Greetings from Russia

It is a great pleasure for me to greet Professor Wade on the occasion of his sixtieth birthday. I am greeting him on behalf of my colleagues in Moscow University who, like myself, have vivid and happy memories of his visit to Russia. It was 26 years ago when I met Bill Wade in Moscow where he spent the academic year 1977-78 as a visiting professor at the Department of Mathematics of Moscow State University. It was rather adventurous of him to come to Russia at that “cold war” period and to bring his family with him. But he came without prejudice, with an open heart, with great sympathy to Russian people, and even with some knowledge of Russian. I am confident that he and his wife Cherri found many friends in Moscow. Above all, he came with a very good knowledge of the Russian literature in Harmonic Analysis. In fact he had used this knowledge in his very first publications. Being a representative of the V.L. Shapiro school in Harmonic Analysis, Wade was the first who managed to bring together methods developed both in this school and in the Russian school of Real Analysis. More precisely we should talk about the Soviet (or Lusin) school because besides Russians, mathematicians from many Soviet republics (now independent states such as Armenia, Kazakhstan, Georgia) formed this school. By the time W.R.Wade started his research in the theory of uniqueness of Walsh and Haar series, some progress was made in this field by both schools. In the middle of the sixties, after pioneering papers by Fine, Vilenkin, Shneider, the uniqueness problem in the case of the Walsh series convergent to the Lebesgue integrable sums was solved independently by Arutunyan and Talalyan (1964) and by Crittenden and Shapiro (1965). By combining the American and Soviet techniques, Wade obtained in his first published paper [1] a theorem which generalized the results of Arutunyan and Talalyan, but which had a form similar to the theorem of Crittenden and Shapiro. In particular, following the idea of Arutunyan and Talalyan he proved first the result for Haar series and obtained the Walsh series result as a corollary. He also adapted successfully the Arutunyan-Talalyan techniques to get in [3] a uniqueness theorem for Cesaro summable Haar series. Another paper by Arutunyan was used by Wade in some of his papers on the uniqueness problem for the Walsh series convergent to sums which are integrable in a more general sense than Lebesgue integration (see [4] and [5]). Scneider elementary sets of uniqueness were used by Wade in [7] to produce some new examples of uniqueness sets for the group of integers of a p-series field.