Approximate integration of modified Riesz potentials Текст научной статьи по специальности «Математика»

УДК 517+518.392

Approximate Integration of Modified Riesz Potentials

Maria I. Medvedeva* Nikolay N. Osipov^

Institute of Space and Information Technology, Siberian Federal University, Kirensky, 26, Krasnoyarsk, 660074

Russia

Received 26.07.2013, received in revised form 09.08.2013, accepted 14.10.2013 The estimates for the errors of quadrature formulas are deduced in the case of integration of potentials such as the Feller potential, which is a modification of the Riesz potential.

Keywords: quadrature formula, error functional, sequence of functionals, approximate calculation, potential.

Introduction

The subject of this article was first studied in the papers [1,2] where the authors considered functionals with a boundary layer on the space of functions that can be represented as the Riesz potentials. It was shown that such functionals have the best power rate of strong convergence among functionals with arbitrary nodes and coefficients as the number of nodes N increases indefinitely. In this paper we prove a similar result for functionals on functions that can be represented as the Feller potentials. In this case it is possible to show that the sequences of functionals with a boundary layer have asymptotically the best rate of convergence among the formulas with arbitrary nodes and coefficients.

The main results of this paper are Theorem 1 and 3. In Theorem 1 we derive upper bounds for the error functionals and lower bounds in Theorem 3. Lemmas 1, 2 and Theorem 2 are of auxiliary character.

Assume that a, b, p, q, a, and N are real numbers such that a < b, 1 < p < to, q = p/(p — 1), 0 < a < 1, N is a positive integer. Suppose that ap > 1 (in particular, this condition guarantees the inclusion of spaces considered here in the space C [a, b] of continuous functions on [a, b]).

Let lN be the error functional of the quadrature formula

b N

f (x) dx — £ cN f (xN), (1)

k= 1

where xN (xN G [a, b], 1 < k < N) and cN stand for the nodes and coefficients of the formula. In what follows, we shall assume that formula (1) is exact for constants.

Recall that the function f (x) satisfies the Holder condition on [a, b] with the exponent A, 0 < A < 1, if

|f(xi) — f(x2)| < K|xi — x2|A

for all x1, x2 G [a, b], where K is a positive constant depending on the function f (x) (for A = 1 we have the Lipschitz condition). For a fixed A, denote the set of such functions by HA([a, b]).

* [email protected] [email protected] © Siberian Federal University. All rights reserved

)= / J a

= / B*T -(2)

Fix the number A > a. For the given a let

rb sgn (x — t)p(t) dt |x -t|

be the fractional integration operator. The equation

(B»(x) = f (x)

is the special case (u = v = 1) of the general equation (see [3, p. 455, (30.79)])

«(/0»(x) + v(I£_p)(x) = f (x),

where u, v are constants and

i

(ia+p)(x)= (x—t)a-ip(t) dt, (3)

J a

r b

(Ib-p)(x)= (t — x)a-1p(t) dt (4)

J x

for a < x < b.

Denote by Ba(Lp(a, b)) the set of functions f (x) on (a, b) of the form (2) with f (x) £ Lp(a, b). As usual, Lp(a, b) is the linear space of measurable functions f : [a, b] ^ R with the norm

1/p

II Pll Lp(a,b) =[ja |P(x)|P <

Definition 1. The numbers x and y from the segment [a, b] are called dual if x + y = a + b. Definition 2. The system of nodes 0 = {xN : 1 ^ k ^ N} of the quadrature formula

r b N

/ f (x) dx cN f (xN)

Ja k=1

is called symmetric if for each x £ 0 its dual y = a + b — x £ 0 too.

Definition 3. A quadrature formula with a symmetric system of nodes is called symmetric if its coefficients at dual nodes are equal.

Theorem 1. There exist sequences of the error functionals for symmetric quadrature formulas {lN} (1) such that

l(lN,f)| < AN-a\\pf \\Lp{afi), where f = Baff for ff £ Lp (a, b), and A is a constant.

Proof. Let the sequence of functionals {lN} satisfy the conditions of the theorem for f (x) of the form (3), i.e., f (x) = (1%+ ff )(x). The existence of such {lN} has been proved in [4] (in the case when a > 0). Then such a sequence of functionals satisfies the theorem for f (x) of the form (4), i.e., f (x) = (If—ff )(x). To see that, replace in the integral (4) the variable t by a + bt

/• b /• a+b-x

/ (t — x)a-1f(t)dt = (a + b — x — t)a-1f(a + b — t)dt

xa

and take into account the symmetry of the quadrature formulas corresponding to lN, i.e.,

fb A ryN

/ dx(Ib-f)(x) cN (yN — t)a-1f(a + b — t)dt,

aa

where y^ = a + b — xN is the dual point for xN. □

Remark. The statement of the theorem is satisfied by a number of well-known sequences of quadrature formulas such as the (complicated) Gregory formula (see [4]).

