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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.