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Representation and approximation of functions via (0,2)-interpolation. (English) Zbl 0614.41001
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
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References:
 [1] Balász, J; Turán, P, Notes on interpolation. II. explicit formulae, Acta math. acad. sci. hungar., 8, 201-215, (1957) · Zbl 0078.05401 [2] Balász, J; Turán, P, Notes on interpolation. III. convergence, Acta math. acad. sci. hungar., 9, 195-214, (1958) · Zbl 0085.05104 [3] Balász, J; Turán, P, Notes on interpolation. IV. inequalities, Acta math. acad. sci. hungar., 9, 243-258, (1958) · Zbl 0085.05202 [4] Boas, R.P, Entire functions, (1954), Academic Press New York [5] Erdös, P; Turán, P, On the role of the Lebesgue function in the theory of the Lagrange interpolation, Acta math. acad. sci. hungar., 6, 47-66, (1955) · Zbl 0064.30101 [6] Freud, G, Bemerkungen über die konvergenz eines interpolationsverfahrens von P. Turán, Acta math. acad. sci. hungar., 9, 337-341, (1958) · Zbl 0085.05201 [7] Freud, G; Scheick, J.T, Polynomial approximation on the real line, Studia sci. math. hungar., 6, 23-25, (1971) · Zbl 0226.41001 [8] Freud, G; Szabados, J, Rational approximation on the whole real axis, Studia sci. math. hungar., 3, 201-209, (1968) · Zbl 0174.35401 [9] Gervais, R; Rahman, Q.I, An extension of Carlson’s theorem for entire functions of exponential type, Trans. amer. math. soc., 235, 387-394, (1978) · Zbl 0373.30025 [10] Gervais, R; Rahman, Q.I; Schmeisser, G, Approximation by (0, 2)-interpolating entire functions of exponential type, J. math. anal. appl., 82, 184-199, (1981) · Zbl 0469.30027 [11] Kiš, O, On trigonometric interpolation, Acta math. acad. sci. hungar., 11, 255-276, (1960), [Russian] · Zbl 0103.28703 [12] Surányi, J; Turán, P, Notes on interpolation. I. on some interpolatorical properties of the ultraspherical polynomials, Acta math. acad. sci. hungar., 6, 67-79, (1955) · Zbl 0064.30005 [13] Timan, A.F, Theory of approximation of functions of a real variable, (1963), Pergamon New York · Zbl 0117.29001
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