Consider the integral equation

l ^x-xparVf (t) dt = f (x), 0 < a < 1, (5)

which is an important special case of the generalized Abel equation on a segment. Its left hand side is called the Feller potential. The equation (5) is solvable for any right hand side from HA ([a, b]) and has a nontrivial solution of the form

= c 1 an d fb f (t) dt cos2 an

Vf (x) = (x — a)(1+a)/2(b — x)(1+a)/2 CtgT dxja |x — t|a ~ 2n2 X

d ( f V , fb sgn(x — y) dy

x A(t - a)(b - t)]1-a/2/(t) dt /6

dx V ./a ./a

(t - y)|x - y|a[(y - a)(b - y)]1-a/2

—i[(i—„>,—()]<1-a>/2f(i) ^ (t—ynx—^—;>* y,]^-.,/^ (,,

with an arbitrary constant c (see [3, p. 457-459]). Further on we shall take c = 0.

Lemma 1. If f (x) G HA([a, b]) satisfies f (a) = f (b) = 0, then the corresponding function Vf (x) from (5) can be represented in the form

( ) =_c__+ sin an d [b f (t) dt

Vf (x) (x — a)(1+a)/2(b — x)(1+a)/2 + 2n dxja |x — t|a ' (7)

Proof. Let f (x) G HA([a, b]) for some A > a. Using the equalities (see [5], c. 530-531)

fb_sgn (x — y) dy_=_n ctg an_

Ja (t — y)|x — y|a[(y — a)(b — y)]1-a/2 = |t — x|a[(t — a)(b — t)]1-a/2 , fb sgn(x — y) dy ( —1)n Ctg ^jpn

J a (t — y)|x — y|a[(y — a)(b — y)](1-a)/2 |x — t|a[(t — a)(b — t)](1-a)/2'

0 < a < 1, a < x < b, a < t < b, we calculate the interior integrals in the second and third summands of (6).

Then the expression (6) turns into

c 1 an d fb f (t) dt

Vf (x)=, „W1+a)/2^ ^(1+a)/2 + ^ Ctg ^T XT

(x - a)(1+a)/2(b - x)(1+«)/2 2n & 2 dxja |x - t|a cos2 Oif ^^an f6 f(t) dt 1 + a_ f6 f(t) dt

n ctg — -.-— + n ctg —— n

2n2 dx y 2 Ja |x - t|a 2 Ja |x -1|

c 1 an d f6 f(t) dt

+ —ctg — — 1 f ( )

(x - a)(1+a)/2(b - x)(1+a)/2 2n 0 2 dxja |x - t|

1 2 an / an an \ d i'b f (t) dt - —- cos2 — ctg — - tg — — ' w -

2n 2 V 2 2 / dx Ja |x - t|a

sin an d fb f (t) dt

+

(x - a)(1+a)/2(b - x)(1+°)/2 2n dxja |x - t|a'

□

a

Put h = h(N) = (b — a)/(2N), Q(N) = |J (a + hi, a + hi + h), where a(N) is the family

iEa(N)

of integers i G [0, 2N — 1] such that the intervals (a + hi, a + hi + h) do not contain the nodes xN,..., xN of formula (1).

Let the function (x) belong to C([0,2N]), vanish outside the segment [0, 2N], and satisfy

^h(x) =

^(x - i), i G a(N), x G (i,i + 1);

0, i G &(N),

where ^(x) G C([0,1]), ^(0) = ^(1) = 0,

^(x) dx = 0,

and is equal to zero outside [0,1].

In what follows, the symbol k with different indices denotes positive constants independent of h, a, b. Let

rd2 h(s) ds

Bd!,d2 (z)

dl |z -

where di, d2 and z some numbers such that 0 ^ di < d2 and 0 ^ z ^ 2N. Lemma 2. There exists a constant k1 independent of z such that

\Bo,2N (z)| < ki. Proof. In Lemma 18 of [4] it was proved that the integral

Bo,z (z) = r ■J 0

^h(s)ds (z - s)a

is bounded

Note that the functions

can be transformed to

|Bo,z(z)| <k2. (ia+v)(x)= [X(x - tr-iv(t) dt

J a

(8)

(Ib-v)(x) = i (t — x)°-l^(t) dt

J x

by a simple change of variables (see [3], p. 42)

QI°a+ = Ib-Q, QIb- = Q, (Qv)(x) = f(a + b — x). Thus, for the integral Bz,2N(z) there is a similar estimate, namely,

|Bz,2N (z)| =

r2N

'^h(s)ds

(s - z)

< k3.

The statement of lemma follows from this inequality, the relations (8) and the following equality

B (z)= r Ms)ds + f2N Bo'2N (z)=io ^^+L

'^h(s)ds (s - z)a .

