.- five Weighted Polynomial Inequalities.- 6 Cubature formulation on Spheres.- 7 Harmonic research linked to mirrored image Groups.- eight Boundedness of Projection Operator and Cesaro Means.- nine Projection Operators and Cesaro skill in L^p Spaces.- 10 Weighted most sensible Approximation by way of Polynomials.- eleven Harmonic research at the Unit Ball.- 12 Polynomial Approximation at the Unit Ball.- thirteen Harmonic research at the Simplex.- 14 Applications.- A Distance, distinction and indispensable Formulas.- B Jacobi and comparable Orthogonal Polynomials.- References.- Index.- image Index
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Extra info for Approximation Theory and Harmonic Analysis on Spheres and Balls
A quick computation shows that [∂ j , Dk,l ] = δ j,k ∂l − δ j,l ∂k , [xi , Dk,l ] = δi,k xl − δi,l xk , from which it is easy to see that for 1 ≤ i < j ≤ d, 1 ≤ k < l ≤ d, we have [Di, j , Dk,l ] = −δi,k D j,l + δi,l D j,k + δ j,k Di,l − δ j,l Di,k . 8 Angular Derivatives and the Laplace–Beltrami Operator 25 Using Eq. 6), a simple computation shows that [D1,2 , D21,l ] = −(D1,l D2,l + D2,l D1,l ) and [D1,2 , D22,l ] = D1,l D2,l + D2,l D1,l , so that [D1,2 , D21,l + D22,l ] = 0 for l ≥ 2. Moreover, by Eq.
XN } is fundamental, the matrix is invertible, by Eq. 1). We can then invert the system to express Yk as a linear combination of P1 , . . , PN , which completes the proof. A word of caution is in order. The polynomial Cnλ ( x, xi ) is, for x ∈ Sd−1 , a linear combination of the spherical harmonics according to the addition formula. It is not, however, a homogeneous polynomial of degree n in x ∈ Rd ; rather, it is the restriction of the homogeneous polynomial x nCnλ ( x/ x , y ) to the sphere.
Using Eq. 4) and that of cos mθ and sin mθ on [0, 2π ), and the formula for hα follows from the normalizing constant of the Gegenbauer polynomial. For d = 2 and the polar coordinates (x1 , x2 ) = (r cos θ , r sin θ ), it is easy to see that ∇0 = ∂θ , where ∂θ = ∂ /∂ θ . Hence by Eq. 8), the Laplace–Beltrami operator for d = 2 is Δ0 = ∂θ2 . Using Eq. 1), Δ0 = ∂ ∂ 1 sind−2 θd−1 ∂ θd−1 sind−2 θd−1 ∂ θd−1 + d−2 1 ∂ ∂ sin j−1 θ j . 2 j−1 ∂ θ ∂ θj θj j d−1 . . 6 Spherical Harmonics in Two and Three Variables Since spherical harmonics in two and three variables are used most often in applications, we state their properties in this section.