CONFORMAL TRANSFORMATIONS IN ELECTRICAL ENGINEERING

By U. HAVAE

CONTENTS IN?NITE ORDER DIFFERENTIAL OPERATORS WITH A GLIMPSE TO APPLICATIONS TO SUPEROSCILLATIONS INTERPOLATION IN MULTIVARIABLE DE BRANGES-ROVNYAK SPACES MONOTONICITY OF CERTAIN LEFT AND RIGHT RIEMANN SUMS A SURVEY ON THE RECENT ADVANCES IN THE SPECTRAL THEORY ON THE S-SPECTRUM STOCHASTICS AND DYNAMICS OF FRACTALS FLUID–PLATE INTERACTION WITH KELVIN-VOIGT DAMPING AND BENDING MOMENT AT THE INTERFACE: WELL-POSEDNESS, SPECTRAL ANALYSIS, UNIFORM STABILITY AUTOMORPHISMS OF HYPER-REINHARDT FREE SPECTRAHEDRA N SIN X N, PART II LIPSCHITZ-TYPE BOUNDS FOR FUNCTIONS OF OPERATORS WITH NONCOMPACT PERTURBATIONS EXTENDED FOCK SPACE FORMALISM AND POLYANALYTIC FUNCTIONS ESTIMATES OF CAUCHY–SZEGÖ KERNEL IN HARDY SPACES ON NILPOTENT LIE GROUPS OF STEP TWO AN(Z) DN DZN WHERE THE FUNCTIONS AN ARE ENTIRE FUNCTIONS ON THE COMPLEX PLANE (POSSIBLY SATISFYING SUITABLE GROWTH CONDITIONS), AND WE WILL STUDY THEIR ACTION ON SUITABLE SPACES OF ENTIRE FUNCTIONS AS WELL AFTER AN INTRODUCTORY SECTION, WHOSE CONTENT IS WELL KNOWN, THAT DESCRIBES IN?NITE ORDER DIFFERENTIAL OPERATORS FROM THE POINT OF VIEW OF THE THEORY OF HYPERFUNCTIONS, WE WILL DESCRIBE AS WELL CONVOLUTORS ARISING FROM ANALYTIC FUNCTIONALS, AND HOW THEY CAN BE REPRESENTED BY IN?NITE SERIES OF DERIVATIVES THE NEXT SECTION IS DEDICATED TO THE WAY IN WHICH THE STUDY OF LONGEVITY PHENOMENA FOR SUPEROSCILLATIONS HAS LED TO A RENEWED INTEREST FOR THE THEORY OF IN?NITE ORDER DIFFERENTIAL OPERATORS, AND WILL PRESENT SOME RECENT RESULTS ON THE CONTINUITY OF SUCH OPERATORS FINALLY, WE WILL SHOW HOW THE IDEA OF IN?NITE ORDER DIFFERENTIAL OPERATORS EXTENDS FRUITFULLY TO THE HYPERCOMPLEX SETTING KEYWORDS IN?NITE ORDER DIFFERENTIAL OPERATORS ENTIRE FUNCTIONS WITH GROWTH CONDITIONS ENTIRE HYPERHOLOMORPHIC FUNCTIONS N! DN DZN , WHICH IS SUCH THAT FOR ANY ENTIRE FUNCTION F (Z) ONE HAS P D DZ F (Z) = F (Z + 1) IN ORDER TO CLARIFY THIS SITUATION WE WILL BEGIN WITH CONSIDERING THE SPACE H{0} OF GERMS OF ONE-VARIABLE HOLOMORPHIC FUNCTIONS AT THE ORIGIN THIS SPACE CAN BE ENDOWED WITH THE INDUCTIVE LIMIT TOPOLOGY THAT DERIVES FROM DE?NING H{0} := LIM ??H(U) OVER A DECREASING FAMILY OF OPEN SETS WHOSE INTERSECTION IS THE ORIGIN ITS DUAL H? {0} IS ISOMORPHIC TO THE SPACE OF ONE-VARIABLE HYPERFUNCTIONS SUPPORTED BY THE ORIGIN, AND AS SUCH IS ISOMORPHIC TO THE QUOTIENT B{0} = H(C \ {0}) H(C) CONFORMAL TRANSFORMATIONS IN ELECTRICAL ENGINEERING AN DN DZN ACTS CONTINUOUSLY ON THE SPACE OF ENTIRE FUNCTIONS ON C IF LIM K?? K K!|AK| = 0 A SECOND INTERESTING QUESTIONS IS TO INQUIRE AS TO WHAT HAPPENS WHEN THE COEF?CIENTS AN ARE NOT CONSTANT BUT, SAY, HOLOMORPHIC FUNCTIONS AN(Z) THIS QUESTION IS ALSO WELL UNDERSTOOD, AND WE STILL HAVE IN?NITE ORDER DIFFERENTIAL OPERATORS (THAT IS OBJECTS THAT ACT ON THE SHEAF OF HOLOMORPHIC FUNCTIONS), AS LONG AS THE SAME KIND OF GROWTH CONFORMAL TRANSFORMATIONS IN ELECTRICAL ENGINEERING VJ ?J(EIX Z)D?J(Z), WHERE THE VJ ARE SUBVARIETIES OF THE ALGEBRAIC VARIETY V = {Z ?CN : P1(Z) = = PR(Z) = 0}, THE ?J ARE DIFFERENTIAL OPERATORS WITH POLYNOMIAL COEF?CIENTS, AND THE D?J ARE MEASURES SUPPORTED ON THE VARIETIES VJ IN THE CASE IN WHICH N = 1 THIS THEOREM IS OBVIOUSLY VERY WELL KNOWN AND IT IS OFTEN REFERRED TO AS THE EULER THEOREM FOR ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEF?CIENTS THERE IS, HOWEVER, A WAY TO REMAIN IN ONE VARIABLE AND YET TAKE ADVANTAGE OF THE DEEP IDEAS UNDERLYING THE FUNDAMENTAL PRINCIPLE IN HIS BOOK [27], EHRENPREIS SUGGESTS THE CONSIDERATION OF A SEQUENCE OF POLYNOMIALS IN ONE OR SEVERAL VARIABLES PJ, AND THEIR CORRESPONDENT LINEAR CONSTANT COEF?CIENTS OPERATORS DJ (I E THE DIFFERENTIAL OPERATORS WHOSE SYMBOLS WERE THE POLYNOMIALS PJ), AND THEN TO CONSIDER SERIES OF SOLUTIONS FJ TO THE EQUATIONS DJ(FJ) = 0 EHRENPREIS PROVIDED SOME GENERAL IDEAS ON THE GEOMETRIC CONDITIONS THAT THE VARIETIES VJ := {Z ?CN : PJ(Z) = 0} NEEDED TO SATISFY, IN ORDER FOR THE SERIES J FJ TO CONVERGE AND HAVE SUITABLE INTEGRAL REPRESENTATION THE STUDY CONFORMAL TRANSFORMATIONS IN ELECTRICAL ENGINEERING J! DJF DZJ , BUT IT IS EASY TO SHOW THAT SUCH FUNCTION CORRESPONDS TO F (Z + 1), SO THAT THE OPERATOR WHOSE SYMBOL IS EZ IS NOT AN IN?NITE ORDER DIFFERENTIAL OPERATOR, BUT RATHER THE SIMPLEST OF CONVOLUTION OPERATORS, THE TRANSLATION BY 1 CONFORMAL TRANSFORMATIONS IN ELECTRICAL ENGINEERING