A Panorama of Hungarian Mathematics in the Twentieth Century, I (Bolyai Society Mathematical Studies Book 14)

A glorious period of Hungarian mathematics started in 1900 when Lipót Fejér discovered the summability of Fourier series.This was followed by the discoveries of his disciples in Fourier analysis and in the theory of analytic functions. At the same time Frederic (Frigyes) Riesz created functional analysis and Alfred Haar gave the first example of wavelets. Later the topics investigated by Hungarian mathematicians broadened considerably, and included topology, operator theory, differential equations, probability, etc. The present volume, the first of two, presents some of the most remarkable results achieved in the twentieth century by Hungarians in analysis, geometry and stochastics. The book is accessible to anyone with a minimum knowledge of mathematics. It is supplemented with an essay on the history of Hungary in the twentieth century and biographies of those mathematicians who are no longer active. A list of all persons referred to in the chapters concludes the volume. Cover......Page 1 Bolyai Society Mathematical Studies Volume 14......Page 3 A Panorama of Hungarian Mathematics in the Twentieth Century I......Page 4 3540289453......Page 5 Table of Contents......Page 8 PREFACE......Page 10 TOPOLOGY......Page 12 REFERENCES......Page 26 Constructive Function Theory......Page 30 1. THE RIESZ-FISCHER THEOREM......Page 32 2. RIESZ TYPICAL MEANS......Page 36 3. T HE HAAR ORTHOGONAL SYSTEM......Page 39 4. THE SATURATION PROBLEM FOR THE FEJER MEANS......Page 42 5. ALMOST EVERYWHERE CONVERGENCE OF ORTHOGONAL SERIES......Page 46 6. CESARO SUMMABILITY OF ORTHOGONAL SERIES......Page 52 7. UNCONDITIONAL CONVERGENCE OF ORTHOGONAL SERIES......Page 54 REFERENCES......Page 55 ORTHOGONAL POLYNOMIALS......Page 58 REFERENCES......Page 72 1. INTRODUCTION......Page 74 2. LAGRANGE INTERPOLATION. LEBESGUE FUNCTION. LEBESGUE CONSTANT. OPTIMAL LEBESGUE CONSTANT. DIVERGENCE OF INTERPOLATION......Page 75 3. ON THE CONVERGENCE OF THE INTERPOLATORY PROCESSES......Page 87 4. HERMITE-FEJER TYPE AND OTHER CONVERGENT INTERPOLATORY PROCESSES......Page 90 5. LACUNARY OR BIRKHOFF INTERPOLATION......Page 96 6. ON THE MEAN CONVERGENCE OF INTERPOLATION......Page 100 7. WEIGHTED LAGRANGE INTERPOLATION , WEIGHTED LEBESGUE FUNCTION, WEIGHTED LEBESGUE CONSTANT......Page 105 8. GETTING CONVERGENCE BY RAISING THE DEGREE......Page 110 9. MEAN CONVERGENCE......Page 111 REFERENCES......Page 112 1. MARKOV- AND BERNSTEIN-TYPE INEQUALITIES......Page 122 2. MUNTZ POLYNOMIALS AND EXPONENTIAL SUMS......Page 130 3. GEOMETRIC PROPERTIES OF POLYNOMIALS......Page 136 REFERENCES......Page 154 Harmonic Analysis......Page 160 1. THE THEOREM OF FEJER......Page 162 2. THE THEOREM OF RIESZ-FISCHER......Page 169 3. BOUNDARY VALUES OF ANALYTIC FUNCTIONS......Page 174 4. RIESZ PRODUCT AND SIDON SETS......Page 178 5. MISCELLANEOUS*......Page 182 REFERENCES......Page 191 NON-COMMUTATIVE HARMONIC ANALYSIS......Page 196 1.2. Invariant Measures a nd Analysis on Locally Compact Groups......Page 197 1.3. Representation Theory and Quantum Physics......Page 199 1.4 . Invariant Means and Almost Periodic Functions and Groups......Page 201 1.5. Von Neumann Algebras......Page 203 1.6. Development of Unitary Representation Theory of Non-Compact Lie Groups......Page 205 2.1. Bela Sz.