arXiv subjects
Publications and source records attributed to John J. F. Fournier.
We modify the standard proof of Paley's theorem about lacunary coefficients of functions in $H^1$ to work without analytic factorization. This leads to the first direct proof of the extension of Paley's theorem that we applied to the former Littlewood conjecture about $L^1$ norms of exponential sums.
Denote the coefficients in the complex form of the Fourier series of a function $f$ on the interval $[-π, π)$ by $\hat f(n)$. It is known that if $p = 2j/(2j-1)$ for some integer $j>0$, then for each function $f$ in $L^p$ there exists another function $F$ in $L^p$ that majorizes $f$ in the sense that $\hat F(n) \ge |\hat f(n)|$ for all $n$, and for which $\|F\|_p \le \|f\|_p$. When $j > 1$, the existence proofs for such small majorants do not provide constructions of them, but there is a unique majorant of minimal $L^p$ norm. We modify previous existence proofs to say more about the form of that majorant.
Denote the coefficients in the complex form of the Fourier series of a function $f$ on the interval $[-π, π)$ by $\hat f(n)$. It is known that if $p = 2j/(2j-1)$ for some integer $j>0$, then for each function $f$ in $L^p$ there exists another function $F$ in $L^p$ that majorizes $f$ in the sense that $\hat F(n) \ge |\hat f(n)|$ for all $n$, but that also satisfies $\|F\|_p \le \|f\|_p$. Rescaling $F$ suitably then gives a majorant with the same $L^p$ norm as $f$. We show how that majorant comes from a variant in $L^{2j}$ of the notion of exact majorant in $L^2$.
Paley's theorem about lacunary coefficients of functions in the classical space $H^1$ on the unit circle is equivalent to the statement that certain Hankel matrices define bounded operators on $\ell^2$ of the nonnegative integers. Since that statement reduces easily to the case where the entries in the matrix are all nonnegative, it must be provable by the Schur test. We give such proofs with interesting patterns in the vectors used in the test, and we recover the best constant in the main case. We use related ideas to reprove the characterization of Paley multipliers from $H^1$ to $H^2$.
We reprove an inequality for Rademacher series with coefficients in the Schatten class $S_1$. Our method yields the same estimate for coefficients after suitable gaps in $S_1$-valued trigonometric series; this was known for scalar-valued functions. A very similar method gives a new proof of the extension to $S_1$-valued $H^1$ functions of Paley's theorem about lacunary coefficients.
Consider the group ${\mathbb{R}}^2$ with the discrete topology, and denote its Fourier algebra by $A({\mathbb{R}_{\rm d}^2})$. We reformulate a theorem of V.A. Yudin as a statement about restrictions of functions in $A({\mathbb{R}_{\rm d}^2})$ to the boundary of a strictly convex domain when those functions vanish outside that boundary. We give visual proofs of that statement and a complementary one.
The following theorem on the circle group $\mathbb{T}$ is due to Norbert Wiener: If $f\in L^{1}\left( \mathbb{T}\right) $ has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then $f\in L^{2}\left( \mathbb{T}\right) $. This result has been extended to even exponents including $p=\infty$, but shown to fail for all other $p\in\left( 1,\infty\right] .$ All of this was extended further (appropriately formulated) well beyond locally compact abelian groups. In this paper we prove Wiener's theorem for even exponents for a large class of commutative hypergroups. In addition, we present examples of commutative hypergroups for which, in sharp contrast to the group case, Wiener's theorem holds for all exponents $p\in\left[ 1,\infty\right] $. For these hypergroups and the Bessel-Kingman hypergroup with parameter $\frac{1}{2}$ we characterise those locally integrable functions that are of positive type and square-integrable near the identity in terms of amalgam spaces.