□

i

0

Consider a continuously differentiable function g(x) G HA([0,1]) such that g(0) = g(1) = g'(0) = g'(1) = 0 and 1

/ g(x)dx > 0. Jo

Extend g(x) to the whole real axis by setting it equal to zero outside the interval [0,1] and denote

gh(x)

g - i) , i G <r(N), x G (a + hi, a + hi + h),

o, i G o-(N).

(9)

By construction, the continuously differentiable function gh(x) belongs to the space HA([a, b]).

Theorem 2. Let the function gh(x) be defined by formula (9). Then there exist functions vh(x) such that

, b

gh(x) = / |x - t|a-1^h(i) dt, 0 < a < 1,

J a

and

lKlk(a,b) < k4h-a,

(10) (11)

where k4 is a positive constant.

Proof. Since gh(x) belongs to the space HA([a, b]) and gh(a) = gh(b) = 0, the inversion formula (7) with f (x) = gh(x) holds. It is known (see [3, p. 43]) that if a function f is continuously differentiable on the segment [a, b], then

d f f (t) dt _ f (a)

dx Ja (x — t)a (x — a) d fb f (t) dt _ f (b)

dx (t — x)a (b — x)a

+

: f '(t) dt

(x — t)a f '(t) dt (t — x)a.

Then from the last equations and formulas (7), (10) with c = 0 and relations

(gh)'(x) =

h-1 g' (^TT — i) , i G <r(N), x G (a + hi, a + hi + h),

o, i G ^(n)

we infer that

^(x)| = k5

k5

(gh )'(t) dt

|x — t|a

h

a+ih+h g' (^ — i) dt ¿eT(N )J a+ih

E

E

/.¿+1

g'(r — i) dr

|x — hr — a|a

k5h

E

|x — t|a /"¿+1 g'(r — i) dr

Applying Lemma 2, with ^h(x) = g'(x) and z = (x — a)/h, to the right hand side of the expression above, we obtain that

bh(x)| < kehTa.

Theorem 2 is completely proved.

□

Theorem 3. For every sequence of functionals of the form (1) there exist a number k7 > 0 and functions Vf (x) G Lp(a, b) such that

|(1N,f)| >krN—a||^||Ma,b)(b — a)1/q+a.

b

b

1

a

a

x —a

— T

h

Proof. Let gh(x) be the function from formula (9). Since gh(xN) =0 (1 < k < N), we infer

(1N,9h)= gh(x) dx = gh(x) dx =

Ja JQ(N)

/a+ih+h

h g

Using (11) and (12), we find

— i^ dx = mesQ(N) J g(x)dx ^

(b — a)

i g(x)dx > 0. (12)

J 0

l(lN, gh)| > k8(b - a)1/qha\\(fg\\Lp{0,bb) > kgN-a(b - a)1/q+a\\^g||ip(0j6), where k8 and kg > 0 are some constants. Theorem 3 is proved.

□

Combining Theorems 1 and 3, we can formulate the following result: there exist sequences of points {xN)N=1 C [a, b], numbers {cNconstants A, B > 0, and functions of the form (2) such that for the error functionals (1) the following inequality holds

BN-a\\Vf \\Lp(o,b) (b - a)1/q+a < |(lN, f )| < AN-ayf \\Lp(o,b)(b - a)1/q+a, 0 < a < 1.

Thus, these sequences give the error functionals for quadrature formulas with the best rate of convergence to zero (on functions of the form (2)) as the number of nodes N increases indefinitely.

x — a

h

2

References

[1] M.I.Medvedeva, On the Order of Convergence of Quadrature Formulae on Functions in the Spaces of Riesz Potential, Journal of Siberian Federal University. Mathematics & Physics, 1(2008), no. 3, 296-307 (in Russian).

[2] M.I.Medvedeva, V.I.Polovinkin, Approximate calculations of Riesz integrals, Siberian Adv. Math., 3(2010), 180-190.

[3] S. G.Samko, A.A.Kilbas, O.I.Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Sci. Publ., London-New York, 1993.

[4] V.I.Polovinkin, Sequences of functionals with boundary layer in spaces of one-dimensional functions with Riemann-Liouville fractional derivatives, Siberian Adv. Math, 13(2003), no. 1, 32-54.

[5] A.P.Prudnikov, Yu.A.Brychkov, O.I.Marichev, Integrals and Series, Gordon and Breach Sci. Publ., New York, 1986.

Приближенное интегрирование модифицированных потенциалов Рисса

Мария И. Медведева Николай Н. Осипов

Выводятся оценки погрешностей квадратурных формул при интегрировании потенциалов типа потенциала Феллера, представляющих собой модификацию потенциалов Рисса.

Ключевые слова: квадратурная формула, функционал ошибки, последовательность функционалов, приближенное вычисление, потенциал.