Abstract Generated abstract
The paper studies constructive characterizations of functions in L2 on the unit sphere in terms of their best approximation by finite spherical harmonic sums and the behavior of spherical Fourier coefficients. It establishes necessary and sufficient conditions, under Lozinskii type assumptions on a modulus function, relating the decay of approximation errors to L2 estimates for iterated spherical Laplacians of differences between a function and its circular means. Further theorems give coefficient estimates, criteria for convergence of weighted series formed from spherical Fourier coefficients, and analogues of results of Lorentz and Bari for absolute convergence. The paper also derives comparison results for two functions whose moduli of smoothness are ordered, showing transfer of summability properties of spherical Fourier coefficients under suitable hypotheses.
Full Text
UDC 517.512.6
MATHEMATICS
A. JAFAROV
A CONSTRUCTIVE CHARACTERIZATION OF A CLASS OF FUNCTIONS ON THE SPHERE WITH INTEGRABLE SQUARE
(Presented by Academician S. L. Sobolev on 29 IV 1968)
Let \(S\) be the sphere of unit radius centered at the origin of three-dimensional Euclidean space; let \(P\) be a point of the sphere \(S\) with spherical coordinates \(\theta,\varphi\). Let \(f(P)\in L_2(S)\), and let \(E_n^{(2)}(f)\) be the best approximation of the function \(f\) on \(S\) by spherical sums of order not exceeding \(n-1\) in the metric \(L_2\), i.e.
\[ E_n^{(2)}(f)=\inf_{S_{n-1}}\|f(P)-S_{n-1}(P)\|_{L_2(S)}, \]
where
\[ S_n(P)=Y_0(P)+Y_1(P)+\dots+Y_n(P), \]
and \(Y_k(P)\) is the usual notation for a spherical harmonic of order \(k\), so that
\[ DY_k(P)=-k(k+1)Y_k(P); \]
here
\[ D \equiv \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\frac{\partial}{\partial\theta}\right) +\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2} \]
is the Laplace operator on the sphere.
I. Consider the question of the structural properties of the class of functions satisfying the condition
\[ E_n^{(2)}(f)\leq \frac{C_1}{n^{2r}}\,\omega\left(\frac{1}{n}\right), \qquad n=1,2,\ldots, \tag{1} \]
where \(C_1=\mathrm{const}\), and \(\omega(\delta)\) is a certain function monotonically tending to zero.
Denote by \(L_2^*(S)\) the set of functions from \(L_2(S)\) whose Fourier series in spherical functions converge uniformly on \(S\). Let \(C(P,h)\) be the circle with center at the point \(P\) and spherical radius \(h\) on \(S\), and let \(f_h(P)\) be the mean value of the function \(f\) along the circle \(C(P,h)\), i.e.
\[ f_h(P)=\frac{1}{2\pi\sin h}\int_{C(P,h)} f(Q)\,dS_Q. \]
Theorem 1. Let \(f\in L_2^*(S)\), and let the function \(\omega(h)\downarrow 0\) as \(h\to 0\) satisfy the \((\xi_2)\)-condition of S. M. Lozinskii \((^2)\): there exists a constant \(C>1\) such that
\[ 1<\varliminf_{h\to 0}\omega(Ch)/\omega(h)\leq \varlimsup_{h\to 0}\omega(Ch)/\omega(h)<C^2. \]
Then, in order that inequality (1) hold, it is necessary and sufficient that
\[ \|D^r(f-f_h)\|_{L_2(S)}=O(\omega(h)), \]
where \(D^r\) denotes the \(r\)-th power of the Laplace operator \(D\) (\(r\) is a nonnegative integer).
An analogous result in the metric of the space \(C(S)\) (\(C(S)\) is the space of functions continuous on \(S\)) was obtained by G. G. Kushnirenko \((^{3})\) for the case \(\omega(h)=h^\alpha\) \((0<\alpha<2)\), and by the author \((^{4})\) for the function \(\omega(h)\) indicated in Theorem 1. The proof of Theorem 1 will follow from Theorems 2 and 3, in which relations are established between the behavior of the Fourier coefficients of the function \(f\) and the differential properties of this function.
