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Applied methods of the theory of random functions, Edition: by Berry, J.; Haller, L.; Sveshnikov, Aram Aruti︠u︡novich

By Berry, J.; Haller, L.; Sveshnikov, Aram Aruti︠u︡novich

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Additional resources for Applied methods of the theory of random functions, Edition: [1st English ed.]

Example text

8) to where X(t) is the function on which the given operator acts, and p(t, tj) is a given function the form of which finally de­ termines the properties of the operator. In the particular case when the function p(t, fx) is a function of the difference of its arguments, that is, P(*,h) =P(*-h)> formula (8) reduces to the form t Y(t) = jpit-tJXtfJdh. 10) to The importance of operators of this form is due to the fact that the finding of the particular integral of a linear homo­ geneous equation the right-hand side of which is the func­ tion X(t) reduces to this.

On the other hand, t0 is arbitrary, con­ sequently the function p(t, t) must vanish identically. Simi­ larly it can be shown t h a t the terms outside the integral of all the derivatives of y^t) except the derivative of the n t h order must also vanish. 15) and for the successive derivatives of y^t) we have ' dPj yiW = t C d? -$■ p{t, hWh) &i> i = o, l, . . 16) x(t) + dt71-1 df ti=t j&pit' h)x(h) dt± After combining the integrals into one, the substitution of these relations in the original equations gives f T dn dn~i 1 ^r+%(*) ^ z r + • • • +«*(*) P(*> h)*(h) <*h+ x(t) = x{t).

2 )]}. 31) 56 BANDOM FUNCTIONS where the operators have been provided with the subscripts t± and t2 in order to indicate t h a t in the first case the operator acts with respect to the variable tx and in the second case with respect to the variable t2 and the asterisk above t h e operator shows t h a t if complex expressions occur in the mathematical expression for this operator, it is necessary to replace them by their complex conjugates. Making use of the linearity of the homogeneous operator L and t h e linearity of the operation of finding the mathematical expectation we obtain instead of (31) Ky(tltt2) t h a t is, = M{L?

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