Read e-book online Approximation of Functions of Several Variables and PDF

By Sergei Mihailovic Nikol’skii (auth.)

Show description

Read or Download Approximation of Functions of Several Variables and Imbedding Theorems PDF

Similar number systems books

The Fractional Laplacian - download pdf or read online

The fractional Laplacian, also referred to as the Riesz fractional by-product, describes an strange diffusion technique linked to random tours. The Fractional Laplacian explores functions of the fractional Laplacian in technology, engineering, and different parts the place long-range interactions and conceptual or actual particle jumps leading to an abnormal diffusive or conductive flux are encountered.

Additional resources for Approximation of Functions of Several Variables and Imbedding Theorems

Sample text

4. Averaging of Functions According to Sobolev This process is due to Sobolev [4J. Denote by a8 = {lxl ~ s}, a1 = a, the ball in IR = IRn of radius s and with center at the origin. Suppose that 1p(t) is an infinitely differentiable even nonnegative function of one variable t(-oo < t <(0), equal to zero for It I ;;S 1 and such that (1 ) f 1p(lxj) dx = 1, 1"'1~1 where "n is the volume of the unit (n - i)-dimensional sphere in IRn. 28 1. Preparatory information I As "P we may choose the function °~ It I < 1, ~ el'~l ' An "P(t) = 0, 1 ~ Itl, where the constant An is chosen so that condition (1) is satisfied.

Fm (for k j = the corresponding sum is extended only to Vi = 0); or - is entered according as k j is positive or negative. Transform a function of period 2n of the type (see 1. 2. (7)) ° + (2) f(x) = ~ Ckeik3J = ~ Ok (f) E Lp* < P< (1 (0) k by means of the Marcinkiewicz multipliers Ak: 0) F(x) = L: }'kckeikaJ = L: ok(F) . k Then F E L: and there exists a constant cp depending only on p such that (4) . Proof. We restrict ourselve;;; to the case n = 2. Moreover, we will suppose that in (2) the series are extended only over k ;::;; 0, which does not affect the generality.

J K(S) I(). - = S) dg = K@ I()· - S) d; = K *1. Since IE 5, then the integral in '1'1 in the top line is a function of u, belonging to 5 c L. ' The product belongs to L, so that after multiplication by ei "''' and integration with respect to u we obtain a /'.. continuous function fcJ of x. The change of the order of integration relative to sand), is legitimate, since K, I E L. Here we use Fubini's theorem. 5. Generalized functions The integral in the next-to last term of these relations is called the convolution of K and I.

Download PDF sample

Rated 4.39 of 5 – based on 22 votes