Let \(C_{\nu m}(f)\) be the Fourier coefficients of the function \(f(P)\) with respect to the orthonormal system of spherical functions \(Y_m^{(\nu)}(P)\) \((\nu=-m,\ldots,-1,0,1,\ldots,m;\ m=0,1,2,\ldots)\) (see, for example, \((^{6})\)). Introduce the notation
\[ R_n^{(r)}(f)=\left\{\sum_{m=n}^{\infty} m^{4r}\rho_m^2(f)\right\}^{1/2}, \qquad \text{where }\quad \rho_m^2(f)=\sum_{\nu=-m}^{m} C_{\nu m}^2(f), \]
and note that, by virtue of the closedness of the indicated system,
\[ R_n^{(0)}(f)=\left\{\sum_{m=n}^{\infty}\rho_m^2(f)\right\}^{1/2}=E_n^{(2)}(f). \tag{2} \]
Lemma. Let \(f\in L_2^*(S)\). Then, for any natural number \(n\), the inequality
\[ \sup_{h\leqslant 1/n}\left\|D^r(f-f_h)\right\|_{L_2(S)} \leqslant \frac{C_2}{n^4}\sum_{k=1}^{n} k^3\{R_k^{(r)}(f)\}^2, \]
holds, where the constant \(C_2\) depends only on \(r\).
Using this lemma, one proves
Theorem 2. Let \(f\in L_2^*(S)\), and let the function \(\omega(h)\) satisfy the \((\mathfrak L_2)\)-condition of S. M. Lozinskii. Then, if
\[ \left\{\sum_{m=n}^{\infty}\rho_m^2(f)\right\}^{1/2} = O\left(n^{-2r}\omega\left(\frac1n\right)\right), \]
then
\[ \left\|D^r(f-f_h)\right\|_{L_2(S)} = O(\omega(h)),\qquad 0<h\leqslant 1. \]
It is said that a function \(\omega(h)\downarrow0\) \((h\to0)\) satisfies the \((B)\)-condition of N. K. Bari \((^{2})\), if
\[ \sum_{k=n+1}^{\infty}\frac1k\,\omega\left(\frac1k\right) = O\left(\omega\left(\frac1n\right)\right). \]
Theorem 3. Let \(f\in L_2^*(S)\), \(\beta\) be arbitrary, \(0<p\leqslant2\), and \(r\geqslant0\) an integer. In order that the relation
\[ \left\{\sum_{k=n}^{\infty} k^\beta \rho_k^p(f)\right\}^{1/p} = O\left(n^{-\varkappa}\omega\left(\frac1n\right)\right), \]
where
\[ \varkappa=2r+\frac12-\frac1p(1+\beta)\geqslant0, \]
hold, it is sufficient that
\[ \left\|D^r(f-f_h)\right\|_{L_2(S)} = O(\omega(h)), \]
where \(\omega(h)\downarrow0\) \((h\to0)\), if \(\varkappa>0\), and \(\omega(h)\) satisfies the \((B)\)-condition of N. K. Bari, if \(\varkappa=0\) and \(p\geqslant1\).
We note that condition \((\mathfrak L_2)\) entails (see \((^{2})\)) condition \((B)\), and by virtue of this Theorem 1 follows from Theorems 2, 3 and equality (2).
Theorem 3 does not cover the case when \(r=\beta=0,\ p=1\). In this case the following theorem holds, which is an analogue of G. Lorentz’s theorem \((^{1})\).
