Representation and approximation of functions via (0,2)-interpolation. (English) Zbl 0614.41001

Gervais, R.; Rahman, Q. I.; Schmeisser, G.

J. Approximation Theory 50, 89-110 (1987).

For \(f\in C^ 2({\mathbb{R}})\) we introduce an interpolation operator \(R(f;z):=\sum^{\infty}_{n=-\infty}(f(n)A_ n(z)+f''(n)B_ n(z))\) with the following properties: R(f;.) is an entire function of exponential type \(2\pi\), \(R(f;n)=f(n)\), \(R''(f;n)=f''(n)\) for all \(n\in {\mathbb{Z}}\); \(R'(f;0)=R\prime''(f;0)=0\). We establish essentially best possible conditions under which an entire function f of exponential type \(\tau <2\pi\) or \(\tau =2\pi\) may be represented as \(f(z)=R(f;z)+c_ 1(f) \sin \pi z+c_ 2(f) \sin 2\pi z\) with explicitly given constants \(c_ 1(f)\), \(c_ 2(f)\) depending on f’(0) and f”’(0). These results are used for approximation of a continuous function by a sequence of (0,2)- interpolating entire functions of exponential type. Defining \[ R_{\tau}(f;\beta,z)\quad:=\quad \sum^{\infty}_{n=- \infty}(f(\frac{n\pi}{\tau})A_ n(\frac{\tau}{\pi}z)+(\frac{\pi}{\tau})^ 2\beta_{\tau n}B_ n(\frac{\tau}{\pi}z)) \] we have an entire function of exponential type \(2\pi\) such that \(R_{\tau}(f;\beta,n\pi /\tau)=f(n\pi /\tau)\) and \(R''_{\tau}(f;\beta,n\pi /\tau)=\beta_{\tau n}\) for all \(n\in {\mathbb{Z}}\). It is then shown that \(\lim_{\tau \to \infty}R_{\tau}(f;\beta,x)=f(x)\) uniformly on every compact subset of \({\mathbb{R}}\) provided f is a bounded continuous function on \({\mathbb{R}}\) such that \(| f(x+h)-2f(x)+f(x-h)| =o(h)\) as \(h\to 0\) uniformly for \(x\in {\mathbb{R}}\) and the ”free parameters” \(\beta_{\tau n}\) satisfy \(\sup_{n}| \beta_{\tau n}| =o(\tau)\) as \(\tau\to \infty\). These results are analogous to the work of P. Turán with J. Balász and J. Surányi, and of G. Freud on (0,2)-interpolation by algebraic polynomials. Furthermore they extend a result of O. Kiš on (0,2)- interpolation of periodic functions by trigonometric polynomials.

MSC:

41A05	Interpolation in approximation theory
41A30	Approximation by other special function classes
30D10	Representations of entire functions of one complex variable by series and integrals
30D15	Special classes of entire functions of one complex variable and growth estimates

Keywords:

interpolation operator; (0,2)-interpolation by algebraic polynomials

References:

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