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International Journal of Mathematics and Mathematical Sciences
Volumeย 2012ย (2012), Article IDย 508721, 28 pages
http://dx.doi.org/10.1155/2012/508721
Research Article

Control Systems and Number Theory

Sanmenxia SuDa Communication Group, Sun Road, Sanmenxia Economical Development Zone, Sanmenxia, Henan, 472000, China

Received 3 October 2011; Accepted 16 November 2011

Academic Editor: Shigeruย Kanemitsu

Copyright ยฉ 2012 Fuhuo Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We try to pave a smooth road to a proper understanding of control problems in terms of mathematical disciplines, and partially show how to number-theorize some practical problems. Our primary concern is linear systems from the point of view of our principle of visualization of the state, an interface between the past and the present. We view all the systems as embedded in the state equation, thus visualizing the state. Then we go on to treat the chain-scattering representation of the plant of Kimura 1997, which includes the feedback connection in a natural way, and we consider the ๐ปโˆž-control problem in this framework. We may view in particular the unit feedback system as accommodated in the chain-scattering representation, giving a better insight into the structure of the system. Its homographic transformation works as the action of the symplectic group on the Siegel upper half-space in the case of constant matrices. Both of ๐ปโˆž- and PID-controllers are applied successfully in the EV control by J.-Y. Cao and B.-G. Cao 2006 and Cao et al. 2007, which we may unify in our framework. Finally, we mention some similarities between control theory and zeta-functions.

1. Introduction and Preliminaries

It turns out there is great similarity in control theory and number theory in their treatment of the signals in time domain (๐‘ก)and frequency domain (๐œ”,๐‘ =๐œŽ+๐‘—๐œ”)which is conducted by the Laplace transform in the case of control theory while, in the theory of zeta-functions, this role is played by the Mellin transform, both of which convert the signals in time domain to those in the right half-plane. For integral transforms, compare Section 11.

Section 5 introduces the Hardy space ๐ป๐‘ which consists of functions analytic in โ„›โ„‹๐’ซโ€”right half-plane ๐œŽ>0.

2. State Space Representation and the Visualization Principle

Let ๐ฑ=๐ฑ(๐‘ก)โˆˆโ„๐‘›, ๐ฎ=๐ฎ(๐‘ก)โˆˆโ„๐‘Ÿ, and ๐ฒ=๐ฒ(๐‘ก)โˆˆโ„๐‘š be the state function, input function, and output function, respectively. We write ฬ‡๐ฑ for (๐‘‘/๐‘‘๐‘ก)๐ฑ. The system of (differential equations) DEsฬ‡๐ฑ=๐ด๐ฑ+๐ต๐ฎ,๐ฒ=๐ถ๐ฑ+๐ท๐ฎ(2.1) is called a state equation for a linear system, where ๐ดโˆˆ๐‘€๐‘›,๐‘›(โ„), ๐ต,๐ถ,๐ท are given constant matrices.

The state state ๐ฑ is not visible while the input and output are so, and the state may be thought of as an interface between the past and the present information since it contains all the information contained in the system from the past. The ๐ฑ being invisible, (2.1) would read๐ฒ=๐ท๐ฎ,(2.2) which appears in many places in the literature in disguised form. All the subsequent systems, for example, (3.1), are variations of (2.2). And whenever we would like to obtain the state equation, we are to restore the state ๐ฑ to make a recourse to (2.1), which we would call the visualization principle. In the case of feedback system, it is often the case that (2.2) is given in the form of (3.8). It is quite remarkable that this controller ๐‘† works for the matrix variable in the symplectic geometry (compare Section 4).

Using the matrix exponential function ๐‘’๐ด๐‘ก, the first equation in (2.1) can be solved in the same way as for the scalar case:๐ฑ=๐ฑ(๐‘ก)=๐‘’๐ด๐‘ก๐ฑ(0)+๐ต๐‘’๐ด๐‘ก๎€œ๐‘ก0๐‘’โˆ’๐ด๐œ๐ฎ(๐‘ก)d๐œ.(2.3)

Definition 2.1. A linear system with the input ๐ฎ=๐จฬ‡๐‘‘๐ฑ=๐‘‘๐‘ก๐ฑ=๐ด๐ฑ,(2.4) called an autonomous system, is said to be asymptotically stable if for all initial values, ๐ฑ(๐‘ก) approaches a limit as ๐‘กโ†’โˆž.

Since the solution of (2.4) is given by๐ฑ=๐‘’๐ด๐‘ก๐ฑ(0),(2.5) the system is asymptotically stable if and only if||||||๐‘’๐ด๐‘ก||||||โŸถ0as๐‘กโŸถโˆž.(2.6)

A linear system is said to be stable if (2.6) holds, which is the case if all the eigenvalues of ๐ดhave negative real parts. Compare Section 5 in this regard. It also amounts to saying that the step response of the system approaches a limit as time elapses, where step response means a response๎€œ๐ฒ(๐‘ก)=๐‘ก0๐‘’๐ด(๐‘กโˆ’๐œ)๐‘ข(๐œ)๐‘‘๐œ,(2.7) with the unit step function ๐‘ข=๐‘ข(๐‘ก) as the input function, which is 0 for ๐‘ก<0 and 1 for ๐‘กโ‰ฅ0.

Up here, the things are happening in the time domain. We now move to a frequency domain. For this purpose, we refer to the Laplace transform to be discussed in Section 11. It has the effect of shifting from the time domain to frequency domain and vice versa. For more details, see, for example, [1]. Taking the Laplace transform of (2.1) with ๐ฑ(0)=๐จ, we obtain ๐‘Œ๐‘ ๐‘‹(๐‘ )=๐ด๐‘‹(๐‘ )+๐ต๐‘ˆ(๐‘ ),(๐‘ )=๐ถ๐‘‹(๐‘ )+๐ท๐‘ˆ(๐‘ ),(2.8) which we solve as๐‘Œ(๐‘ )=๐บ(๐‘ )๐‘ˆ(๐‘ ),(2.9) where๐บ(๐‘ )=๐ถ(๐‘ ๐ผโˆ’๐ด)โˆ’1๐ต+๐ท,(2.10) where ๐ผindicates the identity matrix, which is sometimes denoted by ๐ผ๐‘›to show its size.

In general, supposing that the initial values of all the signals in a system are 0, we call the ratio of output/input of the signal, the transfer function, and denote it by ๐บ(๐‘ ), ฮฆ(๐‘ ), and so forth. We may suppose so because, if the system is in equilibrium, then we may take the values of parameters at that moment as standard and may suppose the initial values to be 0.

Equation (2.10) is called the state space representation (form, realization, description, characterization) of the transfer function ๐บ(๐‘ ) of the system (2.1) and is written asโŽ›โŽœโŽœโŽ๐ด๐บ(๐‘ )=๐ต๐ถ๐ทโŽžโŽŸโŽŸโŽ .(2.11)

According to the visualization principle above, we have the embedding principle. Given a state space representation of a transfer function ๐บ(๐‘ ), it is to be embedded in the state equation (2.1).

Example 2.2. If โŽ›โŽœโŽœโŽ๐ด๐บ(๐‘ )=๐ต๐ถ๐ทโŽžโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽ010โˆ’2โˆ’31โˆ’10โˆ’22โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,(2.12) then it follows from (2.10) that ๐บ๎‚€๎‚โŽ›โŽœโŽœโŽโŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ โˆ’โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ โŽžโŽŸโŽŸโŽ (๐‘ )=โˆ’10,โˆ’2๐‘ 00๐‘ 01โˆ’2โˆ’3โˆ’1โŽ›โŽœโŽœโŽ01โŽžโŽŸโŽŸโŽ =๎‚€๎‚โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ +2โˆ’10,โˆ’2๐‘ โˆ’12๐‘ +3โˆ’1โŽ›โŽœโŽœโŽ01โŽžโŽŸโŽŸโŽ =1+2๎‚€๎‚โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ (๐‘ +1)(๐‘ +2)โˆ’10,โˆ’2๐‘ +31โˆ’2๐‘ โˆ’1โŽ›โŽœโŽœโŽ01โŽžโŽŸโŽŸโŽ +2=โˆ’2๐‘ +5(๐‘ +1)(๐‘ +2)+2=2(๐‘ +3)(๐‘ โˆ’1)(.๐‘ +1)(๐‘ +2)(2.13)

The principle above will establish the most important cascade connection (concatenation rule) [1, (2.13), page 15]. Given two state space representations๐บ๐‘˜(โŽ›โŽœโŽœโŽ๐ด๐‘ )=๐‘˜๐ต๐‘˜๐ถ๐‘˜๐ท๐‘˜โŽžโŽŸโŽŸโŽ ,๐‘˜=1,2,(2.14) their cascade connection ๐บ(๐‘ )=๐บ1(๐‘ )๐บ2(๐‘ ) is given by๐บ(๐‘ )=๐บ1(๐‘ )๐บ2โŽ›โŽœโŽœโŽœโŽœโŽ๐ด(๐‘ )=1๐ต1๐ถ2๐ต1๐ท2๐‘‚๐ด2๐ต2๐ถ1๐ท1๐ถ2๐ท1๐ท2โŽžโŽŸโŽŸโŽŸโŽŸโŽ .(2.15)

Proof of (2.15). We have the input/output relation (2.10) ๐‘Œ(๐‘ )=๐บ1(๐‘ )๐‘ˆ(๐‘ ),๐‘ˆ(๐‘ )=๐บ2(๐‘ )๐‘‰(๐‘ ),(2.16) which means that ฬ‡๐ฑ=๐ด1๐ฑ+๐ต1๐ฎ,๐ฒ=๐ถ1๐ฑ+๐ท1ฬ‡๐ฎ,(2.17)๐ƒ=๐ด2๐ƒ+๐ต2๐ฏ,๐ฎ=๐ถ2๐ƒ+๐ท2๐ฏ.(2.18)
Eliminating ๐ฎ, we conclude thatฬ‡๐ฑ=๐ด1๐ฑ+๐ต1๐ถ2๐œ‰+๐ต1๐ท2๐ฏ,๐ฒ=๐ถ1๐ฑ+๐ท1๐ถ2๐œ‰+๐ท1๐ท2๐ฏ.(2.19) Hence โŽ›โŽœโŽœโŽฬ‡๐ฑฬ‡๐ƒโŽžโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽ๐ด1๐ต1๐ถ2๐‘‚๐ด2โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐ฑ๐ƒโŽžโŽŸโŽŸโŽ +โŽ›โŽœโŽœโŽ๐ต1๐ท2๐ต2โŽžโŽŸโŽŸโŽ ๎‚€๐ถ๐ฏ,๐ฒ=1๐ท1๐ถ2๎‚โŽ›โŽœโŽœโŽ๐ฑ๐ƒโŽžโŽŸโŽŸโŽ +๐ท1๐ท2๐ฏ,(2.20) whence we conclude (2.15).

Example 2.3. Given two state space representations (2.14), their parallel connection ๐บ(๐‘ )=๐บ1(๐‘ )+๐บ2(๐‘ ) is given by ๐บ(๐‘ )=๐บ1(๐‘ )+๐บ2โŽ›โŽœโŽœโŽœโŽœโŽ๐ด(๐‘ )=1๐‘‚๐ต1๐‘‚๐ด2๐ต2๐ถ1๐ถ2๐ท1+๐ท2โŽžโŽŸโŽŸโŽŸโŽŸโŽ .(2.21)
Indeed, we have (2.17), and for (2.18), we have ฬ‡๐ƒ=๐ด2๐ƒ+๐ต2๐ฎ,๐ฒ+๐ณ=๐ถ2๐ƒ+๐ท2๐ฎ.(2.22)
Hence for (2.20), we have(๐ฑ+๐ƒ)โ‹…=๐ด1๐ฑ+๐ด2๎€ท๐ต๐œ‰+1+๐ต2๎€ธ๐ฎ,๐ฒ=๐ถ1๐ฑ+๐ถ2๎€ท๐ท๐œ‰+1+๐ท2๎€ธ๐ฏ,(2.23) whence (2.21) follows.

