UDC 518:517.392

ESTIMATION OF A LOWER BOUND FOR THE ERROR OF NUMERICAL INTEGRATION FORMULAS ON A CLASS OF FUNCTIONS

V. T. KALANDA, V. K. YASEL’SKII

(Submitted by the scientific seminar of the Faculty of Physics and Technology)

If a function of \(n\) variables admits an expansion in a Fourier series

\[ f(x_1,\ldots,x_n)= \sum_{m_1,\ldots,m_n=-\infty}^{\infty} C(m_1,\ldots,m_n)e^{2\pi i(m_1x_1+\cdots+m_nx_n)} \tag{1} \]

and for the coefficients of the expansion \(C(m_1,\ldots,m_n)\) the inequality

\[ C(m_1,\ldots,m_n)\le e^{-h(|m_1|+\cdots+|m_n|)} \tag{2} \]

holds, then the function \(f(x_1,\ldots,x_n)\) belongs to the class of functions \(A_n(h)\) [1].

A particular case of the expansion (1), having great practical application in solving physical problems in a cylindrical coordinate system, is the expansion of a function of one variable in a sine series

\[ f(x)=\sum_{m=1}^{\infty} b_m \sin 2\pi m x. \tag{3} \]

Since in the general case, for a function \(f(x)\) admitting an expansion in a Fourier series, the equality [2]

\[ f(x)=\frac{1}{2}a_0+\sum_{m=1}^{\infty} a_m\cos 2\pi m x+b_m\sin 2\pi m x, \tag{4} \]

holds, where

\[ C_m=\frac{1}{2}(a_m-i b_m),\qquad b_0=0, \]

the condition for a function admitting the expansion (3) to belong to the class \(A_1(h)\) can be written in the form

\[ \left|\frac{b_m}{2}\right|\le e^{-hm}. \tag{5} \]

The purpose of the present work is to estimate a lower bound for the error of quadrature formulas on the class of functions \(A_1(h)\) admitting expansion in a Fourier series according to formula (3).

In deriving expressions characterizing the lower error of formulas of numerical integration on the class of functions described by relations (3), (5), one may use the results of [1].

Let \(N\) points \(x_j\) \((j=1,2,\ldots,N)\) be given on the unit interval \([0,1]\). Then, whatever the method of approximate computation of

\[ \int_0^1 f(x)\,dx \]

from the values of the function \(f(x)\) taken at \(N\) arbitrary points \(x_j\) of the interval \([0,1]\), in the class \(A_1(h)\) there will be found a function for which the error arising under this method of computing the integral will be no less than \(e^{-h(N+1)}\).

Proof. Denote by \(\sigma(M)\) the set of integer points \(m\) in the domain \(m \leq M\), and by \(S(M)\) their number. Let \(M\) be so large that \(S(M) \geq N+1\). On the other hand, the number of points \(S(M)\) in the domain \(\sigma(M)\) is equal to \(M\) \((m=1, m=2,\ldots, m=M)\). Then, in order to satisfy the inequality \(S(M) \geq N+1\), \(M\) must be greater than or equal to \(N+1\). Take \(M=N+1\).

Construct a polynomial of the form

\[ T_0(x)=\sum_{m\in\sigma(M)} b_m^0 \sin 2\pi mx \not\equiv 0 \tag{6} \]

and require that \(T_0(x_j)=0\) \((j=1,2,\ldots,N)\). To determine its Fourier coefficients we obtain a homogeneous system of linear equations with the number of unknowns greater than the number of equations (the number of equations is \(N\), the number of unknowns is \(N+1\)). Therefore such polynomials exist.

In order to ensure that the polynomial under consideration belongs to the class \(A_1(h)\), transform it to the form

\[ T_1(x)=\frac{2e^{-hM}\sin 2\pi m'x}{b_{m'}^0}\,T_0, \tag{7} \]

where

\[ |b_{m'}^0|=\max_m |b_m^0|. \]

It is obvious that

\[ T_1\in A_1(h)\quad \text{and}\quad T_1(x_j)=0, \tag{8} \]

\((j=1,\ldots,N)\). Consequently, the integral value computed for the polynomial \(T_1(x)\) by using quadrature formulas with nodes \(x_j\) \((j=1,2,\ldots,N)\) is equal to zero.

But, integrating (7), we obtain

\[ \int_0^1 T_1(x)\,dx=e^{-h(N+1)}. \tag{9} \]

Consequently,

\[ \int_0^1 T_1(x)\,dx-\sum_{j=1}^N C_jT_1(x_j)=e^{-h(N+1)}. \tag{10} \]

Thus, the lower bound of numerical integration formulas on the class of functions described by relations (3), (5) can be estimated by the quantity

\[ R_n \leq e^{-h(N+1)}. \tag{11} \]

The case of expanding a function in the series

\[ f(x)=\sum_{m=1}^{\infty} B_m \sin \pi mx \tag{12} \]

is also of practical interest. This case reduces to the one already considered by introducing the new variable \(t=x/2\).

By means of the easily proved equality

\[ \int_{0}^{1} f(x)\,dx = 2 \int_{0}^{0.5} f(2t)\,dt \tag{13} \]

the conditions for the membership of a function admitting the expansion (12) in the class of functions \(A_1(h)\) are obtained in the form

\[ |B_m| \leqslant e^{-hm}. \tag{14} \]

Since in this case integration of the function is carried out on the interval \([0,1/2]\), it is easy to show that the lower bound for the error of numerical integration is half as large as on the full interval. Then, taking (4) into account, we obtain

\[ R'_h \leqslant \frac{1}{4} e^{-h(N+1)} . \tag{15} \]

It should be noted that if the amplitude of the first harmonic of the expansion (3), (12) is maximal, while the amplitudes of the remaining harmonics decrease approximately according to an exponential law, then correcting factors must be introduced for the value of the lower bound of the error of the quadrature formulas (11), (15).

Let the coefficients of the expansions (3), (12) decrease according to the laws

\[ |b_m| \leqslant |b_1| e^{-h(m-1)}, \tag{16} \]

\[ |B_m| \leqslant |B_1| e^{-h(m-1)} . \tag{17} \]

This means that the correcting factors introduced into the restrictions (5), (14) are

\[ \left|\frac{b_1}{2}\right| e^{h}, \tag{18} \]

\[ |B_1| e^{h} \tag{19} \]

respectively.

In this case, the values of the lower bound of the error (11) and (15) are also multiplied by the factors (18) and (19). That is, the final result may be represented, respectively, in the form

\[ R_h \leqslant \frac{1}{2} |b_1| e^{-hN}, \tag{20} \]

\[ R'_h \leqslant \frac{1}{4} |B_1| e^{-hN}. \tag{21} \]

REFERENCES

1. N. F. Sharygin. A lower estimate of the error of quadrature formulas on classes of functions. Journal of Computational Mathematics and Mathematical Physics, 1963, vol. 3, No. 2.

2. I. S. Gradshteyn and I. M. Ryzhik. Tables of Integrals, Sums, Series, and Products. Moscow, “Nauka,” 1971.