On estimates of diameter values of classes of functions in the weight space L_2,γ (ℝ²), γ = exp(–x² – y²)

Abstract

For a function f ∈ L_2,γ (ℝ²), the q-averaging of the generalized m-order modulus of continuity Ω_m,γ (f) with the weight function φ, i.e., the value of 𝔐_q (Ω_m,γ (f), φ; t) (t ∈ (0, 1); m ∈ ℕ; q ∈ (0,∞)) has been considered. With its help and with the participation of the majorant Ψ, classes of two-variable functions ${\overset{\sim }{W}}_2\left({\Omega}_{m,\gamma}^q,\varphi; \Psi \right)$ = $\left\{f\in {L}_{2,\gamma}\left({\mathbb{R}}^2\right):{\mathfrak{M}}_q\left({\Omega}_{m,\gamma }(f),\varphi; t\right)\leqslant \Psi (t)\forall t\in \left(0,1\right)\right\}$ were introduced, for which upper and lower estimates of various diameters in the space L_2,γ (ℝ²) were found. A certain condition on the majorant Ψ was found, under which exact diameter values can be calculated. A similar problem has been considered for the classes ${\overset{\sim }{W}}_2^{r,0}\left({\Omega}_{m,\gamma}^q,\varphi, \Psi \right)$ = ${L}_{2,\gamma}^{r,0}\left(D,{\mathbb{R}}^2\right)\cap {\overset{\sim }{W}}_2^r\left({\Omega}_{m,\gamma}^q,\varphi, \Psi \right)$ $\left(r,m\in \mathbb{N};q\in \left(0,\infty \right);D=-\frac{\partial^2}{{\partial x}^2}-\frac{\partial^2}{{\partial y}^2}+2x\right.\frac{\partial }{\partial x}+2y\frac{\partial }{\partial y}$ is the differential operator) consisting of functions f of class the ${L}_{2,\gamma}^{r,0}\left(D,{\mathbb{R}}^2\right)=\left\{f\in {L}_{2,\gamma}\left({\mathbb{R}}^2\right):{c}_{i0}(f)={c}_{0j}(f)={c}_{00}(f)=0\forall i,j\in \mathbb{N}\right\}$ (here c_ij (f) are Fourier–Hermitian coefficients) for which the r-th iterations of D^rf = D (D^r–1f) ∈ L_2,γ (ℝ²) satisfy the condition ${\mathfrak{M}}_q\left({\Omega}_{m,\gamma}\left({D}^rf\right),\varphi; t\right)\leqslant \Psi (t)\forall t\in \left(0,1\right)$. A number of results related to the specification of the exact values of various diameters are given.