Abstract Generated abstract
This note studies differentiability properties of Fourier series in generalized spherical functions for functions on the three parameter domain associated with the sphere. It introduces Sobolev type Hilbert spaces defined through the differential operators B1, B2, and B3, proves density of generalized spherical polynomials in these spaces, and identifies them with the domains of fractional powers of the generalized spherical Laplace operator. Using these identifications and estimates for generalized spherical functions and their derivatives, the paper proves convergence in L2 after termwise differentiation up to the corresponding order, and establishes absolute and uniform convergence of the Fourier series and of derivatives of order less than r minus 2 when r is at least 3.
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MATHEMATICS
S. S. LITVINKOV
ON THE DIFFERENTIABILITY OF FOURIER SERIES IN GENERALIZED SPHERICAL FUNCTIONS
(Presented by Academician V. I. Smirnov on 23 I 1962)
1°. In the article (¹) we considered the question of expanding vector functions on the surface of a sphere into uniformly and pointwise convergent series in generalized spherical functions (²). In the present note, we study the question of the differentiability of a Fourier series in generalized spherical functions and obtain theorems analogous to the theorems of S. G. Mikhlin for Fourier series in spherical functions (³).
Consider functions \(u(\varphi_1,\theta,\varphi_2)\), periodic with period \(2\pi\) in \(\varphi_1\) and \(\varphi_2\), in the parallelepiped \(\sigma=(0\leqslant \varphi_1\leqslant 2\pi;\ 0\leqslant \theta\leqslant \pi;\ 0\leqslant \varphi_2\leqslant 2\pi)\). By \(L_2(\sigma)\) we denote the Hilbert space with scalar product
\[ (u,v)=\frac{1}{8\pi^2}\int_0^{2\pi}\int_0^\pi\int_0^{2\pi} u(\varphi_1,\theta,\varphi_2)\cdot \overline{v_1(\varphi_1,\theta,\varphi_2)}\sin\theta\,d\varphi_1\,d\theta\,d\varphi_2 . \]
By \(B_1\), \(B_2\), and \(B_3\) we denote the operators:
\[ B_1=e^{i\varphi_2}\left(\operatorname{ctg}\theta\,\frac{\partial}{\partial\varphi_2} -\frac{1}{\sin\theta}\frac{\partial}{\partial\varphi_1} +i\frac{\partial}{\partial\theta}\right); \]
\[ B_2=e^{i\varphi_2}\left(-\operatorname{ctg}\theta\,\frac{\partial}{\partial\varphi_2} +\frac{1}{\sin\theta}\frac{\partial}{\partial\varphi_1} +i\frac{\partial}{\partial\theta}\right),\qquad B_3=i\frac{\partial}{\partial\varphi_2}. \]
On the set of functions \(u(\varphi,\theta,\varphi_2)\) for which \(B_1^{k_1}B_2^{k_2}B_3^{k_3}u\) \((k_1+k_2+k_3=1,2,3,\ldots,r)\) are continuous, introduce the norm
\[ \left\|u(\varphi_1,\theta,\varphi_2)\right\|_{W_2^r(B,\sigma)} = \sum_{k=0}^{r} \sum_{k_1+k_2+k_3=k} \left\|B_1^{k_1}B_2^{k_2}B_3^{k_3}u\right\|_{L_2(\sigma)} . \]
The completion of this set of functions with respect to the norm introduced will be called the space \(W_2^r(B,\sigma)\).
From the commutation relations for the operators \(B_1, B_2, B_3\) (²) it follows that every expression of the form
\[ B_{i_1}^{\gamma_1}B_{i_2}^{\gamma_2}\cdots B_{i_m}^{\gamma_m}u \quad (i_1,i_2,\ldots,i_m=1,2,3), \]
where \(\gamma_1+\gamma_2+\cdots+\gamma_m=r\), can be expressed through a linear combination of \(B_1^{k_1}B_2^{k_2}B_3^{k_3}u\) \((k_1+k_2+k_3\leqslant r)\).
In \(W_2^r(B,\sigma)\) introduce the scalar product
\[ [u,v]=\frac{1}{8\pi^2}\int_\sigma \sum_{k=0}^{r} \sum_{k_1+k_2+k_3=k} \left(B_1^{k_1}B_2^{k_2}B_3^{k_3}u\right) \left(\overline{B_1^{k_1}B_2^{k_2}B_3^{k_3}v}\right)\,d\sigma . \]
Here \(d\sigma=\sin\theta\,d\varphi_1\,d\theta\,d\varphi_2\). Thus, \(W_2^r(B,\sigma)\) will be a Hilbert space.