As an example, combining (2.15) and (2.21) we deduce๐ผโˆ’๐บ1(๐‘ )๐บ2(โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘ )=๐ผ๐‘‚๐‘‚โˆ’๐ด1โˆ’๐ต1๐ถ2โˆ’๐ต1๐ท2๐‘‚๐‘‚โˆ’๐ด2โˆ’๐ต2๐‘‚โˆ’๐ถ1โˆ’๐ท1๐ถ2๐‘‰โˆ’1โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .(2.24)

Example 2.4. For (2.1), we consider the inversion๐‘ˆ(๐‘ )=๐บโˆ’1(๐‘ )๐‘Œ(๐‘ ). Solving the second equality in (2.1) for ๐ฎโ€‰โ€‰we obtain ๐ฎ=โˆ’๐ทโˆ’1๐ถ๐ฑ+๐ทโˆ’1๐ฒ.(2.25)
Substituting this in the first equality in (2.1), we obtainฬ‡๎€ท๐ฑ=๐ดโˆ’๐ต๐ทโˆ’1๐ถ๎€ธ๐ฑ+๐ต๐ทโˆ’1๐ฒ.(2.26) Whence ๐บโˆ’1(โŽ›โŽœโŽœโŽ๐‘ )=๐ดโˆ’๐ต๐ทโˆ’1๐ถโˆ’๐ต๐ทโˆ’1โˆ’๐ทโˆ’1๐ถ๐ทโˆ’1โŽžโŽŸโŽŸโŽ .(2.27)

Example 2.5. If the transfer function โŽ›โŽœโŽœโŽฮ˜ฮ˜(๐‘ )=11ฮ˜12ฮ˜21ฮ˜22โŽžโŽŸโŽŸโŽ (2.28) has a state space representation โŽ›โŽœโŽœโŽ๐ดฮ˜(๐‘ )=๐ต๐ถ๐ทโŽžโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽ๐ด๐ต1๐ต2๐ถ1๐ท11๐ท12๐ถ2๐ท21๐ท22โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,(2.29) then we are to embed it in the linear system ฬ‡๎‚€๐ต๐ฑ=๐ด๐ฑ+1๐ต2๎‚โŽ›โŽœโŽœโŽ๐›1๐›2โŽžโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽ๐š๐Ÿ๐š๐ŸโŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐ถ=๐ฒ=1๐ถ2โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐ท๐ฑ+11๐ท12๐ท21๐ท22โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐›1๐›2โŽžโŽŸโŽŸโŽ .(2.30)

3. Chain-Scattering Representation

Following [1, pages 7 and 67], we first give the definition of a chain-scattering representation of a system.

Suppose ๐š1โˆˆโ„๐‘š, ๐š2โˆˆโ„๐‘ž, ๐›1โˆˆโ„๐‘Ÿ, and ๐›2โˆˆโ„๐‘ are related byโŽ›โŽœโŽœโŽ๐š1๐š2โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐›=๐‘ƒ1๐›2โŽžโŽŸโŽŸโŽ ,(3.1) whereโŽ›โŽœโŽœโŽ๐‘ƒ๐‘ƒ=11๐‘ƒ12๐‘ƒ21๐‘ƒ22โŽžโŽŸโŽŸโŽ .(3.2)

According to the embedding principle, this is to be thought of as ๐ฒ=๐‘†๐ฎ corresponding to the second equality in (2.1).

Equation (3.1) means that๐š1=๐‘ƒ11๐›1+๐‘ƒ12๐›2,๐š2=๐‘ƒ21๐›1+๐‘ƒ22๐›2.(3.3)

Assume that ๐‘ƒ21 is a (square) regular matrix (whence ๐‘ž=๐‘Ÿ). Then from the second equality of (3.3), we obtain๐›1=๐‘ƒโˆ’121๎€ท๐š2โˆ’๐‘ƒ22๐›2๎€ธ=โˆ’๐‘ƒโˆ’121๐‘ƒ22๐›2+๐‘ƒโˆ’121๐š2.(3.4) Substituting (3.4) in the first equality of (3.3), we deduce that๐š1=๎€ท๐‘ƒ12โˆ’๐‘ƒ11๐‘ƒโˆ’121๐‘ƒ22๎€ธ๐›2+๐‘ƒ11๐‘ƒโˆ’121๐š2.(3.5)

Hence puttingโŽ›โŽœโŽœโŽœโŽ๐‘ƒฮ˜=CHAIN(๐‘ƒ)=12โˆ’๐‘ƒ11๐‘ƒโˆ’121๐‘ƒ22๐‘ƒ11๐‘ƒโˆ’121โˆ’๐‘ƒโˆ’121๐‘ƒ22๐‘ƒโˆ’121โŽžโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽฮ˜11ฮ˜12ฮ˜21ฮ˜22โŽžโŽŸโŽŸโŽ ,(3.6) which is usually referred to as a chain-scattering representation of ๐‘ƒ, we obtain an equivalent form of (3.1)โŽ›โŽœโŽœโŽ๐š1๐›1โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐›=CHAIN(๐‘ƒ)2๐š2โŽžโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽฮ˜11ฮ˜12ฮ˜21ฮ˜22โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐š2๐›2โŽžโŽŸโŽŸโŽ .(3.7)

Suppose that ๐š2 is fed back to ๐›2 by๐›2=๐‘†๐š2,(3.8) where ๐‘† is a controller. Multiplying the second equality in (3.3) by ๐‘† and incorporating (3.8), we find that๐›2=๐‘†๐š2=๐‘†๐‘ƒ21๐›1+๐‘†๐‘ƒ22๐›2,(3.9)

whence ๐›2=(๐ผโˆ’๐‘ƒ22๐พ)โˆ’1๐‘†๐‘ƒ21๐›1.

Let the closed-loop transfer function ฮฆ be defined by๐š1=ฮฆ๐›1.(3.10)

ฮฆ is given byฮฆ=๐‘ƒ11+๐‘ƒ12๎€ท๐ธโˆ’๐‘ƒ22๐‘†๎€ธโˆ’1๐‘†๐‘ƒ21.(3.11) Equation (3.11) is sometimes referred to as a linear fractional transformation and denoted by๐ฟ๐น(๐‘ƒ;๐พ).(3.12) Substituting (3.8), (3.7) becomesโŽ›โŽœโŽœโŽ๐š1๐›1โŽžโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽฮ˜11๐‘†+ฮ˜12ฮ˜21๐‘†+ฮ˜22โŽžโŽŸโŽŸโŽ ๐š2,(3.13)

whence we deduce that๎€ทฮ˜ฮฆ=11๐‘†+ฮ˜12ฮ˜๎€ธ๎€ท21๐‘†+ฮ˜22๎€ธโˆ’1=ฮ˜๐‘†,(3.14) the linear fractional transformation (which is referred to as a homographic transformation and denoted by HM(ฮฆ;๐‘†)), where in the last equality we mean the action of ฮ˜ on the variable ๐‘†. We must impose the nonconstant condition |ฮ˜|โ‰ 0. Then ฮ˜โˆˆ๐บ๐ฟ๐‘š+๐‘Ÿ(โ„).

If ๐‘† is obtained from ๐‘†โ€ฒ under the action of ฮ˜โ€ฒ, ๐‘†=ฮ˜โ€ฒ๐‘†โ€ฒ, then its composition ๐ฝ with (3.14) yields ๐ฝ๐‘†โ€ฒ=ฮฆฮฆโ€ฒ=ฮ˜ฮ˜โ€ฒ๐‘†โ€ฒ, that is,๎€ท๎€ทฮ˜๐ฝ=ฮ˜ฮ˜โ€ฒ,HMฮ˜;HM๎…ž๎€ท;๐‘†๎€ธ๎€ธ=HMฮ˜ฮ˜๎…ž๎€ธ;๐‘†,(3.15) which is referred to as the cascade connection or the cascade structure of ฮ˜ and ฮ˜โ€ฒ.

Thus the chain-scattering representation of a system allows us to treat the feedback connection as a cascade connection.

Suppose a closed-loop system is given with ๐ณ=๐š1โˆˆโ„๐‘š, ๐ฒ=๐š2โˆˆโ„๐‘ž, ๐ฐ=๐›1โˆˆโ„๐‘Ÿ, and ๐ฎ=๐›2โˆˆโ„๐‘ and ฮฆ given by (3.2).

๐ปโˆž-Control Problem. Find a controller ๐พ such that the closed-loop system is internally stable and the transfer function ฮฆ satisfiesโ€–ฮฆโ€–โˆž<๐›พ,(3.16) for a positive constant ๐›พ. For the meaning of the norm, compare Section 5.

4. Siegel Upper Space

Let โˆ— denote the conjugate transpose of a square matrix: ๐‘†โˆ—=๐‘ก๐‘†, and let the imaginary part of ๐‘† defined by Im๐‘†=(1/2๐‘—)(๐‘†โˆ’๐‘†โˆ—). Let โ„‹๐‘› be the Siegel upper half-space consisting of all the matrices ๐‘† (recall (3.8)) whose imaginary parts are positive definite (Im๐‘†>0โ€”imaginary parts of all eigen values are positive) and satisfies ๐‘†=๐‘ก๐‘†:โ„‹๐‘›=๎‚ป๐‘†โˆˆM๐‘›(โ„‚)โˆฃIm๐‘†>0,๐‘†=๐‘ก๐‘†๎‚ผ,(4.1) and let Sp(๐‘›,โ„) denote the symplectic group of order ๐‘›:โŽงโŽชโŽจโŽชโŽฉโŽ›โŽœโŽœโŽฮ˜Sp(๐‘›,โ„)=ฮ˜=11ฮ˜12ฮ˜21ฮ˜22โŽžโŽŸโŽŸโŽ โˆฃโŽ›โŽœโŽœโŽฮ˜11ฮ˜12ฮ˜21ฮ˜22โŽžโŽŸโŽŸโŽ โˆ’1=โŽ›โŽœโŽœโŽฮ˜22โˆ’๐‘กฮ˜12โˆ’๐‘กฮ˜21ฮ˜11โŽžโŽŸโŽŸโŽ โŽซโŽชโŽฌโŽชโŽญ.(4.2)

The action of Sp(๐‘›,โ„) on โ„‹๐‘› is defined by (3.14) which we restate as๎€ทฮ˜ฮ˜๐‘†=11๐‘†+ฮ˜12ฮ˜๎€ธ๎€ท21๐‘†+ฮ˜22๎€ธโˆ’1(=ฮฆ).(4.3)

Theorem 4.1. For a controller ๐‘† living in the Siegel upper space, its rotation ๐‘=โˆ’๐‘—๐‘† lies in the right half-space โ„›โ„‹๐’ฎ, that is, stable having positive real parts. For the controller ๐‘, the feedback connection โˆ’๐‘—๐›2๎€ท=๐‘โˆ’๐‘—๐š2๎€ธ(4.4) is accommodated in the cascade connection of the chain-scattering representation ฮ˜ (3.15), which is then viewed as the action (3.15) of ฮ˜โˆˆSp(๐‘›,โ„) on ๐‘†โˆˆโ„‹๐‘›: ๎€ทฮ˜ฮ˜๎…ž๎€ธ๎€ทฮ˜๐‘†=ฮ˜๎…ž๐‘†๎€ธ๎€ท๎€ทฮ˜;๐‘œ๐‘ŸHMฮ˜;HM๎…ž๎€ท;๐‘†๎€ธ๎€ธ=HMฮ˜ฮ˜๎…ž๎€ธ;๐‘†,(4.5) where ฮ˜ is subject to the condition ๐‘กฮ˜๐‘ˆฮ˜=๐‘ˆ,(4.6) with ๎‚€๐‘ˆ=๐‘‚๐ผ๐‘›โˆ’๐ผ๐‘›๐‘‚๎‚. An FOPID controller (in Section 6), being a unity feedback connection, is also accommodated in this framework.

