C. Muller, Spherical Harmonics, Springer-Verlag, Berlin (1966).  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... ) where = 1 ; Gamma1 ) by x 1 = cos 1 x 2 = sin 1 cos 2 x 3 = sin 1 sin 2 cos 3 : x Gamma1 = sin 1 sin 2 : sin Gamma2 cos Gamma1 x = sin 1 sin 2 : sin Gamma2 sin Gamma1 : The Laplace operator can be written in the form [14] (8:1) Deltap = p rr Gamma 1 r p r 1 r 2 Delta p where Delta is a second order elliptic operator in ; for = 3, and Delta p = 1 sin 2 (sin p ) 1 sin 2 2 p 2 : Consider a surface S ffl : r = 1 fflf ( with jf j C 2 1, and ....

....(and elliptic estimates) kr Delta k Fk L 2 (B) kFk H 2k 1 (B) K 1 kr Delta k Fk L 2 (B) Thus, it suffices to show that (8.14) holds with kFk H s (B) replaced by k Delta k Fk L 2 (B) and kr Delta k Fk L 2 (B) when s = 2k and s = 2k 1, respectively. On the other hand, since [14] Delta X m;n Fnm (r)Y nm ( X n;m ( 1 r Gamma1 r (r Gamma1 r Fnm ) Gamma n(n Gamma 2)Fnm )Y nm ; we have, using the othogonality properties of Ymn , 8:16) k Delta k Fk L 2 (B) X m;n Z 1 ffi r Gamma1 j[ 1 r Gamma1 r (r Gamma1 r ) Gamma n(n ....

C. Muller, Spherical Harmonics, Springer-Verlag, Berlin (1966).

....Let be a fixed positive integer, and let p j;k 2 S 2 have coordinates ( j ; OE k ) where j = j 2 , OE k = k , and j; k = 0; 2 Gamma 1. We then take u j;k = ffi p j;k . The eigenfunctions for the Laplace Beltrami operator on the 2 sphere are the spherical harmonics [7], Y ;m , where = 0; 1; and m = Gamma ; We will use (p; q) 1 X =0 X m= Gamma a( m)Y ;m (p)Y ;m (q) 11 Again these are of the form (3.7) As before, we assume that the a( m) s decay quickly enough for to be in H 2s (S 2 Theta S 2 ) If is a power of ....

C. Muller, Spherical Harmonics, Springer-Verlag, Berlin, 1966.

....in such cases one may rephrase the results above in terms of N . 4 APPROXIMATION ON THE SPHERE AND TORUS 10 4 Approximation on the Sphere and Torus We deal first with the case of the n sphere. Here, the orthonormal system is based on spherical harmonics, and will be denoted by Y #,m [4, 8]. A function f in L 2 (S n ) has the expansion f = # X #=0 N(n,#) X m=1 f(#, m)Y #,m z P # f The truncated version of f is f L = P L #=0 P # f . We want to estimate #f f L # # . We will start by estimating the L# (S n ) norm of the projection P # f . From the addition ....

....) has the expansion f = # X #=0 N(n,#) X m=1 f(#, m)Y #,m z P # f The truncated version of f is f L = P L #=0 P # f . We want to estimate #f f L # # . We will start by estimating the L# (S n ) norm of the projection P # f . From the addition theorem for spherical harmonics [4, 8], we have N(n,#) X m=1 Y #,m (x) 2 = N(n, #) w n P # (n 1; 1) N(n, #) # n where P # (n 1; is the Legendre polynomial of degree # in n 1 dimensions, normalized by P # (n 1; 1) 1 (cf. 4] and # n denotes the surface area of S n . Using this, we get the following ....

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C. Muller, "Spherical Harmonics", Lecture Notes in Mathematics, Vol. 17, Springer Verlag, Berlin, 1966.

.... scalar functions on Omega Gamma The spherical harmonics of degree n are defined as the everywhere on Omega infinitely differentiable eigenfunctions of the Beltrami operator Delta corresponding to the eigenvalues ( Delta ) n = Gamman(n 1) n = 0; 1; As it is well known (cf. e.g. [14]) the linear space Harm n of all spherical harmonics of degree n is of dimension 2n 1. We denote fY n;j ( n = 0; 1; j = 1; 2; 2n 1)g to be an orthonormalized basis of L 2( Omega Gamma4 where n is called degree and j order of the spherical harmonics. The space Harm 0; m of ....

....differentiable eigenfunctions of the Legendre operator, satisfying the normalization condition P n (1) 1. It should be noted that jP n (t)j 1 and jP 0 n (t)j n(n 1) 2 for all t 2 [ Gamma1; 1] and all n 2 IN 0 . The well known addition theorem of spherical harmonics states (cf. 5] [14]) 2n 1 X j=1 Y n;j ( Y n;j (j) 2n 1 4 P n ( Delta j) j) 2 Omega Theta Omega : 1) In particular, we have 2n 1 X j=1 jY n;j ( j 2 = 2n 1 4 ; jY n;j ( j s 2n 1 4 ; 2 Omega : 2) For s 2 IR consider the space E s ( Omega Gamma = fF 2 C 1 ( Omega Gamma ....

Muller, C.(1966) Spherical Harmonics, Lecture Notes in Mathematics 17, Springer Verlag.

