Speciality:
01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date:
09.09.1971
E-mail: Keywords: Fourier transform; Fourier integral; Fourier multipliers; Hardy classes; harmonical analysis in tube domains over open cones; approximation by operators; approximation by entire functions; entire functions of bounded half-degree; best constants in approximation theory; moduli of smoothness.
Subject:
Definitions of Fourier transform and multiplier in Hardy spaces $H^p$, $p\in(0,1)$, in tube domains over open cones are introduced. These definitions are well-connected to the classical definitions. According to the introduced definition, the Fourier transform of a function $f$ defined on a tube domain $T_\Gamma\subset{\Bbb C}^n$ over open cone $\Gamma\subset{\Bbb R}^n$ is the following function defined for every $t\in{\Bbb R}^n$: $\widehat{f}(t):=e^{2\pi(t,\delta)}\widehat{f_\delta}(t)$. Here $\delta\in\Gamma$ may be choosen arbitrarily and the function $f_\delta(\cdot):=f(\cdot+i\delta)$ belongs to $L_1({\Bbb R}^n)$ for any such $\delta$. It is established that the support of Fourier transform of a function $f\in H^p(T_\Gamma)$, $p\in(0,1]$, is the conjugate cone $\Gamma^*$. The inverse formula: $f(z)=\int_{\Gamma^*}\widehat{f}(t)e^{2\pi i(z,t)}\,dt$ $\forall z\in T_\Gamma$ is also obtained. Effective sufficient conditions for a function defined on the conjugate cone to be a multiplier from $H^p$ to $H^p$, $p\in(0,1]$, are obtained. These sufficient conditions are applied to solution of some problems of approximation theory. In particular, some special moduli of smoothness, used for obtaining of estimates of degree of approximation of a function by its Fourier integral means, are introduced. The equivalence of these moduli is proved. The solution of a classical problem on pointwise approximation of a smooth function defined on real half-axis by entire functions of bounded half-degree with the best constant in the main term of the estimate of degree of approximaton is obtained: Let $f\in W^r({\Bbb R}_+)$, $r\in{\Bbb N}$. Then for every $\sigma>0$ there exists an entire function $g_\sigma$ of bounded half-degree not higher than $\sigma$, so that $|f(x)-g_\sigma(x)|\le {2^rK_r\over{\sigma^r}}x^{r/2}+{\gamma\over{\sigma^{r+1}}}x^{(r-1)/2}$ $\forall x\ge 0$. Here $\gamma$ is some positive constant depending only on $r$, and a constant $2^rK_r$, generally speaking, cannot be reduced.
Main publications:
Tovstolis A. V. Fourier multipliers in Hardy spaces in tube domains over open cones and their applications // Methods of functional analysis and topology, 1998, 4(1), 68–89.
