Subspace Of Entire Functions Research Articles

Recovery of analytic functions that is exact on subspaces of entire functions

A family of optimal recovery methods is developed for the recovery of analytic functions in a strip and their derivatives from inaccurately specified trace of the Fourier transforms of these functions on the real axis. In addition, the methods must be exact on some subspaces of entire functions. Bibliography: 12 titles.

On the recovery of analytic functions that is exact on subspaces of entire functions

Построены семейства оптимальных методов восстановления аналитических в полосе функций и их производных по неточно заданному следу преобразования Фурье этих функций на вещественной оси. При этом от методов дополнительно требуется, чтобы они были точны на подпространствах целых функций. Библиография: 12 названий.

Nikol’skii–Bernstein Constants for Entire Functions of Exponential Spherical Type in Weighted Spaces

We study the exact constant in the Nikol’skii–Bernstein inequality $$\|Df\|_{q}\leq C\|f\|_{p}$$ on the subspace of entire functions $$f$$ of exponential spherical type in the space $$L^{p}(\mathbb{R}^{d})$$ with a power-type weight $$v_{\kappa}$$ . For the differential operator $$D$$ , we take a nonnegative integer power of the Dunkl Laplacian $$\Delta_{\kappa}$$ associated with the weight $$v_{\kappa}$$ . This situation encompasses the one-dimensional case of the space $$L^{p}(\mathbb{R}_{+})$$ with the power weight $$t^{2\alpha+1}$$ and Bessel differential operator. Our main result consists in the proof of an equality between the multidimensional and one-dimensional weighted constants for $$1\leq p\leq q=\infty$$ . For this, we show that the norm $$\|Df\|_{\infty}$$ can be replaced by the value $$Df(0)$$ , which was known only in the one-dimensional case. The required mapping of the subspace of functions, which actually reduces the problem to the radial and, hence, one-dimensional case, is implemented by means of the positive operator of Dunkl generalized translation $$T_{\kappa}^{t}$$ . We prove its new property of analytic continuation in the variable $$t$$ . As a consequence, we calculate the weighted Bernstein constant for $$p=q=\infty$$ , which was known in exceptional cases only. We also find some estimates of the constants and give a short list of open problems.

Exactness and optimality of methods for recovering functions from their spectrum

Optimal methods are constructed for recovering functions and their derivatives in a Sobolev class of functions on the line from exactly or approximately defined Fourier transforms of these functions on an arbitrary measurable set. The methods are exact on certain subspaces of entire functions. Optimal recovery methods are also constructed for wider function classes obtained as the sum of the original Sobolev class and a subspace of entire functions.

Dunkl’s theory and best approximation by entire functions of exponential type in L 2-metric with power weight

In this paper, we study the sharp Jackson inequality for the best approximation of f ∈ L2,κ(ℝd) by a subspace Eκ2(σ) (SEκ2(σ)), which is a subspace of entire functions of exponential type (spherical exponential type) at most σ. Here L2,κ(ℝd) denotes the space of all d-variate functions f endowed with the L2-norm with the weight \(v_\kappa (x) = \prod\nolimits_{\xi \in R_ + } {|(\xi ,x)|^{2\kappa (\xi )} } \), which is defined by a positive subsystem R+ of a finite root system R ⊂ ℝd and a function κ(ξ): R → ℝ+ invariant under the reflection group G(R) generated by R. In the case G(R) = ℤ2d, we get some exact results. Moreover, the deviation of best approximation by the subspace Eκ2(σ) (SEκ2(σ)) of some class of the smooth functions in the space L2,κ(ℝd) is obtained.

De Branges’ theorem on approximation problems of Bernstein type

AbstractThe Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted ${C}_{0} $-space on the real line. A theorem of de Branges characterizes non-density by existence of an entire function of Krein class being related with the weight in a certain way. An analogous result holds true for weighted sup-norm approximation by entire functions of exponential type at most $\tau $ and bounded on the real axis ($\tau \gt 0$ fixed).We consider approximation in weighted ${C}_{0} $-spaces by functions belonging to a prescribed subspace of entire functions which is solely assumed to be invariant under division of zeros and passing from $F(z)$ to $ \overline{F( \overline{z} )} $, and establish the precise analogue of de Branges’ theorem. For the proof we follow the lines of de Branges’ original proof, and employ some results of Pitt.

Weighted -Integral Representations of -Functions in

For -functions , given in the complex space , integral representations of the form are obtained. Here, is the orthogonal projector of the space onto its subspace of entire functions and the integral operator appears by means of explicitly constructed kernel Φ which is investigated in detail.

Density of Polynomial Elements in Invariant Subspaces of Entire Functions of Exponential Type

Translation invariant subspaces in the space of entire functions of exponential type of several complex variables are considered and conditions are presented under which the set of exponential polynomials is dense in such a space. In particular, necessary and sufficient conditions for the polynomial elements to be dense are obtained.

Estimates for the ∂̄-Neumann operator in weighted Hilbert spaces

Estimates for the ∂ ¯ \bar \partial operator are used to derive estimates for the Neumann operator in weighted Hilbert spaces. The technique is similar to that used to prove regularity of solutions of elliptic partial differential equations. A priori estimates are first obtained for smooth compactly supported forms and these estimates are then extended by suitable approximation results. These estimates are applied to give new bounds for the reproducing kernels in the subspaces of entire functions.

Subspaces of entire functions of exponential type which are invariant with respect to shifts

Subspaces of entire functions of exponential type which are invariant with respect to shifts