Remark 4.2. With action, we may introduce the orbit decomposition of โ„‹๐‘› and whence the fundamental domain. We note that, in the special case of ๐‘›=1, we have โ„‹1=โ„‹ and Sp(1,โ„)=SL๐‘›(โ„) and the theory of modular forms of one variable is well known. Siegel modular forms are a generalization of the one variable case into several variables. As in the case of the sushmna principle in [2], there is a need to rotate the upper half-space into the right half-space โ„›โ„‹๐’ฎ, which is a counter part of the right-half plane โ„›โ„‹๐’ซ. In the case of Siegel modular forms, the matrices are constant, while in control theory, they are analytic functions (mostly rational functions analytic in โ„›โ„‹๐’ซ). A general theory would be useful for controlling theory. See Section 7 for physically realizable cases. There are many research problems lying in this direction.

5. Norm of the Function Spaces

The norm ๎‚€๐ฑ=๐‘ฅ1โ‹ฎ๐‘ฅ๐‘›๎‚โˆˆโ„‚๐‘› is defined to be the Euclidean normโ€–๐ฑโ€–=โ€–๐ฑโ€–2=๎„ถ๎„ต๎„ตโŽท๐‘›๎“๐‘—=1||๐‘ฅ๐‘—||2,(5.1) or by the sup norm โ€–๐ฑโ€–=โ€–๐ฑโ€–โˆž๎€ฝ||๐‘ฅ=max1||||๐‘ฅ,โ€ฆ.๐‘›||๎€พ,(5.2) or anything that satisfies the axioms of the norm. They introduce the same topology on โ„‚๐‘›.

The definition of the norm of a matrix should be given in a similar way by viewing its elements as an ๐‘›2-dimensional vector, that is, embedding it in โ„‚๐‘›2. If ๐ด=(๐‘Ž๐‘–๐‘—),1โ‰ค๐‘–,๐‘—โ‰ค๐‘›, thenโ€–๐ดโ€–=โ€–๐ดโ€–2=๎„ถ๎„ต๎„ตโŽท๐‘›๎“๐‘–,๐‘—=1||๐‘Ž๐‘–๐‘—||2,(5.3) or otherwise.

The sup norm is a limit of the ๐‘-norm as ๐‘โ†’โˆž. For ๐š=(๐‘Ž1,โ€ฆ,๐‘Ž๐‘›), lim๐‘โ†’โˆžโ€–๐šโ€–๐‘=lim๐‘โ†’โˆž๎ƒฉ๐‘›๎“๐‘˜=1||๐‘Ž๐‘˜||๐‘๎ƒช1/๐‘=โ€–๐šโ€–โˆž=max1โ‰ค๐‘˜โ‰ค๐‘›๎€ฝ||๐‘Ž๐‘˜||๐‘๎€พ.(5.4)

Supposeโ€‰โ€‰|๐‘Ž1|=max1โ‰ค๐‘˜โ‰ค๐‘›{|๐‘Ž๐‘˜|๐‘}. Then for any ๐‘>0|๐‘Ž1|=(|๐‘Ž1|๐‘)1/๐‘โˆ‘โ‰ค(๐‘›๐‘˜=1|๐‘Ž๐‘˜|๐‘)1/๐‘.

On the other hand, since |๐‘Ž1|โ‰ฅ|๐‘Ž๐‘˜|,1โ‰ค๐‘˜โ‰ค๐‘›, we obtain๎ƒฉ๐‘›๎“๐‘˜=1||๐‘Ž๐‘˜||๐‘๎ƒช1/๐‘=||๐‘Ž1||๎ƒฉ1+๐‘›๎“๐‘˜=2||||๐‘Ž๐‘˜๐‘Ž1||||๐‘๎ƒช1/๐‘โ‰ค||๐‘Ž1||(1+๐‘›โˆ’1)1/๐‘.(5.5)

For ๐‘>1, the Bernoulli inequality gives (1+๐‘›โˆ’1)1/๐‘โ‰ค1+(๐‘›โˆ’1)/๐‘โ†’1 as ๐‘โ†’โˆž. Hence the right-hand side of (5.5) tends to |๐‘Ž1|.

The proof of (5.4) can be readily generalized to givelim๐‘โ†’โˆžโ€–๐‘“โ€–๐‘=โ€–๐‘“โ€–โˆž=sup๐‘กโ‰ฅ0||||.๐‘“(๐‘ก)(5.6)

The ๐‘-norm in (5.6) is defined byโ€–๐‘“โ€–๐‘=๎‚ต๎€œโˆž0โ€–๐‘“(๐‘ก)โ€–๐‘๎‚ถ๐‘‘๐‘ก1/๐‘,(5.7)

where โ€–๐‘“(๐‘ก)โ€– is any Euclidean norm. Note that the functions are not ordinary functions but classes of functions which are regarded as the same if they differ only at measure 0 set. ๐ฟ๐‘ is a Banach space (i.e., a complete metric space), and in particular ๐ฟ2 is a Hilbert space. The 2-norm โ€–โ‹…โ€–2 is induced from the inner product๎€œโŸจ๐‘“,๐‘”โŸฉ=โˆž0๐‘“โˆ—(๐‘ก)๐‘”(๐‘ก)๐‘‘๐‘ก,โ€–๐‘“โ€–2=โˆšโŸจ๐‘“,๐‘“โŸฉ,(5.8) where โˆ— refers to the transposed complex conjugation.

The Parseval identity holds true if and only if the system is complete.

However, the restriction that โ€–๐‘“(๐‘ก)โ€–โ†’0 as ๐‘กโ†’โˆž excludes signals of infinite duration such as unit step signals or periodic ones from ๐ฟ๐‘. To circumvent the inconvenience, the notion of averaged norm, ๐‘€2(๐‘“)=๐‘€2โˆซ(๐‘“,๐‘‡)=(1/๐‘‡)๐‘‡0โ€–๐‘“(๐‘ก)โ€–2๐‘‘๐‘ก or similar, is important and the power norm has been introduced: power(๐‘“)=lim๐‘‡โ†’โˆž๐‘€2(๐‘“,๐‘‡)1/2=lim๐‘‡โ†’โˆž๎‚ต1๐‘‡๎€œ๐‘‡0โ€–๐‘“(๐‘ก)โ€–2๎‚ถ๐‘‘๐‘ก1/2.(5.9)

Remark 5.1. In mathematics and in particular in analytic number theory, studying the mean square in the form of a sum or an integral is quite common. Especially, this idea is applied to finding out the true order of magnitude of the error term on average. Such an average result will give a hint on the order of the error term itself.

Example 5.2. Let ๐œ(๐‘ ) denote the Riemann zeta-function defined for ๐œŽ>1(๐‘ =๐œŽ+๐‘–๐‘ก), in the first instance, where it is analytic and then continued meromorphically over the whole complex plane with a simple pole at ๐‘ =1. It is essential that it does not vanish on the line ๐œŽ=1 for the prime number theorem (PNT) to hold. The plausible best bound for the error term for the PNT is equivalent to the celebrated Riemann hypothesis (RH) to the effect that the Riemann zeta-function does not vanish on the critical line ๐œŽ=1/2. Since the values on the critical line are expected to be small, the averaged norm ๐‘€2(๐œ) or ๐‘€4(๐œ), that is, the mean value โˆซ(1/๐‘‡)๐‘‡0|๐œ((1/2)+๐‘–๐‘ก)|2๐‘˜๐‘‘๐‘ก for ๐‘˜=1,2 is of great interest and there have appeared a great deal of research on the subject. The first result for ๐‘€4(๐œ) is due to Ingham who used the approximate functional equation for the Riemann zeta-function to obtain ๐‘€41(๐œ)=๐‘‡๎€œ๐‘‡0|||๐œ๎‚€12๎‚|||+๐‘–๐‘ก41๐‘‘๐‘ก=4๐œ‹2log4๐‘‡(1+๐‘œ(1)),(5.10) for ๐‘‡โ†’โˆž. See, for example, [3]. The main interest in such estimates as (5.10) lies in the fact that estimates for all ๐‘˜โˆˆโ„•๐‘€2๐‘˜1(๐œ)=๐‘‡๎€œ๐‘‡0|||๐œ๎‚€12๎‚|||+๐‘–๐‘ก2๐‘˜๐‘‘๐‘ก=๐‘‚(๐‘‡๐‘Ž)(5.11) would imply the weak Lindelรถf hypothesis (LH) in the form ๐œ๎‚€12๎‚๎€ท๐‘‡+๐‘–๐‘ก=๐‘‚(๐‘Ž/2๐‘˜)+๐œ€๎€ธ,(5.12) for every ๐œ€>0. It is apparent that the RH implies the LH.

The Hardy space ๐ป๐‘ (cf. e.g., [1, page 39]) is well known. It consists of all ๐‘“(๐‘ ) which are analytic in โ„›โ„‹๐’ซโ€”right half-plane ๐œŽ>0 such that ๐‘“(๐‘—๐œ”)โˆˆ๐ฟ๐‘, in particular, ๐ปโˆž with sup norm. Thus ๐ปโˆž-control problem is about those (rational) functions which are analytic in โ„›โ„‹๐’ซ, a fortiori stable, with regard to the sup norm. Thus the above-mentioned mean-value problem for the Riemann zeta-function is related to the ๐ป2๐‘˜-control problem with finite Dirichlet series (main ingredients in the approximate functional equation). Since the ๐ปโˆž-control problem asks for all individual values, it flows afar from the ๐ป2๐‘˜-control problem and goes up to the LH or the RH.

6. (Unity) Feedback System

The synthesis problem of a controller of the unity feedback system, depicted inโ€‰โ€‰Figure 1, refers to the sensitivity reduction problem, which asks for the estimation of the sensitivity function ๐‘†=๐‘†(๐‘ ) multiplied by an appropriate frequency weighting function ๐‘Š=๐‘Š(๐‘ ):๐‘†=(๐ผ+๐‘ƒ๐ถ)โˆ’1(6.1) is a transfer function from ๐‘Ÿ to ๐‘’, where ๐ถ=๐พ is a compensator and ๐‘ƒ is a plant. The problem consists in reducing the magnitude of ๐‘† over a specified frequency range ฮฉ, which amounts to finding a compensator ๐ถ stabilizing the closed-loop system such thatโ€–๐‘Š๐‘†โ€–โˆž<๐›พ(6.2) for a positive constant ๐›พ.

508721.fig.001
Figure 1: Unity feedback system.

