By Prof. RNDr Karel Rektorys DrSc (auth.)

The impulse which resulted in the writing of the current ebook has emerged from my a long time of lecturing in detailed classes for chosen scholars on the collage of Civil Engineering of the Tech nical collage in Prague, from event received as manager and advisor to graduate students-engineers within the box of utilized arithmetic, and - final yet now not least - from widespread consultations with technicians in addition to with physicists who've requested for suggestion in overcoming problems encountered in fixing theoretical difficulties. even supposing a diversified mixture of difficulties of the main various nature was once usually in query, the issues mentioned during this publication stood forth because the such a lot necessary to this classification of experts. the numerous discussions i've got had gave upward thrust to issues on writing a publication which should still fill the really unlucky hole in our literature. The ebook is designed, within the first position, for experts within the fields of theoretical engineering and technological know-how. despite the fact that, it used to be my objective that the ebook can be of curiosity to mathematicians to boot. i've been good acutely aware what an ungrateful activity it can be to write down a e-book of the current sort, and what difficulties such an attempt can carry: Technicians and physicists at the one facet, and mathematicians at the different, are usually of diametrically opposing critiques so far as books con ceived for either those different types are concerned.

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**Example text**

32) is a complete orthonormal system in L 2 (a, b). Analogous assertions are valid in special N-dimensional domains (N > 1). For instance, if G is the rectangle 0 < Xl < a, 0 < X2 < b, then the system of functions 2 . m1tx l . Jab sm-a-smb-, m = 1,2, ... ,n = 1,2, ... 33) is a complete orthonormal system in L 2 ( G). 13) is a complete orthonormal system in L 2 { G), then a function u E L 2 { G) may be expressed - in accordance with the above text - by its Fourier series u(x) 00 = L OCt IPk(X) , k=1 OCk = (u, IPk)' k = 1,2, ....

45) that c = -(lP2' 1/13). Similarly as above, we can show that IIg311 oF o. 43), orthogonal to the functions In this way, we continue till we arrive at the orthonormal system lPl(X), lP2(X). 46) constitutes an orthonormal base in L2 { G): The orthonormality of the system is obvious from its construction. It remains to prove that the system is complete. 35), and vice-versa. 46). From here, it follows (cf. 46) is complete in L2{G). Hence, the above orthonormalization process really gives an orthonormal base in L 2 { G).

18) was explained in the directly preceding text. Similarly, the simple rule for calculations with the inner product (a l u1 + a2 u2, a3 u3 + a4u4) = = a1a3(u 1, u3) + a2a3(u2, U3) + ~1a4(uh U4) + a2a4(u2, U4) remains valid as derived in Chap. 2. 14) is frequently quoted in the literature as the Schwarz (or the Holder, or also the Cauchy-Buniakovski) inequality. Written out in full, for the space Lz(G), it reads as follows: In what follows, the inner product, the norm, and the distance of functions from the space Lz( G) are denoted by the above introduced symbols (u, v), Ilull, and e(u, v), respectively.