R. M. Trigub. On multiply monotone functions ... P. 257-271.

The subject and the method of this paper belong to classical analysis. The Wiener Banach algebra (the normed ring) $A(\mathbb{R}^d)$, $d\in\mathbb N$, is the space of Fourier transforms of functions from $L_1(\mathbb{R}^d)$ (with pointwise product). The membership in this algebra is essential for Fourier multipliers from $L_1$ to $L_1$ and principal for the convergence on the space $L_1$ of summation methods for Fourier series and integrals given by one factor function. A function $f$ is called $m$-multiply monotone on $\mathbb{R}_+=(0,+\infty)$ if $(-1)^{\nu}f^{(\nu)}(t)\ge 0$ for $t\in \mathbb{R}_+$ and $0\le\nu\le m+1$. For such functions, Shoenberg's integral presentation has long been known, which becomes Bernstein's formula for monotone functions as $m\to \infty$. Denote by $V_0(\mathbb{R}_+)$ the set of functions of bounded variation on $\mathbb{R}_+$, i.e., the set of functions representable as the difference of two bounded monotone functions. Denote by $V_m(\mathbb{R}_+)$, $m\in\mathbb N$, the space of functions $f$ from $V_{0,\mathrm{loc}}(\mathbb{R}_+)$ such that $\|f\|_{V_m}=\sup_{t\in \mathbb{R}_+}|f(t)|+\int_0^\infty t^m|df^{(m)}(t)|<\infty$. This is a Banach algebra. A function $f$ belongs to $V_m(\mathbb{R}_+)$ if and only if $f$ can be represented as the difference of two bounded functions with convex derivatives of order $m-1$ (Theorem~1). We also study conditions under which functions of the form $f_0(|x|_{p,d})$, where $|x|_{p,d}=\big(\sum_{j=1}^d |x_j|^p\big)^{1/p}$, $x=(x_1,\ldots,x_d)$, for $p\in (0,\infty)$ and $ |x|_\infty=\max\limits_{1\le j\le d}|x_j|$, belong to $A(\mathbb{R}^d)$. The case $p=2$ (radial functions) is well studied, including the P\'olya-Askey criterion of the positive definiteness of functions on $\mathbb {R}^d$. We prove Theorem 2, which has the following corollaries.

(1) If $f_0\in C_0[0,\infty)$ and $f_0\in V_d(\mathbb{R}_+)$, then $f_0(|x|_{p,d})\in A(\mathbb{R}^d)$ for $p\in [1,\infty]$.

(2) If $f_0\in C_0[0,\infty)$ and $f_0\in V_{d+1}(\mathbb{R}_+)$, then $f_0(|x|_{p,d})\in A(\mathbb{R}^d)$ for $p\in (0,1)$.

We give some examples, including an example with an oscillating function.

Keywords: function of bounded variation, convex function, multiply monotone function, completely monotone function, positive definite function, Fourier transform.

The paper was received by the Editorial Office on April 14, 2017.

Roald Mikhailovich Trigub, Dr. Phis.-Math. Sci., Prof., Sumy State University, Sumy, 40007, Ukraine, e-mail: roald.trigub@gmail.com


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