To accommodate this in the ๐ปโˆž control problem (3.1), we choose the matrix elements ๐‘ƒ๐‘–๐‘— of ๐‘ƒ in such a way that the closed-loop transfer function ฮฆ in (3.11) coincides with ๐‘Š๐‘†. First we are to choose ๐‘ƒ22=โˆ’๐‘ƒ. Then we would choose ๐‘ƒ12๐‘ƒ21=๐‘Š๐‘ƒ. Then ฮฆ becomes ๐‘ƒ11+๐‘Š๐‘ƒ๐ถ(๐ผ+๐‘ƒ๐ถ)โˆ’1=๐‘ƒ11โˆ’๐‘Š+๐‘Š(๐ผ+๐‘ƒ๐ถ)โˆ’1. Hence choosing ๐‘ƒ11=๐‘Š, we have ฮฆ=๐‘Š๐‘†. Hence we may choose, for example,โŽ›โŽœโŽœโŽ๐‘ƒ๐‘ƒ=11๐‘ƒ12๐‘ƒ21๐‘ƒ22โŽžโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ๐‘Š๐‘Š๐‘ƒ๐ธโˆ’๐‘ƒ.(6.3)

Example 6.1. First we treat the case of general feedback scheme. Denoting the Laplace transforms by the corresponding capital letters, we have ๐‘Œ=๐‘ƒ๐‘…+๐‘ƒ๐‘ˆ,๐‘ˆ=๐พ๐ธ,(6.4) whence ๐‘Œ=๐‘ƒ๐‘…+๐‘ƒ๐พ๐ธ. Now if it so happens that ๐‘’=๐‘Ÿโˆ’๐‘ฆ and ๐‘ƒ is replaced by ๐‘ƒ๐พ, that is, in the case of unity FD, we derive (6.1) directly fromโ€‰โ€‰Figure 2. We have ๐ธ=๐‘…โˆ’๐‘Œ, so that ๐‘Œ=๐‘ƒ๐‘…+๐‘ƒ๐พ(๐‘…โˆ’๐‘Œ). Solving in ๐‘Œ, we deduce that (๐ผ+๐‘ƒ๐พ)โˆ’1๐‘ƒ๐พ๐‘….
We take into account the disturbance ๐‘‘,โ€‰โ€‰andโ€‰โ€‰we obtain since ๐‘ˆ=๐ถ๐ธ=๐ถ(๐‘…โˆ’๐‘Œ)๐‘Œ=๐‘ƒ๐‘ˆ+๐‘ƒ๐ท=๐‘ƒ๐ถ(๐‘…โˆ’๐‘Œ)+๐‘ƒ๐ท,(6.5) whence ๐‘Œ=๐‘ƒ๐ถ(๐‘…โˆ’๐‘Œ)+๐‘ƒ๐ท. It follows that ๐‘Œ=(๐ผ+๐‘ƒ๐ถ)โˆ’1๐‘ƒ๐ถ๐‘…+(๐ผ+๐‘ƒ๐ถ)โˆ’1๐‘ƒ๐ท. In the case where ๐‘‘=0, ๐‘ƒ๐ถ being the open-loop transfer function, we have ๐‘†๐‘… is the tracking error for the input ๐‘…. Hence (6.1) holds true.

508721.fig.002
Figure 2

7. ๐ฝ-Lossless Factorization and Dualization

In this section we mostly follow Helton ([4โ€“6]) who uses the unit ball in place of โ„›โ„‹๐’ซ. They shift to each other under the complex exponential map. For conventional control theory, the unit ball is to be replaced by the critical line (๐œŽ=0). In practice what appears is the algebra of functions ([5, page 2]), (Table 1)โ„›={functionsde๏ฌnedontheunitballhavingtherationalcontinuationtothewholespace},(7.1) or still larger algebra ๐œ“ consisting of those functions which have (pseudo)meromorphic continuations ([5, footnote 6, page 27]). The occurrence of the gamma function [5, Figureโ€‰โ€‰2.5, page 17] justifies our incorporation of more advanced special functions and ultimately zeta-functions in control theory (see Section 13).

tab1
Table 1: Correspondence between control systems and zeta-functions.

Along with the algebra โ„›, one considersโ„ฌ๐ปโˆž={๐นโˆฃanalyticontheunitballhavingthesupremumnorm<1}.(7.2)

Then the only mapping ฮ˜โˆˆโ„›๐‘ˆ(๐‘š,๐‘›) acting on โ„ฌ๐ปโˆž must satisfy the ๐ฝ-lossless property. Let ฮ˜ denote an (๐‘š+๐‘›)ร—(๐‘š+๐‘›) matrix.

Thenฮ˜โˆ—๐ฝ๐‘š๐‘›ฮ˜โ‰ค๐ฝ๐‘š๐‘›,(7.3) which is interpreted to be the power preservation of the system in the chain-scattering representation (3.6) ([1, page 82]).

We now briefly refer to the dual chain-scattering representation of the plant ๐‘ƒ in (3.2). We assume ๐‘ƒ12 is a square invertible matrix (whence ๐‘š=๐‘). Then the argument goes in parallel to that leading to (3.7). Defining the dual chain-scattering matrix byโŽ›โŽœโŽœโŽœโŽ๐‘ƒDCHAIN(๐‘ƒ)=โˆ’112๐‘ƒ11๐‘ƒโˆ’112โˆ’๐‘ƒโˆ’112๐‘ƒ22๐‘ƒ21โˆ’๐‘ƒ22๐‘ƒโˆ’112๐‘ƒ11โŽžโŽŸโŽŸโŽŸโŽ ,(7.4) we obtainCHAIN(๐‘ƒ)โ‹…DCHAIN(๐‘ƒ)=๐ธ.(7.5)

8. FOPID

โ€œFOโ€ means โ€œFractional orderโ€ and โ€œPIDโ€ refers to โ€œProportional, Integral, Differential,โ€ whence โ€œProportionalโ€ means just constant times the input function ๐‘’(๐‘ก), โ€œIntegralโ€ means the fractional order integration ๐ผ๐œ†๐‘ก๐ท๐‘กโˆ’๐œ† of ๐‘’(๐‘ก) (๐œ†>0), and โ€œDifferentialโ€ the fractional order differentiation ๐ท๐›ฟ๐‘ก of ๐‘’(๐‘ก) (๐›ฟ>0).

The FO ๐‘ƒ๐ผ๐œ†๐ท๐›ฟ controller (control signal in the time domain) is one of the most refined feed-forward compensators defined by๎€ท๐พ๐‘ข(๐‘ก)=๐‘+๐พ๐‘–๐ท๐‘กโˆ’๐œ†+๐พ๐‘‘๐ท๐›ฟ๐‘ก๎€ธ๐‘’(๐‘ก),(8.1) where ๐‘ข is the input function, ๐‘’ is the deviation, and ๐พ๐‘,๐พ๐‘–,๐พ๐‘‘ are constant parameters which are to be specified (๐พ๐‘: the position feedback gain, ๐พ๐‘‘: the velocity feedback gain). DE (8.1) translates into the state equation ๐‘Œ(๐‘ )=๐ถ(๐‘ )๐ธ(๐‘ ),(8.2) where ๐‘ˆ,๐‘Œ indicate the Laplace transforms of ๐‘ข,๐‘ฆ, respectively, and ๐บ is the compensators continuous transfer function ๐ถ(๐‘ )=๐พ๐‘+๐พ๐‘–๐‘ โˆ’๐œ†+๐พ๐‘‘๐‘ ๐›ฟ.(8.3)

The derivation of (8.3) from (8.1) depends on the following. The general fractional calculus operator๐‘Ž๐ท๐›ผ๐‘ก is symbolically stated as๐‘Ž๐ท๐›ผ๐‘ก=โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘‘๐›ผ๐‘‘๐‘ก๐›ผ๎€œ,Re๐›ผ>0,1,Re๐›ผ=0,๐‘ก๐‘Ž1๐‘‘๐‘ก๐›ผ,Re๐›ผ<0,(8.4) where ๐‘Ž and ๐‘ก are the lower and upper limits of integration and ๐›ผ is the order of calculus.

More precisely, the definition of the fractional differintegral is given by the Riemann-Liouville expression๐‘Ž๐ท๐›ผ๐‘ก1๐‘“(๐‘ก)=๎‚€๐‘‘ฮ“(1โˆ’{๐›ผ})๎‚๐‘‘๐‘ก๐›ผโˆ’{๐›ผ}+1๎€œ๐‘ก๐‘Ž(๐‘กโˆ’๐œ)โˆ’{๐›ผ}๐‘“(๐œ)๐‘‘๐œ,(8.5) where {๐›ผ}=๐›ผโˆ’[๐›ผ] indicates the fractional part of ๐›ผ, with [๐›ผ] the integral part of ๐›ผ. Thus we are also led to the Riemann-Liouville fractional integral transform:[๐‘“]=1โ„›๐ฟ๎€œฮ“(๐œ‡)๐‘ฆ0(๐‘ฆโˆ’๐‘ฅ)๐œ‡โˆ’1๐‘“(๐‘ฅ)๐‘‘๐‘ฅ.(8.6)

For applications, compare Section 13.

When ๐›ผโˆˆโ„•, (8.5) reads๐‘Ž๐ท๐›ผ๐‘ก๎‚€๐‘‘๐‘“(๐‘ก)=๎‚๐‘‘๐‘ก๐›ผ+1๎€œ๐‘ก๐‘Ž๐‘“(๐œ)๐‘‘๐œ=๐‘“(๐›ผ)(๐‘ก),(8.7) the ๐›ผth derivative of ๐‘“.

We will see that the definition (8.5) is a natural outcome of the general formula for the difference operator of order ๐›ผโˆˆโ„• with difference ๐‘ฆโ‰ฅ0:ฮ”๐›ผ๐‘ฆ๐‘“(๐‘ฅ)=๐›ผ๎“๐œˆ=0(โˆ’1)๐›ผโˆ’๐œˆโŽ›โŽœโŽœโŽ๐›ผ๐œˆโŽžโŽŸโŽŸโŽ ๐‘“(๐‘ฅ+๐œˆ๐‘ฆ).(8.8)

If ๐‘“ has the ๐›ผ-th derivative ๐‘“(๐›ผ), thenฮ”๐›ผ๐‘ฆ๎€œ๐‘“(๐‘ฅ)=๐‘ฅ๐‘ฅ+๐‘ฆ๐‘‘๐‘ก1๎€œ๐‘ก1๐‘ก+๐‘ฆ1๐‘‘๐‘ก2โ‹ฏ๎€œ๐‘ก๐›ผโˆ’1๐‘ก+๐‘ฆ๐›ผโˆ’1๐‘“(๐›ผ)๎€ท๐‘ก๐›ผ๎€ธ๐‘‘๐‘ก๐›ผ.(8.9)

The special case of (8.9) with ๐‘ก๐œˆ=๐‘Ž,๐‘Ž+๐‘ฆโ†’๐‘ฅ(๐œ‘(๐‘ก)=๐‘“(๐›ผ)(๐‘ก)) readsฮ”๐›ผ๐‘ฅโˆ’๐‘Ž๎€œ๐œ‘(๐‘ฅ)=๐‘ฅ๐‘Ž๎€œd๐‘ก๐‘ฅ๐‘Ž๎€œd๐‘กโ‹ฏ๐‘ฅ๐‘Ž1๐œ‘(๐‘ก)๐‘‘๐‘ก=๎€œฮ“(๐›ผ)๐‘ฅ๐‘Ž(๐‘ฅโˆ’๐‘ก)๐›ผโˆ’1๐œ‘(๐‘ก)๐‘‘๐‘ก,(8.10) whose far-right hand side is โ„›๐ฟ[๐œ‘].