The generalized spherical functions \(T_{mn}^l(\varphi_1,\theta,\varphi_2)\) constitute a complete system of eigenfunctions of the self-adjoint operator
\[ \Delta_2=-\left[ \frac{\partial^2}{\partial\theta^2} +\operatorname{ctg}\theta\,\frac{\partial}{\partial\theta} +\frac{1}{\sin^2\theta} \left( \frac{\partial^2}{\partial\varphi_1^2} -2\cos\theta\,\frac{\partial^2}{\partial\varphi_1\partial\varphi_2} +\frac{\partial^2}{\partial\varphi_2^2} \right) \right], \]
corresponding to the eigenvalues \(l(l+1)\).
Definition. Every finite linear combination of generalized spherical functions will be called a generalized spherical polynomial.
Since the system of generalized spherical functions is complete in \(L_2(\sigma)\), the set of generalized spherical polynomials is everywhere dense in \(L_2(\sigma)\).
Lemma 1. The set of generalized spherical polynomials is everywhere dense in \(W_2^r(B,\sigma)\).
Let \(u\in W_2^r(B,\sigma)\) be orthogonal to all generalized spherical polynomials in \(W_2^r(B,\sigma)\), i.e. \([u,v]=0\), if \(v\) is a generalized spherical polynomial. Integrating by parts, we obtain
\[ [u,v]=\frac{1}{8\pi^2}\int_\sigma \sum_{k=0}^{r}(-1)^k \sum_{k_1=k_2=k_3=k} u\,(B_3^{k_3}B_2^{k_2}B_1^{k_1}v)\, \overline{(B_1^{k_1}B_2^{k_2}B_3^{k_3}v)}\,d\sigma . \]
For generalized spherical functions,
\[ \left[ B_3^{k_3}B_2^{k_2}B_1^{k_1} \left(\overline{E_1^{k_1}B_2^{k_2}B_3^{k_3}T_{mn}^l}\right) \right] = (-1)^{k_1k_2k_3} (\lambda_{mn}^l)_{k_1+k_2+k_3}\, \overline{T_{mn}^l}, \]
where \((\lambda_{mn}^l)_{k_1+k_2+k_3}\ge 0\) and \((\lambda_{mn}^l)_0=1\).
Denote
\[ \sum_{k=0}^{r} \sum_{k_1+k_2+k_3=k} (\lambda_{mn}^l)_{k_1+k_2+k_3} = (\gamma_{mn}^l)_r . \]
Then
\[ \sum_{k=0}^{r}(-1)^k \sum_{k_1+k_2+k_3=k} \left[ B_3^{k_3}B_2^{k_2}B_3^{k_3} \left(\overline{B_1^{k_1}B_2^{k_2}B_3^{k_3}T_{mn}^l}\right) \right] = \]
\[ = \sum_{k=0}^{r} \sum_{k_2+k_2+k_3=k} (\lambda_{mn}^l)_{k_1+k_2+k_3}\, \overline{T_{mn}^l} = (\gamma_{mn}^l)_r\,\overline{T_{mn}^l}, \]
where \((\gamma_{mn}^l)_r\ge 1\).
Using this identity, for every generalized spherical polynomial \(w\) one can construct such a polynomial \(v\) that
\[ \sum_{k=0}^{r}(-1)^k \sum_{k_1+k_2+k_3=k} B_3^{k_3}B_2^{k_2}B_1^{k_1} \left(\overline{B_1^{k_1}B_2^{k_2}B_3^{k_3}v}\right) = w . \]
Then
\[ [u,v]=\frac{1}{8\pi^2}\int_\sigma u\cdot\overline{w}\,d\sigma=(u,w). \]
But \([(u,w)]=0\), and this means that \(u=0\), i.e. there does not exist a nonzero element \(u\in W_2^r(B,\sigma)\) which would be orthogonal to the set of generalized spherical polynomials. Lemma 1 is proved.
