By Silvanus P. Thompson
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Extra resources for Analysis leicht gemacht
The e r r o r bounds for the trapezoid formula a r e studied in many papers. See (4) (5), (21), (24)/ and others. We will now analyse the proF blem of the choice of the quadrature formula according to the information we mentioned previously. In our considerations we shall confine the class of possible formulae to the linear one, The choice of the quadrature formula means, in our case, to determine of the sequence of linear functionals I in the form n with the requirement that Jn(f) + J(f) (weak) for all functions f(x) of the given class of functions.
Then Kcos x 6n =IJ(f) accordance be seen. This error (19), (20) - for the principal new computationof problem the ~ ~ 1( f ) between the estimation . 1) used . In Fourier . Similar coefficients when we want to compute simulta- .. j = 1,. k Obviously the simplest way is to P. J compute these values independently. There is a question if it is possible J neously the values to gain something when I will we will show it compute both we make the computations simultaneously. in the simplest case. values J 0 and J 1 Let us assume that simultaneously.
Nn ; n = 1,2,. n b . 1,. . 4) . , 0 and to the sequence f ( n ) will be called solution if lim sup ~4 o 1 & and C does We will s p e a k about not (n) s u p \ ;("Ixin)( 4 c ns 16 ai L , i=-pn' ' " Nn (irP 14, depend on n . Bs -solution if So = inf s . The investigations of concrete given processes have been done in the previously mentioned way in many cases. g. (12), (27)) (32)) (41)) (44)) and others. I shall now give some examples previous definitions. 1 change will and Let further 4 . 2 .