Let ๐น(๐‘ ) be the Laplace transform of the input function ๐‘“(๐‘ก). Then ๐ฟ๎€บ0๐ท๐›ผ๐‘ก๐‘“๎€ป(๐‘ก)=๐‘ ๐›ผ๐น(๐‘ )โˆ’0๐ท๐‘ก๐›ผโˆ’1๐‘“(๐‘ก)โˆฃ๐‘ก=0,๐ฟ๎€บ0๐ท๐‘กโˆ’๐›ผ๐‘“๎€ป(๐‘ก)=๐‘ โˆ’๐›ผ๐น(๐‘ ).(8.11)

9. Fourier, Mellin, and (Two-Sided) Laplace Transforms

We state the Mellin, (two-sided) Laplace, and the Fourier transforms. If ๐‘“(๐‘ฅ)=๐‘‚(๐‘ฅ๐›ผ),๐›ผโˆˆโ„ for ๐‘ฅ>0, then its Mellin transform ๐‘€[๐‘“] is defined by๐‘€[๐‘“]๎€œ(๐‘ )=โˆž0๐‘ฅ๐‘ ๐‘“(๐‘ฅ)๐‘‘๐‘ฅ๐‘ฅ,๐œŽ>๐›ผ(9.1)

Under the change of variable ๐‘ฅ=๐‘’โˆ’๐‘ก, the Mellin transform and the two-sided Laplace transform shift each other:๐ฟยฑ[๐œ‘]๎€œ(๐‘ )=โˆžโˆ’โˆž๐‘’โˆ’๐‘ ๐‘ก๐œ‘(๐‘ก)๐‘‘๐‘ก,๐œŽ>๐›ผ,(9.2) where we write ๐œ‘(๐‘ก)=๐‘“(๐‘’โˆ’๐‘ก).

The ordinary Laplace transform (one-sided Laplace transform) is obtained by multiplying the integrand by the unit step function ๐‘ข=๐‘ข(๐‘ก) (cf. the passage immediately after (2.7)): ๐ฟ[๐‘“](๐‘ )=๐ฟยฑ[]๎€œ๐‘“๐‘ข(๐‘ )=โˆž0๐‘’โˆ’๐‘ ๐‘ก๐‘“๎€ท๐‘’โˆ’๐‘ก๎€ธ๐‘‘๐‘ก,๐œŽ>๐›ผ,(9.3) compare Definition 11.1.

If we fix ๐œ˜>๐›ผ and write ๐‘ =๐œ˜+๐‘—๐œ”, ๐บ(๐‘ฆ)=๐ฟยฑ[๐‘“](๐œ˜+๐‘—๐œ”),๐‘”(๐‘ก)=๐‘’โˆ’๐œ˜๐‘ก๐‘“(๐‘’โˆ’๐‘ก) in (9.2), then it changes into[๐‘”]๎€œ๐บ(๐œ”)=๐น(๐œ”)=โˆžโˆ’โˆž๐‘’โˆ’๐‘—๐œ”๐‘ก๐‘”(๐‘ก)๐‘‘๐‘ก=๐ฟยฑ[๐œ‘](๐‘—๐œ”),(9.4) the Fourier transform of ๐‘”.

We explain Plancherelโ€™s theorem for functions in ๐ฟ2(โ„). Let๎๐‘“๐‘‡1(๐‘ฅ)=โˆš๎€œ2๐œ‹๐‘‡โˆ’๐‘‡๐‘’โˆ’๐‘–๐‘ฅ๐‘ก๐‘“(๐‘ก)๐‘‘๐‘ก.(9.5) Then ๎๐‘“๐‘‡(๐‘ฅ) is convergent to a function ๎๐‘“ in ๐ฟ2:โ€–โ€–๎๐‘“๐‘‡โˆ’๎๐‘“โ€–โ€–โŸถ0,๐‘‡โŸถโˆž,lim๐‘‡โ†’โˆž๎๐‘“๐‘‡๎๐‘“(๐‘ฅ)=(๐‘ก),(9.6) whereโ€‰โ€‰lim is a short-hand for โ€œlimit in the mean.โ€ The Parseval identity readsโ€–โ€–๎๐‘“โ€–โ€–2=โ€–๐‘“โ€–2,๎€œโˆž0||๎||๐‘“(๐‘ก)2๎€œ๐‘‘๐‘ก=โˆž0||||๐‘“(๐‘ก)2๐‘‘๐‘ก.(9.7)

If we apply (9.7) to a causal function ๐‘“, then it leads to [1, (3.19)]๎€œโˆžโˆ’โˆž||๐ฟยฑ[๐‘“]||(๐‘–๐œ”)2๎€œ๐‘‘๐œ”=โˆž0||||๐‘“(๐‘ก)2๐‘‘๐‘ก.(9.8)

Hence we see that [1, (3.19)] is indeed the Parseval identity for the Fourier (or Plancherel) transform for ๐‘“โˆˆ๐ฟ2(โ„).

10. Examples of Second-Order Systems

10.1. Electrical Circuits

The electric current ๐‘–=๐‘–(๐‘ก) flowing an electrical circuit which consists of four ingredients, electromotive-force ๐‘’=๐‘’(๐‘ก), resistance ๐‘…, coil ๐ฟ, and condenser ๐ถ, satisfies๐ฟ๐‘‘2๐‘–๐‘‘๐‘ก2+๐‘…๐‘‘๐‘–+1๐‘‘๐‘ก๐ถ๐‘–=๐‘’๎…ž(๐‘ก).(10.1)

10.2. Newtonโ€™s Equation of Motion (cf. [7])

One has๐‘€๐‘‘2๐‘ฆ๐‘‘๐‘ก2+๐‘…๐‘‘๐‘ฆ๐‘‘๐‘ก+๐พ๐‘ฆ=๐‘’(๐‘ก)=๐น,(10.2) where ๐‘€ is the inertance of mass, ๐‘… is the viscous resistance of the dashpot, and ๐พ is the spring stiffness.

Introducing the new parameters๐œ”๐‘›=๎‚™๐พ๐‘€๐‘…โˆถnaturalangularfrequence,๐œ=โˆš2๐พ/๐‘€โˆถdampingratio,(10.3)

(10.2) becomes1๐œ”2๐‘›๐‘‘2๐‘ฆ๐‘‘๐‘ก2+2๐œ๐œ”๐‘›๐‘‘๐‘ฆ1๐‘‘๐‘ก+๐‘ฆ=๐พ๐น.(10.4)

11. Laplace Transforms

To solve (10.1), we use the Laplace transform which has been defined by (9.3) and we state its definition independently.

Definition 11.1. Suppose ๐‘ฆ(๐‘ก)=๐‘‚(๐‘’๐‘Ž๐‘ก),๐‘กโ†’โˆž for an aโˆˆโ„. The Laplace transform ๐‘Œ(๐‘ )=๐ฟ[๐‘ฆ](๐‘ ) of ๐‘ฆ=๐‘ฆ(๐‘ก) is defined by ๐ฟ[๐‘ฆ]๎€œ(๐‘ )=โˆž0๐‘’โˆ’๐‘ ๐‘ก๐‘ฆ(๐‘ก)๐‘‘๐‘ก,Re๐‘ >๐‘Ž.(11.1) The integral converges absolutely in Re๐‘ >๐‘Ž and represents an analytic function there.

Example 11.2. Letโ€‰โ€‰๐›ผโˆˆโ„‚. Then ๐ฟ๎€บ๐‘’๐›ผ๐‘ก๎€ป1(๐‘ )=,๐‘ โˆ’๐›ผ(11.2) valid for Re๐‘ >๐‘…๐‘’๐›ผ in the first instance. The right-hand side of (11.2) gives a meromorphic continuation of the left-hand side to the punctured domain โ„‚โงต{๐›ผ}. Furthermore, (11.2) with ๐›ผ replaced by ๐‘–๐›ผ reads ๐ฟ[]๐›ผsin๐›ผ๐‘ก(๐‘ )=๐‘ 2+๐›ผ2๐ฟ[]๐‘ ,(11.3)cos๐›ผ๐‘ก(๐‘ )=๐‘ 2+๐›ผ2.(11.4)
For ๐›ผ=๐œ”โˆˆโ„ they reduce to familiar formulas: ๐ฟ[]๐œ”sin๐œ”๐‘ก(๐‘ )=๐‘ 2+๐œ”2,๐ฟ[]๐‘ cos๐œ”๐‘ก(๐‘ )=๐‘ 2+๐œ”2.(11.5)

Proof. By definition (11.2) clearly holds true. Since the right-hand side is analytic in โ„‚โงต{๐›ผ}, the consistency theorem establishes the last assertion. Once (11.2) is established, we have ๐ฟ๎€บ๐‘’๐‘–๐›ผ๐‘ก๎€ป1(๐‘ )=๎€บ๐‘’๐‘ โˆ’๐‘–๐›ผ,๐ฟโˆ’๐‘–๐›ผ๐‘ก๎€ป1(๐‘ )=,๐‘ +๐‘–๐›ผ(11.6) whence, for example, ๐ฟ[]1cos๐›ผ๐‘ก(๐‘ )=2๎€ท๐ฟ๎€บ๐‘’๐‘–๐›ผ๐‘ก๎€ป๎€บ๐‘’(๐‘ )+๐ฟโˆ’๐‘–๐›ผ๐‘ก๎€ป๎€ธ=๐‘ (๐‘ )๐‘ 2+๐›ผ2,(11.7)by Eulerโ€™s identity, that is, (11.4).

12. Partial Fraction Expansion and Examples

As long as the input function is a sinusoidal function, Example 11.2 will suffice to compute its Laplace transform. To go back to the time domain from the frequency domain, we need to solve the DE and, for most purposes, the following partial fraction expansion will give the answer almost automatically.

The following theorem, which is well known, provides us with the partial fraction expansion.

Theorem 12.1. If the denominator ๐ถ(๐‘ง) of the rational function ๐‘†(๐‘ง)=๐‘ƒ(๐‘ง)/๐ถ(๐‘ง) is given by ๐ถ(๐‘ง)=๐‘0๐‘ž๎‘๐‘–=1๎€ท๐‘งโˆ’๐›ฝ๐‘–๎€ธ๐œŽ๐‘–,๐‘ž๎“๐‘–=1๐œŽ๐‘–=deg๐ถ=๐ฟ,(12.1) where ๐›ฝ๐‘–=๐œŽ๐‘–โˆ’1, then ๐‘†(๐‘ง)=๐‘ž๎“๐œŽ๐‘–=1๐‘–โˆ’๐‘—๎“๐‘—=0๐‘Ž๐‘˜,๐œŽ๐‘˜โˆ’๐‘—1๎€ท๐‘งโˆ’๐›ฝ๐‘–๎€ธ๐œŽ๐‘–โˆ’๐‘—,(12.2) where the coefficients are given by ๐‘Ž๐‘–,๐œŽ๐‘–โˆ’๐‘—=1๐‘—!lim๐‘งโ†’๐›ฝ๐‘–d๐‘—d๐‘ง๐‘—๎€ท๎€ท๐‘งโˆ’๐›ฝ๐‘–๎€ธ๐œŽ๐‘–๎€ธ.๐‘…(๐‘ง)(12.3)

Proof. By (12.1), for each ๐‘–, 1โ‰ค๐‘–โ‰ค๐‘ž, we may write ๐‘ƒ๐‘†(๐‘ง)=๐‘–(๐‘ง)๎€ท๐‘งโˆ’๐›ฝ๐‘–๎€ธ๐œŽ๐‘–,๐‘ƒ๐‘–(๐‘ง)โˆˆโ„‚(๐‘ง),(12.4) and ๐‘†๐‘–(๐‘ง) has no pole at ๐‘ง=๐›ฝ๐‘–. We write ๎€ท๎€ท๐‘งโˆ’๐›ฝ๐‘–๎€ธ๐œŽ๐‘–๎€ธ๐‘ƒ๐‘†(๐‘ง)=๐‘–(๐‘ง)=๐œŽ๐‘–โˆ’1๎“๐‘—=0๐‘Ž๐‘–,๐œŽ๐‘–โˆ’๐‘—๎€ท๐‘งโˆ’๐›ฝ๐‘–๎€ธ๐‘—+๎€ท๐‘งโˆ’๐›ฝ๐‘–๎€ธ๐œŽ๐‘–๐ป๐‘–(๐‘ง),(12.5) where ๐ป๐‘–(๐‘ง)โˆˆโ„‚(๐‘ง) has no pole at ๐‘ง=๐›ฝ๐‘–. By successively differentiating and setting ๐‘ง=๐›ฝ๐‘–, we obtain (12.3).
Now, the rational function ๐น(๐‘ง)โˆถ=๐‘†(๐‘ง)โˆ’๐‘ž๎“๐œŽ๐‘–=1๐‘–โˆ’1๎“๐‘—=0๐‘Ž๐‘˜,๐‘Ÿ๐‘˜โˆ’๐‘—1๎€ท๐‘งโˆ’๐›ฝ๐‘–๎€ธ๐œŽ๐‘–โˆ’๐‘—(12.6) has no pole, so that it must be a polynomial. But, since lim๐‘งโ†’โˆž๐น(๐‘ง)=0 (where we use the assumption deg๐‘ƒ<deg๐ถ), it follows that ๐น(๐‘ง) must be zero.