\(2^\circ\). As is known (see (4)), the domain of definition of the self-adjoint operator \(\Delta_2^{r/2}\) consists of those and only those functions
\[ u=\sum_{l=0}^{\infty}\sum_{m,n=-l}^{l} C_{mn}^l\sqrt{2l+1}\,T_{mn}^l(\varphi_1,\theta,\varphi_2), \tag{1} \]
for which
\[ \sum_{l=0}^{\infty}\sum_{m,n=-l}^{l}\left|C_{mn}^{l}\right|^{2}[l(l+1)]^{r}<\infty . \tag{2} \]
Let us show that \(D(\Delta_{2}^{r/2})\subset W_{2}^{r}(B,\sigma)\). From the relations
\[ B_{1}T_{mn}^{l}=\alpha_{n+1}T_{m,n+1}^{l};\quad B_{2}T_{mn}^{l}=\alpha_{n}T_{m,n-1}^{l};\quad B_{3}T_{mn}^{l}=nT_{mn}^{l}, \]
where \(\alpha_n=\sqrt{(l+n)(l-n+1)}\), it follows that
\[ \left\|T_{mn}^{l}(\varphi_1,\theta,\varphi_2)\right\|_{W_{2}^{r}(B,\sigma)} \le Cl^{r}\left\|T_{mn}^{l}\right\|_{L_2(\sigma)} = C\frac{l^{r}}{\sqrt{2l+1}} . \tag{3} \]
The constant \(C\) depends only on \(r\). Let \(u\in D(\Delta_{2}^{r/2})\). Denote by \(u_N\) the sum of all terms of the series (1) with indices \(l\le N\). Taking (3) into account, we obtain
\[ \left\|u-u_N\right\|_{W_{2}^{r}(B,\sigma)}^{2} \le \sum_{l=N}^{\infty}\sum_{m,n=-l}^{l} (2l+1)\left\|C_{mn}^{l}T_{mn}^{l}\right\|_{W_{2}^{r}(B,\sigma)}^{2} \le \]
\[ \le C\sum_{l=N}^{\infty}\sum_{m,n=-l}^{l}l^{2r}\left|C_{mn}^{l}\right|^{2}. \]
By virtue of the convergence of the series (2), the right-hand side of the inequality tends to zero as \(N\to\infty\). Consequently, \(u\in W_{2}^{r}(B,\sigma)\), and \(D(\Delta_{2}^{r/2})\subset W_{2}^{r}(B,\sigma)\).
The operator \(\Delta_2\) can be obtained by closure from the set of generalized spherical polynomials, on which \(\Delta_2=(B_1B_2-B_3+B_3^2)\). It then follows from Lemma 1 that the domain of definition of \(\Delta_2\) contains the whole space \(W_{2}^{2}(B,\sigma)\): \(D(\Delta_2)\supset W_{2}^{2}(B,\sigma)\). Combining this conclusion with the preceding one, we obtain: \(D(\Delta_2)=W_{2}^{2}(B,\sigma)\).
Let us now consider the operator \(\Delta_{2}^{1/2}\) and show that \(D(\Delta_{2}^{1/2})=W_{2}^{1}(B,\sigma)\). It is easy to verify that, on generalized spherical polynomials,
\[ \left\|\Delta_{2}^{1/2}u\right\|_{L_2(\sigma)}^{2} = \left\|\frac{\partial u}{\partial\theta}\right\|_{L_2(\sigma)}^{2} + \left\|\frac{1}{\sin\theta}\left(\frac{\partial u}{\partial\varphi_1} -\cos\theta\,\frac{\partial u}{\partial\varphi_2}\right)\right\|_{L_2(\sigma)}^{2} + \left\|\frac{\partial u}{\partial\varphi_2}\right\|_{L_2(\sigma)}^{2}. \]
Taking into account that
\[ \frac{\partial u}{\partial\theta} = \frac{1}{2i}\left(e^{i\varphi_2}B_1+e^{-i\varphi_2}B_2\right)u;\quad \frac{\partial u}{\partial\varphi_2}=-iB_3u;\quad \frac{1}{\sin\theta}\left(\frac{\partial u}{\partial\varphi_1} -\cos\theta\,\frac{\partial u}{\partial\varphi_2}\right) = \frac{1}{2}\left(e^{-i\varphi_2}B_2-e^{i\varphi_2}B_1\right)u, \]
we obtain
\[ \left\|\Delta_{2}^{1/2}u\right\|_{L_2(\sigma)} \le C\left\|u\right\|_{W_{2}^{1}(B,\sigma)}. \]
Hence it follows that \(D(\Delta_{2}^{1/2})\supset W_{2}^{1}(B,\sigma)\), and therefore \(D(\Delta_{2}^{1/2})=W_{2}^{1}(B,\sigma)\). The arguments which we have carried out for \(r=1\) and \(r=2\) are easily repeated for any integer \(r>2\). This implies the validity of the following theorem:
Theorem 1. The sets \(D(\Delta_{2}^{r/2})\) and \(W_{2}^{r}(B,\sigma)\) coincide.
