## On convergence and divergence of Fourier-Bessel series.(English)Zbl 1042.42024

In this interesting paper, the author investigates maximal inequalities for orthogonal expansions involving Bessel functions, and then deduces results on mean convergence, convergence almost everywhere, and divergence. Let $$\nu>-1$$ and let $$J_\nu$$ denote the Bessel function of the first kind of order $$\nu$$. Let $$\{\lambda_n\}^\infty_{n= 1}$$ denote its positive zeros in increasing order. There is the classical orthogonality result $\int^1_0 J_\nu(\lambda_n x)J_\nu(\lambda_m x)\,x\,dx= 0,\quad m\neq n.$ Form the orthonormal functions \begin{aligned} \psi_n(x) &= c_n J_\nu(\lambda_n x)/x^\nu;\\ \phi_n(x) &= c_n J_\nu(\lambda_n x);\\ \varphi_n(x) &= c_n\sqrt{x} J_{n, \nu}(x),\end{aligned} $$n\geq 1$$, where $$c_n$$ is a positive normalizing constant. These are orthonormal, respectively, in $$L^2(0, 1)$$ with weight $$x^{2\nu+1}$$, $$L^2(0,1)$$ with weight $$x$$ and $$L^2(0,1)$$ with weight 1. For a measurable function $$f$$ on $$(0, 1)$$, form its orthonormal expansion in the $$\{\varphi_n\}$$, $\sum^\infty_{n=1} a^\nu_n \varphi_n(x)$ and the partial sums $S_Nf(x)= \sum^N_{n=1} a^\nu_n \varphi_n(x).$ The author gives a new proof of a result of Guadalupe, Perez, Ruiz and Varona, that if $$\nu\geq -{1\over 2}$$, $$1< p< \infty$$, $$-1<\alpha< p-1$$, $\biggl\|\sup_{N\geq 1}| S_N f|\biggl\|_{p,\alpha}\leq C\| f\|_{p,\alpha},$ with $$C$$ independent of $$f$$, and with $\| f\|_{p,\alpha}= \Biggl(\int^1_0 | f(x)|^p x^\alpha \,dx\Biggr)^{1/p}.$ From this maximal function inequality, it follows that as $$N\to\infty$$, $$S_N f(x)\to f(x)$$ a.e. in $$(0,1)$$.
The author proves similar results for orthonormal expansions in the $$\{\phi_n\}$$ and $$\{\psi_n\}$$. For example, if $$\nu> -1/2$$, and $$4{v+1\over 2\nu+3}< p\leq\infty$$, then the partial sums of the orthonormal expansions in $$\{\psi_n\}$$ converge a.e. for each function $$g$$ in $$L^p(x^{2n+1}dx, (0,1))$$. The author shows the bound for $$p$$ is sharp: for $$p= 4{v+1\over 2\nu+3}$$, there is a function $$g$$ for which the orthonormal expansion in the $$\{\psi_n\}$$ diverges a.e. Mean convergence is also examined. The paper is well written and will be of interest to anyone studying orthonormal expansions.

### MSC:

 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 33F99 Computational aspects of special functions
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