FEJÉR KERNEL: ITS ASSOCIATED POLYNOMIALS

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1 Boletín de Matemáticas Nueva Serie, Volumen XV No. (008), pp EJÉR KERNEL: ITS ASSOCIATE POLYNOMIALS MARTHA GALAZ-LARIOS (*) RICARO GARCÍA-OLIVO (**) JOSÉ LUIS LÓPEZ-BONILLA (***) Abstract. We show that the ejér kernel generates the fifth-kind Chebyshev polynomials. Palabras claves. Kernels in ourier series, Chebyshev polynomials. 000 Mathematics Subject Classification: 4A16, 199. Resumen. Se demuestra que el núcleo de ejér genera polinomios de Chebyshev de quinto orden. Key words and phrases. Núcleos en series de ourier, polinomios de Chebyshev. 1. Introduction In the original approach to ourier series, it is convenient to consider the following partial sums for the interval [, π]: (1) f n (y) = 1 a 0 + a 1 cos y + + a n cos(ny)+ + b 1 sin(y) + + b n sin(ny), assuming for a r, b r the values: () a r = 1 π π f(t) cos(rt)dt, b r = 1 π f(t) sin(rt)dt, and investigate what happens if n increases to infinity. rom (1) and () we obtain: (3) f n (y) = f(t) K n (t y)dt, (*) Martha Galaz-Larios. Instituto Politécnico Nacional, México. (**) Ricardo García-Olivo. Instituto Politécnico Nacional, México. (***) José Luis López-Bonilla. SEPI-ESIME-Zacatenco, Instituto Politécnico Nacional. Edif. Z-4, 3er. Piso, Col. Lindavista CP 07738, México. jlopezb@ipn.mx. 14

2 EJÉR KERNEL: ITS ASSOCIATE POLYNOMIALS 15 with the irichlet kernel [1-3]: (4) K n (t y) = 1 sin [( ) ] n + 1 (t y) π sin ( ) t y. Then we hope that with n increasing to infinity, f n (y) approaches f(y) with an error which can be made arbitrarily small. This requires a very strong focusing power of K n (t y), that is, we would like to have the strict property: (5) lim n K n (t y) = δ(t y), however, (4) simulates a irac delta only until certain approximation, then the convergence: (6) lim n f n(y) = f(y) has to be restricted to a definite class of functions f(y) which are conveniently smooth to counteract the insufficient focusing power of K n (t y); the corresponding restrictions on f(y) are the known irichlet conditions [1-3] for infinite convergent ourier series. rom (4) we see that K n (θ) is an even function, then here we consider it for θ [0, π]: (7) K n (θ) = 1 sin ( n + ) 1 θ π sin ( ), θ thus (8) K 0 (θ) = 1 π, K 1(θ) = 1 π (1 + cos θ), K (θ) = 1 π ( 1 + cos θ + 4 cos θ), K 3 (θ) = 1 π ( 1 4 cos θ + 4 cos θ + 8 cos 3 θ), etc. then it is natural to introduce the polynomials: (9) W n (x) = W n (cos θ) = π K n (θ), x [ 1, 1], which were named fourth-kind Chebyshev polynomials by Gautschi [4,5]. Therefore, see ig. 1: (10) W 0 (x) = 1, W 1 (x) = x + 1, W (x) = 4x + x 1, W 3 (x) = 8x 3 + 4x 4x 1, W 4 (x) = 16x 4 + 8x 3 1x 4x + 1, etc. In the next Section we exhibit a set of associated polynomials to ejér kernel [1-3].

3 16 M. GALAZ-LARIOS, R. GARCÍA-OLIVO Y J. LÓPEZ-BONILLA igure 1. Some fourth-kind Chebyshev polynomials. Chebyshev-ejér polynomials ejér [5] invented a new method of summing the ourier series by which he greatly extended the validity of the series. Using the arithmetic means of the partial sums (1), instead of the f n (y) themselves, he could sum series which were divergent. The only condition the function still has to satisfy is the natural restriction that f(y) shall be absolutely integrable. Then, in the ejér approach we construct the sequence: (11) g 1 (y) = f 0 (y), g (y) = 1 [(f 0(y) + f 1 (y)], g 3 (y) = 1 3 [(f 0(y) + f 1 (y) + f (y)],..., g n (y) = 1 n [(f 0(y) + f 1 (y) + + f n 1 (y)], accepting the expressions (1) and (), therefore: (1) g n (y) = f(t) K n (t y)dt, thus we see that ejér results come about by the fact that his method is related with the following kernel [1-3]: (13) K n (t y) = 1 πn sin [ n (t y)], sin t y which possesses a strong focusing power, that is, it satisfies (5), then a f(y) absolutely integrable in [, π] guarantees the convergence of g n (y) towards f(y).

4 EJÉR KERNEL: ITS ASSOCIATE POLYNOMIALS 17 Now we consider the ejér kernel: (14) K n (θ) = 1 sin ( ) n θ, θ [0, π], πn that is: K 0 sin θ (θ) = 0, K 1 (θ) = 1 π, K (θ) = 1 π (1 + cos θ), (15) K 3 (θ) = 1 6π (1 + 4 cos θ + 4 cos θ), etc., then it is natural the introduction of the functions: (16) Wn (x) = W n (cos θ) = π n + 1 K n+1(θ), x [ 1, 1], that we name fifth-kind Chebyshev polynomials, which are not explicitly in the literature. Therefore: (17) W 0 (x) = 1, W1 (x) = 1 (x + 1), ( W (x) = 1 9 4x + 4x + 1 ), W 3 (x) = 1 ( x 3 + x ), W4 (x) = 1 5 thus W n (1) = 1, see the following igure: ( 16x x 3 4x 4x + 1 ), etc. igure. Some fifth-kind Chebyshev polynomials which are solutions of the non-homogeneous differential equation: [ (18) (1 x) (1 x ) W n (3x + ) W ] n + (n + 1) Wn + x W n = 1.

5 18 M. GALAZ-LARIOS, R. GARCÍA-OLIVO Y J. LÓPEZ-BONILLA In other paper we will study topics as recurrence, Rodrigues formula, interpolation properties, orthonormality, generating function, etc., for fifth-kind Chebyshev polynomials introduced in this work. References [1] C. Lanczos, Applied analysis, over NY (1988) [] R. Rodrigues del Río and E. Zuazua, Cubo Mat. Educ. 5, No. (003) [3] C. Lanczos, iscourse on ourier series, Oliver & Boyd, Edinburgh (1966) [4] W. Gautschi, J. Comp. Appl. Math. 43 (199) [5] J. C. Mason and. H. Handscomb, Chebyshev polynomials, Chapman & Hall CRC Press (00)