[Paper Review] Convergence of series of dilated functions and spectral norms of GCD matrices

Kristian Seip, Christoph Aistleitner+2 more|arXiv (Cornell University)|Jul 21, 2014

Holomorphic and Operator Theory28 references16 citations

TL;DR

This paper establishes sharp convergence conditions for series of dilated functions $\sum c_k f(kx)$ and $\sum c_k f(n_k x)$ in $L^2$ and almost everywhere, by linking their $L^2$ norms to spectral norms of greatest common divisor (GCD) matrices. It proves that for functions $f \in C^\alpha$ with $\alpha \in (1/2, 1)$, convergence holds if the coefficients $c_k$ satisfy exponential decay conditions involving $\log k$ and $\log \log k$, with optimal constants derived from recent bounds on GCD matrix spectral norms. The results resolve the sharpness of these conditions for both $L^2$ and a.e. convergence, except for the precise constant in the $L^2$ case of the standard dilated series.

ABSTRACT

We establish a connection between the $L^2$ norm of sums of dilated functions whose $j$th Fourier coefficients are $\\mathcal{O}(j^{-\\alpha})$ for some $\\alpha \\in (1/2,1)$, and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in $L^2$ and for the almost everywhere convergence of series of dilated functions.

Motivation & Objective

• To establish sharp sufficient and necessary conditions for $L^2$ and almost everywhere convergence of series $\sum c_k f(kx)$ and $\sum c_k f(n_k x)$, where $f$ is a periodic function in $C^\alpha$ for $\alpha \in (1/2, 1)$.
• To connect the $L^2$ norm of sums of dilated functions to the spectral norms of greatest common divisor (GCD) matrices, leveraging recent bounds on these norms.
• To determine the optimal growth rate of the coefficient sequence $(c_k)$ for convergence, showing that the required decay is significantly stronger than $\ell^2$ summability ($\sum c_k^2 < \infty$) when $\alpha < 1$.
• To prove the optimality of the derived convergence conditions, both for $L^2$ and almost everywhere convergence, by constructing counterexamples that violate the conditions while still satisfying weaker $\ell^2$-type assumptions.

Proposed method

• The authors relate the $L^2$ norm of $\sum c_k f(kx)$ to the spectral norm of a GCD matrix with entries $\gcd(k,\ell)^{2\alpha}/(k\ell)^\alpha$, using the Fourier decay of $f$ in $C^\alpha$.
• They apply recent sharp bounds on the spectral norms of GCD matrices, particularly involving the divisor function $\sigma_s(k)$, to derive exponential-type decay conditions on $c_k$.
• For proving optimality, they construct sequences $(c_k)$ and $(n_k)$ such that the coefficient condition is satisfied with a slightly smaller constant but the series still diverges in $L^2$ or a.e., using probabilistic methods and the Borel–Cantelli lemma.
• They use a dyadic decomposition of the index set, grouping indices into blocks $\Gamma_i$ and $\Delta_i$, to control the growth of the $L^2$ norm and apply the central limit theorem via Lyapunov’s condition.
• The proof of Theorem 3 uses the Cauchy–Schwarz inequality and properties of the divisor function $\sigma_s(k)$ to bound the $L^2$ norm of partial sums, showing that $\sum c_k^2 \sigma_{1-2\alpha+\varepsilon}(k) < \infty$ implies $L^2$ convergence.
• For the a.e. convergence optimality, they use conditional expectations and moment bounds to construct independent random variables $Y_i$ that stochastically dominate the partial sums, then apply the central limit theorem to show divergence a.s.

Experimental results

Research questions

• RQ1 What is the sharp condition on the coefficient sequence $(c_k)$ for the $L^2$ convergence of $\sum c_k f(kx)$ when $f \in C^\alpha$, $\alpha \in (1/2, 1)$?
• RQ2 What is the sharp condition on $(c_k)$ for almost everywhere convergence of $\sum c_k f(n_k x)$, where $(n_k)$ is a sequence of distinct positive integers?
• RQ3 How do the spectral norms of GCD matrices relate to the $L^2$ norms of dilated function series?
• RQ4 Can the convergence condition be improved beyond $\sum c_k^2 < \infty$ for $f \in C^\alpha$, and if so, what is the optimal rate?
• RQ5 Is the derived condition for $L^2$ convergence of $\sum c_k f(kx)$ also optimal for almost everywhere convergence?

Key findings

• For $f \in C^\alpha$, $\alpha \in (1/2, 1)$, the series $\sum c_k f(kx)$ converges in $L^2$ and a.e. if $\sum c_k^2 \exp\left( K (\log k)^{1-\alpha} / \log \log k \right) < \infty$ with $K = 3/(1-\alpha) + 4/\sqrt{2\alpha - 1}$, and this condition is optimal for $L^2$ convergence.
• For the series $\sum c_k f(n_k x)$, convergence in $L^2$ and a.e. holds if $\sum c_k^2 \exp\left( K (\log k)^{1-\alpha} / (\log \log k)^\alpha \right) < \infty$ with $K = 6/(1-\alpha) + 7(|\log(2\alpha - 1)|^{1/2} + 1)$, and this condition is optimal for both $L^2$ and a.e. convergence.
• The condition $\sum c_k^2 \sigma_{1-2\alpha+\varepsilon}(k) < \infty$ for some $\varepsilon > 0$ implies $L^2$ convergence of $\sum c_k f(kx)$, showing that divisor function growth can replace the exponential factor in some cases.
• The paper constructs counterexamples showing that for any $\varepsilon > 0$, there exist $f \in C^\alpha$ and $(c_k)$ such that $\sum c_k^2 \exp\left( (1-\varepsilon)/(1-\alpha) \cdot (\log k)^{1-\alpha} / \log \log k \right) < \infty$ but the series $\sum c_k f(kx)$ does not converge in $L^2$, proving the sharpness of the $L^2$ condition.
• For almost everywhere convergence of $\sum c_k f(n_k x)$, the condition is optimal: there exist $f \in C^\alpha$, $(c_k)$, and $(n_k)$ such that $\sum c_k^2 \exp\left( \hat{K} (\log k)^{1-\alpha} / (\log \log k)^\alpha \right) < \infty$ for some $\hat{K}$, but the series diverges a.e., proving optimality of the exponential factor.
• The blowup of the constant $K$ in the convergence condition as $\alpha \to 1^-$ is of order $(1-\alpha)^{-1}$, matching both the sufficiency and optimality results.