Now we will give examples of (2.2) for the second-order systems which do not appear anywhere else save for [2].

Example 12.2. We find the output signal (current) ๐‘ฆ=๐‘ฆ(๐‘ก) described by the DE ๐‘ฆ๎…ž๎…ž+๐‘ฆ๎…ž+๐‘ฆ=๐‘’โˆ’(1/2)๐‘กโˆšsin32๐‘ก,(12.7) where the initial values are assumed to be 0: ๐‘ฆ(0)=0,๐‘ฆ๎…ž(0)=0.

Proof. Let ๐‘Œ(๐‘ )=๐ฟ[๐‘ฆ](๐‘ ) be the Laplace transform of ๐‘ฆ(๐‘ก). Then we have [๐‘ข]โˆš๐‘Œ(๐‘ )=ฮฆ(๐‘ )๐‘ˆ(๐‘ ),๐‘ˆ(๐‘ )=๐ฟ(๐‘ )=3/2๐‘ 2,2+๐‘ +1โˆš3๐ฟ[๐‘ฆ]1(๐‘ )=๎€ท๐‘ 2๎€ธ+๐‘ +12,(12.8) and we may obtain the partial fraction expansion 1๎€ท๐‘ 2๎€ธ+๐‘ +12=โˆ’1/3(๐‘ โˆ’๐œŒ)2+โˆ’๎‚€๎‚€3โˆš2/3๐‘–๎‚๎‚+๐‘ โˆ’๐œŒโˆ’1/3๎€ท๐‘ โˆ’๐œŒ๎€ธ2+๎‚€๎‚€3โˆš2/3๐‘–๎‚๎‚๐‘ โˆ’๐œŒ,(12.9)where ๐œŒ=๐‘’(2๐œ‹๐‘–/3)โˆš=(โˆ’1+3๐‘–)/2 is the first primitive cube root of 1. Hence 2โˆš32๐‘ฆ(๐‘ก)=โˆš3๐ฟโˆ’1[๐ฟ[๐‘ฆ2]](๐‘ก)=โˆ’3๐‘ก๐‘’โˆ’(1/2)๐‘กโˆšcos322๐‘ก+3โˆš3๐‘’โˆ’(1/2)๐‘กโˆšsin32๐‘ก.(12.10) As a transfer function, the function in (12.8) 1ฮฆ(๐‘ )=๐‘ 2+๐‘ +1(12.11) is stable.

Example 12.3. The following integral may be evaluated by the partial fraction expansion above or by the residue calculus: ๎€œโˆžโˆ’โˆž1๎€ท๐‘ฅ2๎€ธ+๐‘ฅ+12๎ƒฉโˆ’2๐‘‘๐‘ฅ=2๐œ‹๐‘–3โˆš3๐‘–๎ƒช=43โˆš3๐œ‹.(12.12)

Example 12.4. In the same vein as with Example 12.2, we may find the solution of the DE ๐‘ฆ๎…ž๎…žโˆ’๐‘ฆ๎…ž+๐‘ฆ=๐‘ข(๐‘ก)=๐‘’(1/2)๐‘กโˆšsin32๐‘ก,(12.13)where the initial values are assumed to be 0: ๐‘ฆ(0)=0,๐‘ฆ๎…ž(0)=0.

We have๐‘Œ(๐‘ )=ฮฆ1[๐‘ข]โˆš(๐‘ )๐‘ˆ(๐‘ ),๐‘ˆ(๐‘ )=๐ฟ(๐‘ )=3/2๐‘ 2,โˆ’๐‘ +1(12.14) or2โˆš3๐ฟ[๐‘ฆ]1(๐‘ )=๎€ท๐‘ 2๎€ธโˆ’๐‘ +12.(12.15)

Hence2โˆš32๐‘ฆ(๐‘ก)=โˆš3๐ฟโˆ’1[๐ฟ[๐‘ฆ]](๐‘ก)=โˆ’๐‘ก๐‘’(1/2)๐‘กโˆšcos32โˆš๐‘ก+3๐‘’(1/2)๐‘กโˆšsin32๐‘ก,(12.16)

and the transfer function in (12.14)ฮฆ11(๐‘ )=๐‘ 2โˆ’๐‘ +1(12.17)

is unstable.

13. The Product of Zeta-Functions: ฮ“ฮ“-Type

In this section, we illustrate the use of fractional integrals by proving a slight generalization of the result of Chandrasekharan and Narasimhan ([8]) involving the ฮ“ฮ“-type functional equation, which is the first instance beyond Hecke theory of the functional equation with a single gamma factor. First we state the basic settings.

13.1. Statement of the Situation

Let {๐œ†๐‘˜},{๐œ‡๐‘˜} be increasing sequences of positive numbers tending to โˆž, and let {๐›ผ๐‘˜},{๐›ฝ๐‘˜} be complex sequences. We form the Dirichlet series๐œ‘(๐‘ )=โˆž๎“๐‘˜=1๐›ผ๐‘˜๐œ†๐‘ ๐‘˜,๐œ“(๐‘ )=โˆž๎“๐‘˜=1๐›ฝ๐‘˜๐œ‡๐‘ ๐‘˜(13.1) and suppose that they have finite abscissas of absolute convergence ๐œŽ๐œ‘, ๐œŽ๐œ“, respectively.

We suppose the existence of the meromorphic function ๐œ’ satisfying the functional equation (of ฮ“ฮ“-type) of the form with ๐‘Ÿ a real number and having a finite number of poles ๐‘ ๐‘˜(1โ‰ค๐‘˜โ‰ค๐ฟ): โŽงโŽชโŽจโŽชโŽฉฮ“๎‚€๐œˆ๐œ’(๐‘ )=๐‘ +2๎‚ฮ“๎‚€๐œˆ๐‘ โˆ’2๎‚๎‚ต๐œ‘(๐‘ ),Re๐‘ >๐œŽ๐œ‘๎‚ถ,ฮ“๎‚€๐œˆ๐‘Ÿโˆ’๐‘ +2๎‚ฮ“๎‚€๐œˆ๐‘Ÿโˆ’๐‘ โˆ’2๎‚๎‚ต๐œ“(๐‘Ÿโˆ’๐‘ ),Re๐‘ <๐‘Ÿโˆ’๐œŽ๐œ“๎‚ถ(13.2) We introduce the processing gamma factorฮ“๎‚€๎€ฝ๐‘ฮ”(๐‘ค)=๐‘—+๐ต๐‘—๐‘ค๎€พ๐‘š๐‘—=1๎‚ฮ“๎‚€๎€ฝ๐‘Ž๐‘—โˆ’๐ด๐‘—๐‘ค๎€พ๐‘›๐‘—=1๎‚ฮ“๎‚€๎€ฝ๐‘Ž๐‘—+๐ด๐‘—๐‘ค๎€พ๐‘๐‘—=๐‘›+1๎‚ฮ“๎‚€๎€ฝ๐‘๐‘—โˆ’๐ต๐‘—๐‘ค๎€พ๐‘ž๐‘—=๐‘š+1๎‚๎€ท๐ด๐‘—,๐ต๐‘—๎€ธ>0(13.3) and suppose that for any real numbers ๐‘ข1,๐‘ข2(๐‘ข1<๐‘ข2)lim|๐‘ฃ|โ†’โˆžฮ”(๐‘ข+๐‘–๐‘ฃโˆ’๐‘ )๐œ’(๐‘ข+๐‘–๐‘ฃ)=0,(13.4) uniformly in ๐‘ข1โ‰ค๐‘ขโ‰ค๐‘ข2.

In the ๐‘ค-plane we take two deformed Bromwich paths๐ฟ1(๐‘ )โˆถ๐›พ1โˆ’๐‘–โˆžโŸถ๐›พ1+๐‘–โˆž,๐ฟ2(๐‘ )โˆถ๐›พ2โˆ’๐‘–โˆžโŸถ๐›พ2๎€ท๐›พ+๐‘–โˆž2<๐›พ1๎€ธ(13.5)

such that they squeeze a compact set ๐’ฎ with boundary ๐’ž for which ๐‘ ๐‘˜โˆˆ๐’ฎ(1โ‰ค๐‘˜โ‰ค๐ฟ) and all the poles ofฮ“๎‚€๎€ฝ๐‘๐‘—โˆ’๐ต๐‘—๐‘ +๐ต๐‘—๐‘ค๎€พ๐‘š๐‘—=1๎‚ฮ“๎‚€๎€ฝ๐‘Ž๐‘—โˆ’๐ด๐‘—๐‘ +๐ด๐‘—๐‘ค๎€พ๐‘๐‘—=๐‘›+1๎‚ฮ“๎‚€๎€ฝ๐‘๐‘—+๐ต๐‘—๐‘ โˆ’๐ต๐‘—๐‘ค๎€พ๐‘ž๐‘—=๐‘š+1๎‚(13.6)

lie to the left of ๐ฟ2(๐‘ ) and those ofฮ“๎‚€๎€ฝ๐‘Ž๐‘—+๐ด๐‘—๐‘ โˆ’๐ด๐‘—๐‘ค๎€พ๐‘›๐‘—=1๎‚ฮ“๎‚€๎€ฝ๐‘Ž๐‘—โˆ’๐ด๐‘—๐‘ +๐ด๐‘—๐‘ค๎€พ๐‘๐‘—=๐‘›+1๎‚ฮ“๎‚€๎€ฝ๐‘๐‘—+๐ต๐‘—๐‘ โˆ’๐ต๐‘—๐‘ค๎€พ๐‘ž๐‘—=๐‘š+1๎‚(13.7)

lie to the right of ๐ฟ1(๐‘ ).

