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We show that the Fejer kernel generates the fth-kind Chebyshev polynomials.

Tipo de documento: Artículo - Article

Palabras clave: Kernels in Fourier series, Chebyshev polynomials.





Source: http://www.bdigital.unal.edu.co


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Boletı́n de Matemáticas Nueva Serie, Volumen XV No.
2 (2008), pp.
124–128 FEJÉR KERNEL: ITS ASSOCIATED POLYNOMIALS MARTHA GALAZ-LARIOS (*) RICARDO GARCÍA-OLIVO (**) JOSÉ LUIS LÓPEZ-BONILLA (***) Abstract.
We show that the Fejér kernel generates the fifth-kind Chebyshev polynomials. Palabras claves.
Kernels in Fourier series, Chebyshev polynomials. 2000 Mathematics Subject Classification: 42A16, 12D99. Resumen.
Se demuestra que el núcleo de Fejér genera polinomios de Chebyshev de quinto orden. Key words and phrases.
Núcleos en series de Fourier, polinomios de Chebyshev. 1.
Introduction In the original approach to Fourier series, it is convenient to consider the following partial sums for the interval [−π, π]: (1) fn (y) = 12 a0 a1 cos y · · · an cos(ny) b1 sin(y) · · · bn sin(ny), assuming for ar , br the values: Rπ (2) ar = π1 −π f (t) cos(rt)dt, br = 1 π Rπ −π f (t) sin(rt)dt, and investigate what happens if n increases to infinity.
From (1) and (2) we obtain: Z π (3) fn (y) = f (t) Kn (t − y)dt, D −π (*) Martha Galaz-Larios.
Instituto Politécnico Nacional, México DF. (**) Ricardo Garcı́a-Olivo.
Instituto Politécnico Nacional, México DF. (***) José Luis López-Bonilla.
SEPI-ESIME-Zacatenco, Instituto Politécnico Nacional. Edif.
Z-4, 3er.
Piso, Col.
Lindavista CP 07738, México DF. E-mail: jlopezb@ipn.mx. 124 FEJÉR KERNEL: ITS ASSOCIATED POLYNOMIALS 125 with the Dirichlet kernel [1-3]: 1 sin Kn (t − y) = 2π D (4)  n sin 1 2 (t −  t−y 2   y) . Then we hope that with n increasing to infinity, fn (y) approaches f (y) with an error which can be made arbitrarily small.
This requires a very strong focusing power of Kn (t − y), that is, we would like to have the strict property: D (5) lim Kn (t n→∞ D − y) = δ(t − y), however, (4) simulates a Dirac delta only until certain approximation, then the convergence: (6) lim fn (y) = f (y) n→∞ has...






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