-Nagy......Page 207 2.2. Lajos Pukanszky......Page 208 REFERENCES......Page 210 A PANORAMA OF THE HUNGARIAN REAL AND FUNCTIONAL ANALYSIS IN THE 20TH CENTURY......Page 214 REFERENCES......Page 243 Summary.......Page 248 At the turn of the century.......Page 249 Lajos Schlesinger and his work.......Page 250 Fejer summation theorem and the Dirichlet problem on the unit disc.......Page 252 The work of Fejer in mechanics and his habilitation lecture.......Page 253 F. Riesz' subharmonic functions.......Page 256 The works of F. Riesz and Haar on linear integral equations.......Page 257 Haar's inequality for partial equations of the first order.......Page 258 Haar's existence and uniqueness theorem in the calculus of variations.......Page 259 T. Rado's regularity Lemma.......Page 261 Haar's Lemma on the variation of double integrals.......Page 262 An early paper on billiards.......Page 264 Neumann's method of stability analysis.......Page 265 Lax equivalence theorem, the theoretical result behind.......Page 269 The work of Lax on a single conservation law.......Page 270 The work of Lax on systems of conservation laws......Page 273 The Lax-Milgr am Lemma......Page 274 A cross-section in 1928. The state of art.......Page 275 Polya and Szego on isoperimetric inequalities.......Page 276 M. Riesz' fractional potentials.......Page 280 The work of Egervary.......Page 282 A cross-section in 1953. The state of art.......Page 284 Bihari inequality.......Page 285 The contributions of Makai.......Page 287 Epilogue......Page 291 REFERENCES......Page 292 2. THE JENSEN FORMULA......Page 298 3. POLYNOMIALS......Page 300 4. TRIGONOMETRIC POLYNOMIALS, TOEPLITZ FORMS AND A PROBLEM OF CARATHEODORY......Page 309 5. THE FEJER-RIESZ INEQUALITY......Page 319 6. BOUNDARY VALUES, HP SPACES......Page 321 7. KAKEYA'S THEOREM, POWER SERIES WITH MONOTONE COEFFICIENTS......Page 326 8. POWER SERIES: SINGULARITIES AND ANALYTIC CONTINUATION......Page 337 9. TURAN'S "NEW METHOD"......Page 350 10. POWER SERIES: BEHAVIOR ON THE CIRCLE OF CONVERGENCE......Page 354 11. POLYA-SCHUR FUNCTIONS......Page 358 12. CONFORMAL MAPPING, COMPLEX INTERPOLATION......Page 367 REFERENCES......Page 372 THEODORE VON KARMAN......Page 376 The von Karman equations for plates.......Page 377 The von Karman vortex street.......Page 379 Mathematical methods in engineering.......Page 380 Aeronautics and astronautics.......Page 381 REFERENCES......Page 382 Geometry......Page 386 DIFFERENTIAL GEOMETRY......Page 388 REFERENCES......Page 413 1. INTRODUCTION......Page 418 2. THE PROBLEM OF MOTION......Page 419 3. THE LANCZOS POTENTIAL......Page 422 4. THE LORENTZIAN SIGNATURE......Page 423 REFERENCES......Page 427 1. INTRODUCTION......Page 430 2. THE BEGINNINGS......Page 431 3. PACKINGS AND COVERINGS BY CIRCLES......Page 434 4. PACKINGS AND COVERINGS BY INCONGRUENT CIRCLES......Page 435 5. PACKINGS AND COVERINGS BY CONVEX SETS IN THE PLANE......Page 436 6. PACKINGS AND COVERINGS ON THE SPHERE......Page 438 7. PACKINGS AND COVERINGS IN THE HYPERBOLIC PLANE......Page 440 8. PACKINGS AND COVERINGS IN HIGHER DIMENSIONS......Page 441 9. APPROXIMATION......Page 442 10. THE ERDOS-SZEKERES THEOREM......Page 443 11. REPEATED DISTANCES, DISTINCT