....R n by a single infinite point. It is known [11] that the following relations hold j Gamma oej = 2jx Gamma yj (1 jxj 2 ) 1=2 (1 jyj 2 ) 1=2 ; 22) doe = 2 n dy (1 jyj 2 ) n ; 23) where = s(x) oe = s(y) x; y 2 R n 1 : 6) Spherical harmonics. We refer e.g. to [12], 20] for harmonic analysis on S n and recall only some notations and the Funk Hekke formula . By Ym (oe) oe 2 S n , we denote an arbitrary spherical harmonic of order m , that is, the restriction of any homogeneous harmonic polynomial in R n 1 onto S n : The space Hm of all such ....

....order m , that is, the restriction of any homogeneous harmonic polynomial in R n 1 onto S n : The space Hm of all such harmonics has a dimension d n 1 (m) n 2m Gamma 1 n m Gamma 1 m n Gamma 1 m : 24) LetfYm g =1; d n 1 (m) be an orthonormal basis in Hm . It is known ([12], 20] that the sequence fYm g =1; d n 1 (m) m2N0 is a basis in L 2 (S n ) The formula Z S n k( oe)Y m (oe)doe = mYm ( 2 S n ; 25) known as the Funk Hekke formula [9] states that an arbitrary spherical harmonic Ym ( is an eigen function for any operator K ; defined as ....

Muller C. Spherical Harmonics. Lect. Notes in Math., 17(1966), Springer, Berlin.

....as vectors of unit length in R n 1 , such that the angle between two points p and q satisfies p Delta q = cos . The resulting functions of the form Phi(cos( p; q) are called zonal. It is useful to expand Phi(p Delta q) in terms of the spherical harmonics Y j on S n (cf. [11, 12]) This we can do by employing the famous Addition Theorem for spherical harmonics [11] P (n 1; p Delta q) n d n ( dn ( X j=1 Y j (p) Y j (q) 3.2) Here, d n ( is the dimension of the space of n 1 dimensional harmonic polynomials homogeneous of degree and n is ....

....q satisfies p Delta q = cos . The resulting functions of the form Phi(cos( p; q) are called zonal. It is useful to expand Phi(p Delta q) in terms of the spherical harmonics Y j on S n (cf. 11, 12] This we can do by employing the famous Addition Theorem for spherical harmonics [11]: P (n 1; p Delta q) n d n ( dn ( X j=1 Y j (p) Y j (q) 3.2) Here, d n ( is the dimension of the space of n 1 dimensional harmonic polynomials homogeneous of degree and n is the volume of S n . This results in the expansion Phi(p Delta q) 1 X =0 dn ....

C. Muller. Spherical Harmonics. Springer-Verlag, Berlin, 1966.

.... 1 X k=0 c k (Z) k X m= Gammak c k;m ( Y (k) m : 3) We also have Young s inequality k Zk q k k p kZk r ; 4) where 1 p; q; r 1; 1=q = 1=p 1=r Gamma 1: For more information on harmonic analysis on the sphere the reader should consult the following references: 2] 3] 6] [7], 8] 10] 11] 12] We shall introduce a wide range of smooth functions on a sphere in terms of multiplier operators, which, via (3) can often be realised as convolution operators. Given a sequence = f k g k2IN , we shall say that the function f is in U p Phi IR if f c 1 X k=1 ....

C. Muller, "Spherical Harmonics", Springer-Verlag, Berlin, 1966.

....of the right hand side of (3.5) by 2. This leads to Z S d nC(x;r) jx0yj 0s doe(y) 1 s 0 d fl d r d0s o(r d0s ) r 0) s d; 3:6) and Z S d nC(x;r) jx 0 yj 0d doe(y) fl d [0 log r] O(1) r 0) s = d: 3:7) We also need some basic facts from spherical harmonics; see [16]. Following [21] we denote the ultrapherical polynomials by P n and we recall the Rodrigues formula P n (t) 02) n n 0(n )0(n 2) 0( 0(2n 2) 1 0 t 2 ) 1=20 d dt n (1 0 t 2 ) n 01=2 : 3:8) From the addition formula for spherical harmonics [16, p.20] we get P ....

C. Muller, Spherical Harmonics, Lect. Notes. Math. 17, Springer-Verlag, Berlin, 1966.

....inf y2L kux Gamma ykY : It is well known (see e.g. 15] 18] that c n (u) dn (u ) where u denotes the conjugate operator. 2 Basic elements of harmonic analysis and sets of differentiable functions on the sphere All of the material discussed in this section may be found in [1] 2] [11], 16] 17] Let IR d be d dimensional Euclidean space endowed with the usual scalar product hx; yi = d X k=1 x k y k ; and S d be the d dimensional unit sphere in IR d 1 , i.e. S d = fx j x 2 IR d 1 ; d 1 X k=1 jx k j 2 = 1g; d 2: The sphere inherits a rotation invariant ....

C. Muller, "Spherical Harmonics", Springer-Verlag, Berlin, 1966.

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C. Muller, Spherical Harmonics, Springer-Verlag, Berlin, 1966.

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Muller, C., Spherical Harmonics, Springer-Verlag, Berlin, 1966.

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Muller, C. (1966). Spherical Harmonics. Springer-Verlag, New York, NY.

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Muller, C., Spherical Harmonics, Springer, Berlin, 1966.

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Claus Muller. Spherical Harmonics. Springer-Verlag, New York, NY, 1966.

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Muller, C., Spherical Harmonics, Springer, Berlin, 1966.

No context found.

Claus Muller. Spherical Harmonics. Springer-Verlag, New York, NY, 1966.

No context found.

C. Muller, Spherical Harmonics, Springer-Verlag, Berlin, 1966.

No context found.

Muller, C. (1966). Spherical Harmonics. Springer-Verlag, New York, NY.

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