\(3^\circ\). The derivatives of the function \(u(\varphi_1,\theta,\varphi_2)\) with respect to \(\varphi_1,\theta,\varphi_2\) are expressible in terms of \(B_1,B_2\) and \(B_3\) by the formulas
\[ \frac{\partial^{r}u}{\partial\theta^{r}} = \left(-\frac{i}{2}\right)^{r} \left(e^{i\varphi_2}B_1+e^{-i\varphi_2}B_2\right)^{r}u;\quad \frac{\partial^{r}u}{\partial\varphi_2^{r}} = (-i)^{r}B_3^{r}u; \]
\[ \frac{\partial^{r}u}{\partial\varphi_1^{r}} = \frac{1}{2^{r}} \left(\sin\theta\cdot e^{-i\varphi_2}B_2-\sin\theta\cdot e^{i\varphi_2}B_1-2i\cos\theta\cdot B_3\right)^{r}u. \]
In computing \(\partial^{r}u/\partial\theta^{r}\), the factors \(e^{i\varphi_2}\) and \(e^{-i\varphi_2}\) are carried outside the sign of the operator
\[ \frac{\partial}{\partial\theta} = -\frac{i}{2}\left(e^{i\varphi_2}B_1+e^{-i\varphi_2}B_2\right), \]
therefore \(\partial^{r}u/\partial\theta^{r}\) will be equal only
linear combination of expressions of the form
\[ e^{i\gamma_1\varphi_2}e^{-i\gamma_2\varphi_2}B_1^{k_1}B_2^{k_2}B_3^{k_3}u = e^{i(\gamma_1-\gamma_2)\varphi_2}B_1^{k_1}B_2^{k_2}B_3^{k_3}u, \]
where \(\gamma_1+\gamma_2=r,\ k_1+k_2+k_3\leq r\). Similarly, \(\partial^r u/\partial\varphi_1^r\) is a linear combination of expressions of the form
\[ e^{i(\gamma_1-\gamma_2)\varphi_2}\sin^{\gamma_1+\gamma_2}\theta\,\cos^{\gamma_3}\theta\cdot B_1^{k_1}B_2^{k_2}B_3^{k_3}u \quad (\gamma_1+\gamma_2+\gamma_3=r;\ k_1+k_2+k_3\leq r). \]
Hence we obtain estimates of the norms of the derivatives with respect to \(\varphi_1,\theta,\varphi_2\):
\[ \left\| \frac{\partial^r u}{\partial\varphi_1^{k_1}\partial\varphi_2^{k_2}\partial\theta^{k_3}} \right\|_{L_2(\sigma)} < C\|u\|_{W_2^r(B,\sigma)}. \]
From Theorem 1 and these estimates the following theorem follows:
Theorem 2. If the function \(u(\varphi_1,\theta,\varphi_2)\in W_2^r(B,\sigma)\), then the series obtained from the expansion of the function \(u\) in a Fourier series in generalized spherical functions, after \(r\)-fold termwise differentiation with respect to \(\varphi_1,\theta,\varphi_2\), converges in \(L_2(\sigma)\).
By the symbol \(D^r\) we shall denote any derivative of order \(k\) with respect to \(\varphi_1,\theta,\varphi_2\).
Lemma 2. For generalized spherical functions and their derivatives the estimate
\[ \left|D^kT_{mn}^l(\varphi_1,\theta,\varphi_2)\right| \leq O\left(l^{1/2+k}\right)\|T_{mn}^l\|_{L_2(\sigma)} \]
is valid.
We shall not give the proof of Lemma 2, in view of its cumbersomeness.
Theorem 3. If the function \(u(\varphi_1,\theta,\varphi_2)\in W_2^r(B,\sigma)\) and \(r\geq 3\), then its Fourier series in generalized spherical functions and the series obtained from it by termwise differentiation with respect to \(\varphi_1,\theta,\varphi_2\) of order \(<r-2\) converge absolutely and uniformly.
It follows from Lemma 2 that the series
\[ \sum_{l=0}^{\infty}\sqrt{2l+1}\sum_{m,n=-l}^{l} \left|C_{mn}^{l}D^kT_{mn}^{l}(\varphi_1,\theta,\varphi_2)\right| \]
is majorized by the series
\[ C\sum_{l=0}^{\infty}\sum_{m,n=-l}^{l} \left|l^rC_{mn}^{l}\right|\,l^{-(r-k-1/2)} \leq \frac{C}{2}\sum_{l=0}^{\infty}\sum_{m,n=-l}^{l}\left|l^rC_{mn}^{l}\right|^2 + \frac{C}{2}\sum_{l=0}^{\infty}\beta_l l^{-(2r-2k-1)}, \tag{4} \]
where \(C=\mathrm{const},\ \beta_l=(2l+1)^2=O(l^2)\) is the number of linearly independent generalized spherical functions of order \(l\). From (2) and Theorem 1 it follows that the first series in (4) converges. For \(r\geq 3\) and \(k<r-2\) the second series also converges. Theorem 3 is proved.
Remark. Theorems 2 and 3 remain valid if termwise differentiation of the Fourier series is replaced by termwise application of the operators \(B_1,\ B_2\), and \(B_3\) to the Fourier series.
The author expresses his deep gratitude to Prof. S. G. Krein for valuable advice and constant supervision of the work.
Voronezh Technological
Institute
Received
18 I 1962
References
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- S. G. Mikhlin, DAN, 126, No. 2 (1959).
- V. I. Smirnov, A Course of Higher Mathematics. 5, 1959.