Theorem 4. Let \(f\in L_2^*(S)\) and \(\omega_f^{(2)}(\delta)=O(\delta^\alpha)\), \(\alpha>1/p-1/2\) \((0<p\leq 2)\), where
\[ \omega_f^{(2)}(\delta)=\sup_{h\leq \delta}\|f-f_h\|_{L_2(S)}. \]
Then
\[ \left\{\sum_{k=n}^{\infty}\rho_k^p(f)\right\}^{1/p} =O\left(n^{-\alpha-1/2+1/p}\right). \tag{3} \]
If \(\alpha>1/2\), then Theorem 4 can be applied with \(p=1\), and since in this case the right-hand side of (3) tends to zero as \(n\to\infty\), Theorem 4 implies
Corollary 1. Let, for \(f\in C(S)\), \(\omega_f(\delta)=O(\delta^\alpha)\), \(\alpha>1/2\), where
\[ \omega_f(\delta)=\sup_{h\leq \delta}\|f-f_h\|_{C(S)}. \]
Then
\[ \sum_{n=0}^{\infty}\sum_{\nu=-n}^{n}|C_{\nu n}(f)|<\infty. \tag{4} \]
The latter can also be proved in another way (see Corollary 2). We note that from the convergence of the series (4) there follows the absolute convergence almost everywhere on \(S\) of the Laplace series of the function \(f\), i.e. convergence almost everywhere on \(S\) of the series
\[ \sum_{n=0}^{\infty}|Y_n(P)|,\quad \text{where}\quad Y_n(P)=\frac{2n+1}{4\pi}\iint_S f(Q)P_n(\cos PQ)\,dQ; \]
\(P_n(x)\) is the Legendre polynomial. This result generalizes and refines the corresponding result from (5), obtained by another method.
II. Consider the question of conditions for convergence of series of the form
\[ \sum_{n=1}^{\infty} n^\beta \rho_n^\alpha(f). \tag{5} \]
Theorem 5. Let \(f\in L_2^*(S)\) and let \(\beta\) be arbitrary, \(0<\alpha<2\), \(N>1\) an integer. Then the convergence of the series
\[ \sum_{n=1}^{\infty} N^{\,n(\beta+1-\alpha/2)} \left\{\omega_f^{(2)}\left(\frac{1}{N^n}\right)\right\}^{\alpha} \]
implies the convergence of the series (5).
Corollary 2. Let \(f\in L_2^*(S)\) and let \(\beta\) be any number, \(0<\alpha<2\). Then the convergence of the series
\[ \sum_{n=1}^{\infty} n^{\beta-\alpha/2} \left\{\omega_f^{(2)}\left(\frac{1}{n}\right)\right\}^{\alpha} \]
implies the convergence of the series (5).
The result formulated in this corollary (which contains Corollary 1), for \(\beta=0,\ \alpha=1\), is an analogue of the corresponding result on the absolute convergence of trigonometric Fourier series (see (1), p. 612).
III. Suppose the behavior is known of the Fourier series of a function \(f\) with respect to an orthonormal system of spherical functions. Let us consider how the Fourier series of a function \(g\), connected with \(f\) by the relation
\[ \|g-g_h\|_{L_2(S)}\leq \|f-f_h\|_{L_2(S)}. \tag{6} \]
Theorem 6. Let \(f\) and \(g\) be functions from \(L_2^*(S)\), connected with each other by relation (6), and let \(\rho_n(f) \leq \gamma_n\). Then, if for some \(\alpha\) \((0<\alpha<2)\)
\[ \sum_{n=1}^{\infty} n^{-5/2\alpha}\left\{\sum_{k=1}^{n} k^4\gamma_k^2\right\}^{\alpha/2} + \sum_{n=1}^{\infty} n^{-\alpha/2}\left\{\sum_{k=n+1}^{\infty}\gamma_k^2\right\}^{\alpha/2} <\infty, \]
then
\[ \sum_{n=1}^{\infty}\rho_n^\alpha(g)<\infty. \]
On the basis of this theorem one proves
Theorem 7. Let \(f\) and \(g\) be functions from \(L_2^*(S)\), connected with each other by relation (6), and let \(\rho_n(f)\leq\gamma_n\), where
\[ \gamma_n\downarrow 0,\qquad \sum_{n=1}^{\infty}\gamma_n^\alpha<\infty,\qquad \frac{2}{5}<\alpha<2. \]
Then
\[ \sum_{n=1}^{\infty}\rho_n^\alpha(g)<\infty. \]
In addition to the works mentioned above, we indicate works (7–12), where analogous questions are considered for the one-dimensional case. In these same works one can find references to extensive literature.
Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR
Received
20 II 1968
CITED LITERATURE
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