Then we define the ๐ป-function by (0โ‰ค๐‘›โ‰ค๐‘,0โ‰ค๐‘šโ‰ค๐‘ž,๐ด๐‘—,๐ต๐‘—>0)๐ป๐‘š,๐‘›๐‘,๐‘žโŽ›โŽœโŽœโŽ๎€ท๐‘งโˆฃ1โˆ’๐‘Ž1,๐ด1๎€ธ๎€ท,โ€ฆ,1โˆ’๐‘Ž๐‘›,๐ด๐‘›๎€ธ,๎€ท๐‘Ž๐‘›+1,๐ด๐‘›+1๎€ธ๎€ท๐‘Ž,โ€ฆ,๐‘,๐ด๐‘๎€ธ๎€ท๐‘1,๐ต1๎€ธ๎€ท๐‘,โ€ฆ,๐‘š,๐ต๐‘š๎€ธ,๎€ท1โˆ’๐‘๐‘š+1,๐ต๐‘š+1๎€ธ๎€ท,โ€ฆ,1โˆ’๐‘๐‘ž,๐ต๐‘ž๎€ธโŽžโŽŸโŽŸโŽ =1๎€œ2๐œ‹๐‘–๐ฟฮ“๎€ท๐‘1+๐ต1๐‘ ,โ€ฆ,๐‘๐‘š+๐ต๐‘š๐‘ ๎€ธฮ“๎€ท๐‘Ž1โˆ’๐ด1๐‘ ,โ€ฆ,๐‘Ž๐‘›โˆ’๐ด๐‘›๐‘ ๎€ธฮ“๎€ท๐‘Ž๐‘›+1+๐ด๐‘›+1๐‘ ,โ€ฆ,๐‘Ž๐‘+๐ด๐‘๐‘ ๎€ธฮ“๎€ท๐‘๐‘š+1โˆ’๐ต๐‘š+1๐‘ ,โ€ฆ,๐‘๐‘žโˆ’๐ต๐‘ž๐‘ ๎€ธ๐‘งโˆ’๐‘ ๐‘‘๐‘ .(H-1)

In the special case, where ๐ด๐‘—=๐ต๐‘—=1, the ๐ป-function reduces to ๐บ-functions and denoted by ๐บ with other parameters remaining the same. We also define the ๐œ’-function X(๐‘ง,๐‘ ) by1X(๐‘ง,๐‘ )=๎€œ2๐œ‹๐‘–๐ฟ1(๐‘ )ฮ”(๐‘คโˆ’๐‘ )๐œ’(๐‘ค)๐‘งโˆ’๐‘ค๐‘‘๐‘ค,(13.8) which is for ๐œ’=1 one of ๐ป-functions. Hereafter we always assume that ๐‘ง>0, which may be extended to Re๐‘ง>0.

Then we have1X(๐‘ง,๐‘ )=๎€œ2๐œ‹๐‘–๐ฟ2(๐‘ )ฮ”(๐‘คโˆ’๐‘ )๐œ’(๐‘ค)๐‘งโˆ’๐‘ค1๐‘‘๐‘ค+๎€œ2๐œ‹๐‘–๐’žฮ”(๐‘คโˆ’๐‘ )๐œ’(๐‘ค)๐‘งโˆ’๐‘ค๐‘‘๐‘ค,(13.9) which amounts to the following.

Theorem 13.1 ([9]). One has the modular relation equivalent to (13.2):=X(๐‘ง,๐‘ )โˆž๎“๐‘˜=1๐›ผ๐‘˜๐œ†๐‘ ๐‘˜ร—๐ป๐‘š+2,๐‘›๐‘,๐‘ž+2โŽ›โŽœโŽœโŽ๐‘ง๐œ†๐‘˜โˆฃ๎€ฝ๎€ท1โˆ’๐‘Ž๐‘—,๐ด๐‘—๎€ธ๎€พ๐‘›1,๐‘Ž๎€ฝ๎€ท๐‘—,๐ด๐‘—๎€ธ๎€พ๐‘๐‘›+1๎‚€๐œˆ๐‘ +2๎‚,๎‚€๐œˆ,1๐‘ โˆ’2๎‚,๐‘,1๎€ฝ๎€ท๐‘—,๐ต๐‘—๎€ธ๎€พ๐‘š1,๎€ฝ๎€ท1โˆ’๐‘๐‘—,๐ต๐‘—๎€ธ๎€พ๐‘ž๐‘š+1โŽžโŽŸโŽŸโŽ =โˆž๎“๐‘˜=1๐›ฝ๐‘˜๐œ‡๐‘˜๐‘Ÿโˆ’๐‘ ร—๐ป๐‘›+2,๐‘š๐‘ž,๐‘+2โŽ›โŽœโŽœโŽ๐œ‡๐‘˜๐‘งโˆฃ๎€ฝ๎€ท1โˆ’๐‘๐‘—,๐ต๐‘—๎€ธ๎€พ๐‘š1,๐‘๎€ฝ๎€ท๐‘—,๐ต๐‘—๎€ธ๎€พ๐‘ž๐‘š+1๎‚€๐œˆ๐‘Ÿโˆ’๐‘ +2๎‚,๎‚€๐œˆ,1๐‘Ÿโˆ’๐‘ โˆ’2๎‚,๐‘Ž,1๎€ฝ๎€ท๐‘—,๐ด๐‘—๎€ธ๎€พ๐‘›1,๎€ฝ๎€ท1โˆ’๐‘Ž๐‘—,๐ด๐‘—๎€ธ๎€พ๐‘๐‘›+1โŽžโŽŸโŽŸโŽ +๐ฟโˆ‘๐‘˜=1๎€ทResฮ”(๐‘คโˆ’๐‘ )๐œ’(๐‘ค)๐‘ง๐‘ โˆ’๐‘ค,๐‘ค=๐‘ ๐‘˜๎€ธ.๎ƒฉ๐‘›โˆ‘๐‘—=1๐ด๐‘—+๐‘šโˆ‘๐‘—=1๐ต๐‘—+2โ‰ฅ๐‘โˆ‘๐‘—=๐‘›+1๐ด๐‘—+๐‘žโˆ‘๐‘—=๐‘š+1๐ต๐‘—๎ƒช.(13.10)

In the special case, where ๐ด๐‘—=๐ต๐‘—=1, we have the following.

Theorem 13.2. One has ๐‘ง๐‘ X(๐‘ง,๐‘ )=โˆž๎“๐‘˜=1๐›ผ๐‘˜๐œ†๐‘ ๐‘˜๐บ๐‘š+2,๐‘›๐‘,๐‘ž+2โŽ›โŽœโŽœโŽ๐‘ง๐œ†๐‘˜โˆฃ1โˆ’๐‘Ž1,โ€ฆ,1โˆ’๐‘Ž๐‘›,๐‘Ž๐‘›+1,โ€ฆ,๐‘Ž๐‘๐œˆ๐‘ +2๐œˆ,๐‘ โˆ’2,๐‘1,โ€ฆ,๐‘๐‘š,1โˆ’๐‘๐‘š+1,โ€ฆ,1โˆ’๐‘๐‘žโŽžโŽŸโŽŸโŽ =โˆž๎“๐‘˜=1๐›ฝ๐‘˜๐œ‡๐‘˜๐‘Ÿโˆ’๐‘ ร—๐บ๐‘›+2,๐‘š๐‘ž,๐‘+2โŽ›โŽœโŽœโŽ๐œ‡๐‘˜๐‘งโˆฃ1โˆ’๐‘1,โ€ฆ,1โˆ’๐‘๐‘š,๐‘๐‘š+1,โ€ฆ,๐‘๐‘ž๐œˆ๐‘Ÿโˆ’๐‘ +2๐œˆ,๐‘Ÿโˆ’๐‘ โˆ’2,๐‘Ž1,โ€ฆ,๐‘Ž๐‘›,1โˆ’๐‘Ž๐‘›+1,โ€ฆ,1โˆ’๐‘Ž๐‘โŽžโŽŸโŽŸโŽ +๐ฟ๎“๐‘˜=1๎€ทResฮ”(๐‘คโˆ’๐‘ )๐œ’(๐‘ค)๐‘ง๐‘ โˆ’๐‘ค,๐‘ค=๐‘ ๐‘˜๎€ธ,(2๐‘›+2๐‘š+2โ‰ฅ๐‘+๐‘ž).(13.11)

For many important applications, compare [9].

13.2. The Riesz Sum (๐บ2,24,4โ†”๐บ4,02,6).

Formula (13.11) in the special case of the title readsโˆž๎“๐‘˜=1๐›ผ๐‘˜๐œ†๐‘ ๐‘˜๐บ2,24,4โŽ›โŽœโŽœโŽ๐‘ง๐œ†๐‘˜โˆฃ๐œˆ๐‘Ž,๐‘,๐‘,๐‘‘๐‘ +2๐œˆ,๐‘ โˆ’2โŽžโŽŸโŽŸโŽ =,๐‘’,๐‘“โˆž๎“๐‘˜=1๐›ฝ๐‘˜๐œ‡๐‘Ÿโˆ’๐‘ ๐บ4,02,6โŽ›โŽœโŽœโŽ๐œ‡๐‘˜๐‘งโˆฃ๐œˆ1โˆ’๐‘’,1โˆ’๐‘“๐‘Ÿโˆ’๐‘ +2๐œˆ,๐‘Ÿโˆ’๐‘ โˆ’2โŽžโŽŸโŽŸโŽ +,1โˆ’๐‘Ž,1โˆ’๐‘,1โˆ’๐‘,1โˆ’๐‘‘๐ฟ๎“๐‘˜=1๎€ทResฮ”(๐‘คโˆ’๐‘ )๐œ’(๐‘ค)๐‘ง๐‘ โˆ’๐‘ค,๐‘ค=๐‘ ๐‘˜๎€ธ,(13.12) whereฮ”(๐‘ค)=ฮ“(1โˆ’๐‘Žโˆ’๐‘ค)ฮ“(1โˆ’๐‘โˆ’๐‘ค).ฮ“(๐‘+๐‘ค)ฮ“(๐‘‘+๐‘ค)ฮ“(1โˆ’๐‘’โˆ’๐‘ค)ฮ“(1โˆ’๐‘“โˆ’๐‘ค)(13.13)

We treat the case ๐‘Ÿ=1/2. Assuming ๐œ† is a nonnegative integer, we put ๐‘Ž=๐‘ +(๐œˆ/2)+(๐œ†/2)+(1/2), ๐‘=๐‘ +(๐œˆ/2)+(๐œ†/2)+1, ๐‘=๐‘ โˆ’(๐œˆ/2), ๐‘‘=๐‘ +(๐œˆ/2)+๐œ†+1, ๐‘’=๐‘ +(๐œˆ/2)+(1/2), ๐‘“=๐‘ +(๐œˆ/2)+๐œ†+1. Then (13.12) becomes โˆž๎“๐‘˜=1๐›ผ๐‘˜๐œ†๐‘ ๐‘˜๐บ2,24,4โŽ›โŽœโŽœโŽ๐‘ง๐œ†๐‘˜โˆฃ๐œˆ๐‘ +2+๐œ†2+12๐œˆ,๐‘ +2+๐œ†2๐œˆ+1,๐‘ โˆ’2๐œˆ,๐‘ +2๐œˆ+๐œ†+1๐‘ +2๐œˆ,๐‘ โˆ’2๐œˆ,๐‘ +2+12๐œˆ,๐‘ +2โŽžโŽŸโŽŸโŽ =+๐œ†+1โˆž๎“๐‘˜=1๐›ฝ๐‘˜๐œ‡(1/2)โˆ’๐‘ ๐บ4,02,6โŽ›โŽœโŽœโŽ๐œ‡๐‘˜๐‘งโˆฃ๐œˆโˆ’๐‘ โˆ’2+12๐œˆ,โˆ’๐‘ โˆ’2โˆ—โŽžโŽŸโŽŸโŽ +โˆ’๐œ†๐ฟ๎“๐‘˜=1๎€ทResฮ”(๐‘คโˆ’๐‘ )๐œ’(๐‘ค)๐‘ง๐‘ โˆ’๐‘ค,๐‘ค=๐‘ ๐‘˜๎€ธ,(13.14) where โˆ— indicates โˆ’๐‘ +๐œˆ/2+1/2, โˆ’๐‘ โˆ’๐œˆ/2+1/2, โˆ’๐‘ โˆ’๐œˆ/2โˆ’๐œ†/2+1/2, โˆ’๐‘ โˆ’๐œˆ/2โˆ’๐œ†/2, โˆ’๐‘ +๐œˆ/2+1, โˆ’๐‘ โˆ’๐œˆ/2โˆ’๐œ†.