DISTANCES IN THE PLANE......Page 445 12. REPEATED AND DISTINCT DISTANCES ELSEWHERE......Page 447 13. INCIDENCES......Page 448 14. MISCELLANEOUS RESULTS IN COMBINATORIAL GEOMETRY......Page 450 15. FINITE GEOMETRIES......Page 452 16. STOCHASTIC GEOMETRY......Page 453 17. MISCELLANEOUS RESULTS IN CONVEX GEOMETRY......Page 454 REFERENCES......Page 457 Stochastics......Page 458 PROBABILITY THEORY......Page 460 REFERENCES......Page 488 1. INTRODUCTION......Page 494 2. EARLY STATISTICS IN HUNGARY......Page 495 3. SEQUENTIAL ANALYSIS......Page 499 4. STATISTICAL DECISION FUNCTIONS......Page 501 5. ASYMPTOTIC THEORY OF TESTING AND ESTIMATION......Page 503 6. RANDOMNESS......Page 504 7. NONPARAMETRIC TESTS, ORDER STATISTICS......Page 505 8. GOODNESS OF FIT TESTS......Page 514 9. CRAMER-FRECHET-RAO INEQUALITY......Page 517 REFERENCES......Page 519 STOCHASTICS: INFORMATION THEORY......Page 526 INFORMATION THEORY IN HUNGARY......Page 527 DIMENSIONAL ENTROPY......Page 528 RENYI INFORMATIONS......Page 530 AXIOMATIC CHARACTERIZATIONS......Page 532 RANDOM SEARCH......Page 533 INFORMATION THEORETIC METHODS IN STATISTICS......Page 534 REFERENCES......Page 536 CONTRIBUTION OF HUNGARIAN MATHEMATICIANS TO GAME THEORY......Page 540 REFERENCES......Page 550 A SHORT GUIDE TO THE HISTORY OF HUNGARY IN THE 20TH CENTURY......Page 552 1900-1920......Page 558 1920-1945......Page 560 1945-2000......Page 562 A SHORT NOTE ON THE USE OF NAMES OF PERSONS AND INSTITUTIONS, AND ON PRONUNCIATION:......Page 566 BIOGRAPHIES......Page 568 REFERENCES......Page 612 NAME INDEX......Page 626 A glorious period of Hungarian mathematics started in 1900 when Lipót Fejér discovered the summability of Fourier series. This was followed by the discoveries of his disciples in Fourier analysis and in the theory of analytic functions. At the same time Frederic (Frigyes) Riesz created functional analysis and Alfred Haar gave the first example of wavelets. Later the topics investigated by Hungarian mathematicians broadened considerably, and included topology, operator theory, differential equations, probability, etc. The present volume, the first of two, presents some of the most remarkable results achieved in the twentieth century by Hungarians in analysis, geometry and stochastics. The book is accessible to anyone with a minimum knowledge of mathematics. It is supplemented with an essay on the history of Hungary in the twentieth century and biographies of those mathematicians who are no longer active. A list of all persons referred to in the chapters concludes the volume A glorious period of Hungarian mathematics started in 1900 when Lipot Fejer discovered the summability of Fourier series. This was followed by the discoveries of his disciples in Fourier analysis and in the theory of analytic functions. At the same time Frederic (Frigyes) Riesz created functional analysis and Alfred Haar gave the first example of wavelets. Later the topics investigated by Hungarian mathematicians broadened considerably, and included topology, operator theory, differential equations, probability, etc. The present volume, the first of two, presents some of the most remarkable re János Horváth (ed.). The Present Volume, The First Of Two ...--p. [4] Of Cover. Includes Bibliographical References And Index.