F A S C I C U L I M A T H E M A T I C I
F A S C I C U L I M A T H E M A T I C I Nr 35 2005 Ali Aral ON A GENERALIZED λ-GAUSS WEIERSTRASS SINGULAR INTEGRAL Abstract: In the present note we consider the integral ∫ Wλ s c(n, λ, s) (f; x, α) := f (x + t) exp (− ||t|| s |λ| α |λ| λ /4α s) n dt, R n where x ∈ R n , s > 0, α > 0 and λ 1 , λ 2 , · · · , λ n are positive numbers with |λ| = λ 1 + λ 2 + · · · + λ n . The integrals Wλ s (f; x, α) will be called a generalized λ−Gauss Weierstrass singular integral. We study some approximation properties of this integral in the nonisotropic exponential weighted space. Key words: λ−Gauss Weierstrass integral, exponential weighted space. 1. Introduction Let λ 1 , λ 2 , · · · , λ n be positive numbers with |λ| = λ 1 + λ 2 + · · · + λ n and ||x|| λ = (|x 1 | 1 λ 1 + . . . + |x n | 1 λn ) |λ| n , x ∈ R n . The expression ||x − y|| λ , x, y ∈ R n is called the nonisotropic distance between the x and y. It can be seen that nonisotropic distance become ordinary Euclidean distance |x − y| for λ j = 1 2 , j = 1, 2, . . . n. Nonisotropic distance has the following property. Using the inequality (a + b) m ≤ 2 m (a m + b m ) , m > 1 we obtain (1) ||x − y|| λ ≤ M λ (||x|| λ + ||y|| λ ) , where M λ = 2 (1+ 1 λ min ) |λ| n and λ min = min(λ 1 , λ 2 , . . . λ n ). This integral operators with the kernels depending on nonisotropic distance have important application in theory of partial differential equations and imbedding theorems. ( , ) .