We note that the ๐บ-functions in (13.14) reduce to๐บ2,24,4โŽ›โŽœโŽœโŽ๐œˆ๐‘งโˆฃ๐‘ +2+๐œ†2+12๐œˆ,๐‘ +2+๐œ†2๐œˆ+1,๐‘ โˆ’2๐œˆ,๐‘ +2๐œˆ+๐œ†+1๐‘ +2๐œˆ,๐‘ โˆ’2๐œˆ,๐‘ +2+12๐œˆ,๐‘ +2โŽžโŽŸโŽŸโŽ +๐œ†+1=2๐œ†๐บ1,01,1โŽ›โŽœโŽœโŽโˆšโŽžโŽŸโŽŸโŽ =โŽงโŽชโŽจโŽชโŽฉ2๐‘งโˆฃ2๐‘ +๐œˆ+๐œ†+12๐‘ +๐œˆ๐œ†๐‘งฮ“(๐œ†+1)๐‘ +(๐œˆ/2)๎‚€โˆš1โˆ’๐‘ง๎‚๐œ†,(|๐‘ง|<1)0,(|๐‘ง|>1)(13.15)

(by the formula in [10]) and๐บ4,02,6โŽ›โŽœโŽœโŽ๐œˆ๐‘งโˆฃโˆ’๐‘ โˆ’2+12๐œˆ,โˆ’๐‘ โˆ’2โŽžโŽŸโŽŸโŽ โˆ’๐œ†โˆ—โˆ—=๐บ4,02,6โŽ›โŽœโŽœโŽ๐œˆ๐‘งโˆฃโˆ’๐‘ โˆ’2๐œˆ+1,โˆ’๐‘ โˆ’2โ€ โŽžโŽŸโŽŸโŽ โˆ’๐œ†,(13.16)

where โˆ—โˆ— indicates โˆ’๐‘ +๐œˆ/2+1/2, โˆ’๐‘ โˆ’๐œˆ/2+1/2, โˆ’๐‘ โˆ’๐œˆ/2โˆ’๐œ†/2+1/2, โˆ’๐‘ โˆ’๐œˆ/2โˆ’๐œ†/2, โˆ’๐‘ +๐œˆ/2+1, โˆ’๐‘ โˆ’๐œˆ/2โˆ’๐œ† and โ€ , โˆ’๐‘ +๐œˆ/2+1/2, โˆ’๐‘ +๐œˆ/2+1, โˆ’๐‘ โˆ’๐œˆ/2โˆ’๐œ†/2, โˆ’๐‘ โˆ’๐œˆ/2โˆ’๐œ†/2+1/2, โˆ’๐‘ +๐œˆ/2+1, โˆ’๐‘ โˆ’๐œˆ/2โˆ’๐œ†. Hence it reduces further to๐‘งโˆ’๐‘ โˆ’(๐œ†/2)+(1/4)๎‚†2๐œ‹cos((๐œˆ+๐œ†+1)๐œ‹)๐พ2๐œˆ+๐œ†+1๎‚€44โˆš๐‘ง๎‚+cos((๐œˆ+1)๐œ‹)๐‘Œ2๐œˆ+๐œ†+1๎‚€44โˆš๐‘ง๎‚+sin((๐œˆ+1)๐œ‹)๐ฝ2๐œˆ+๐œ†+1๎‚€44โˆš๐‘ง๎‚๎‚‡=โˆ’๐‘งโˆ’๐‘ โˆ’(๐œ†/2)+(1/4)๎‚†(โˆ’1)๐œ†2๐œ‹cos(๐œˆ๐œ‹)๐พ2๐œˆ+๐œ†+1๎‚€44โˆš๐‘ง๎‚+cos(๐œˆ๐œ‹)๐‘Œ2๐œˆ+๐œ†+1๎‚€44โˆš๐‘ง๎‚+sin(๐œˆ๐œ‹)๐ฝ2๐œˆ+๐œ†+1๎‚€44โˆš๐‘ง๎‚๎‚‡=๐‘งโˆ’๐‘ โˆ’(๐œ†/2)+(1/4)๐บ๐œ†2๐œˆ+๐œ†+1๎‚€44โˆš๐‘ง๎‚,(13.17)

say, where, slightly more general than Wiltonโ€™s (1.22) [11], we put๐บ๐œ†๐œˆ(๐‘ง)=โˆ’(โˆ’1)๐œ†2๐œ‹๎‚€sin๐œˆโˆ’๐œ†2๐œ‹๎‚๐พ๐œˆ๎‚€(๐‘ง)โˆ’sin๐œˆโˆ’๐œ†2๐œ‹๎‚๐‘Œ๐œˆ๎‚€(๐‘ง)+cos๐œˆโˆ’๐œ†2๐œ‹๎‚๐ฝ๐œˆ(๐‘ง).(13.18) Hence (13.14) reads๐‘ง๐‘ +๐œˆ/22๐œ†๎“ฮ“(๐œ†+1)๐œ†๐‘˜<1/๐‘ง๐›ผ๐‘˜๐œ†๐‘˜๐œˆ/2๎‚€โˆš1โˆ’๐‘ง๐œ†๐‘˜๎‚๐œ†(13.19)=๐‘งโˆž๐‘ +๐œ†/4โˆ’1/4๎“๐‘˜=1๐›ฝ๐‘˜๐œ‡๐‘˜๐œ†/4+1/4๐บ๐œ†2๐œˆ+๐œ†+1๎ƒฉ44๎‚™๐œ‡๐‘˜๐‘ง๎ƒช+๐ฟ๎“๐‘˜=1๎€ทResฮ”(๐‘คโˆ’๐‘ )๐œ’(๐‘ค)๐‘ง๐‘ โˆ’๐‘ค,๐‘ค=๐‘ ๐‘˜๎€ธ,(13.20) which gives a more general form of Wiltonโ€™s Theoremโ€‰โ€‰1 [11].

Rewriting (13.19) slightly, we deduce an analogue of Chandrasekharan and Narasimhan result [8, Theoremโ€‰โ€‰7.1(a)],

Theorem 13.3. For ๐‘ฅ>0, the functional equation (13.2) implies the identity 1ฮ“๎“(๐œ†+1)๐œ†๐‘˜<๐‘ฅ๐›ผ๐‘˜๐œ†๐œˆ๐‘˜๎€ท๐‘ฅโˆ’๐œ†๐‘˜๎€ธ๐œ†=๐‘ฅ๐œ†/2+๐œˆ+1/22โˆžโˆ’๐œ†๎“๐‘˜=1๐›ฝ๐‘˜๐œ‡๐‘˜๐œ†/2+1/2๐บ๐œ†2๐œˆ+๐œ†+1๎€ท4โˆš๐œ‡๐‘˜๐‘ฅ๎€ธ+P๐œ†(๐‘ฅ),(13.21) where P๐œ†(๐‘ฅ)=๐‘ฅ๐ฟ2๐‘ +๐œ†+๐œˆ๎“๐‘˜=1๎€ทRe๐‘ ฮ”(๐‘คโˆ’๐‘ )๐œ’(๐‘ค)๐‘ฅ2(๐‘คโˆ’๐‘ ),๐‘ค=๐‘ ๐‘˜๎€ธ(13.22) and where ฮ”(๐‘ค)=ฮ“(1โˆ’๐‘ โˆ’๐œ†/2โˆ’๐œˆ/2โˆ’๐‘ค)ฮ“(โˆ’๐‘ โˆ’๐œ†/2โˆ’๐œˆ/2โˆ’๐‘ค),ฮ“(๐‘ โˆ’๐œˆ/2+๐‘ค)ฮ“(๐‘ +๐œ†+๐œˆ/2+1+๐‘ค)ฮ“(1โˆ’๐‘ โˆ’๐œˆ/2โˆ’1/2โˆ’๐‘ค)ฮ“โ„จ(13.23) where โ„จ denotes (โˆ’๐‘ โˆ’๐œ†โˆ’๐œˆ/2โˆ’๐‘ค), with ๐บ๐œ†2๐œˆ+๐œ†+1 being given by (13.18).

Corollary 13.4. For ๐‘ฅ>0, the functional equation (13.2) implies the identity ๐ด๐œ†1(๐‘ฅ)โˆถ=ฮ“๎“(๐œ†+1)๐œ†๐‘˜<๐‘ฅ๐›ผ๐‘˜๎€ท๐‘ฅโˆ’๐œ†๐‘˜๎€ธ๐œ†=โˆ’2โˆžโˆ’๐œ†๎“๐‘˜=1๐›ฝ๐‘˜๎‚ต๐‘ฅ๐œ‡๐‘˜๎‚ถ๐œ†/2+1/2๐น๐œ†+1๎€ท4โˆš๐œ‡๐‘˜๐‘ฅ๎€ธ+P๐œ†(๐‘ฅ),(13.24) where ๐น๐œ†+1(๐‘ง)=โˆ’๐บ๐œ†๐œ†+1(๐‘ง)=๐‘Œ๐œ†+1(๐‘ง)+(โˆ’1)๐œ†2๐œ‹๐พ๐œ†+1(๐‘ง).(13.25)

We are now in a position to prove an analogue of [8, Theorem 7.1(b)] (although Theorem 13.3 contains [8, Theorem 7.2], too) by the Riemann-Liouville fractional integral transform.

Lemma 13.5 (Riemann-Liouville integral of Bessel functions). For the well-known Bessel functions ๐ฝ and ๐‘Œ, one has 1๎€œฮ“(๐œ‡)๐‘ฆ0(๐‘ฆโˆ’๐‘ฅ)๐œ‡โˆ’1๐‘ฅ(1/2)๐œˆ๐ฝ๐œˆ๎€ท๐‘Ž๐‘ฅ1/2๎€ธ๐‘‘๐‘ฅ=2๐œ‡๐‘Žโˆ’๐œ‡๐‘ฆ(1/2)๐œ‡+(1/2)๐œˆ๐ฝ๐œ‡+๐œˆ๎€ท๐‘Ž๐‘ฆ1/2๎€ธ๎‚ต๎‚ถRe๐œ‡>0,Re๐œˆ>0(13.26)1๎€œฮ“(๐œ‡)๐‘ฆ0(๐‘ฆโˆ’๐‘ฅ)๐œ‡โˆ’1๐‘ฅ(1/2)๐œˆ๐‘Œ๐œˆ๎€ท๐‘Ž๐‘ฅ1/2๎€ธ๐‘‘๐‘ฅ=2๐œ‡๐‘Žโˆ’๐œ‡๐‘ฆ(1/2)๐œ‡+(1/2)๐œˆ๐ฝ๐œ‡+๐œˆ๎€ท๐‘Ž๐‘ฆ1/2๎€ธ+ฮ“(๐œˆ+1)ฮ“2(๐œ‡)๐œ‹๐œˆ+2๐‘Žโˆ’๐œ‡๐‘ฆ(1/2)๐œ‡+(1/2)๐œˆ๐‘†๐œ‡โˆ’๐œˆโˆ’1,๐œ‡+๐œˆ๎€ท๐‘Ž๐‘ฆ1/2๎€ธ๎‚ต๎‚ถ,Re๐œ‡>0,Re๐œˆ>โˆ’1(13.27) where ๐‘† stands for the Lommel function.

Equation (13.26) is [12, (63), page 194] and (13.27) is [12, page 196]. We only need (13.27) and (13.27) is for treating the ๐ฝ-Bessel function.

Arguing in the same way as in [8], we may prove the following.

Theorem 13.6. With a ๐ถโˆž-function ๐‘…๐œŒ and a certain constant ๐‘ one has ๐ด๐œŒ(๐‘ฅ)โˆ’๐‘…๐œŒ(๐‘ฅ)=๐‘โˆž๎“๐‘˜=1๐›ฝ๐‘˜๎‚ต๐‘ฅ๐œ‡