Topics in Classical Analysis and Applications In Honor of Daniel Waterman


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It seemed that Detroit had several gifted students who, for various reasons, were unable to leave the area. We also had some very good foreign students.


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In my years there, six students completed their doctoral dissertations with me. Meanwhile, our family had grown; Mudite and I and our three lovely children went to Berkeley for a sabbatical during the academic year. On our return, we decided we needed more living space and purchased another house in Detroit.

I accepted it, of course, and purchased another house in the Syracuse area. We now had two houses too many in the very depressed market which followed the Detroit riots of It took some ingenuity to dispose of them, but we did. Leaving Detroit was not so easy; we had made good friends there whom we would miss, particularly the Nishiuras. However Syracuse had many advantages including a superb library, an excellent environment for our children, and the presence of Wolfgang Jurkat, whose work I greatly admired.

Syracuse was an interesting institution.

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This made for many lean years. As a frequent member of the department executive committee and ultimately the chairman, I learned much about the futility of making predictions concerning future research productivity based on early performance. Perhaps in another department culture they would have fared better. My time at Syracuse was very satisfying. My research and my family thrived. My two daughters went to Cornell and Syracuse. They both became physicians. My son studied electrical and computer engineering at Berkeley and obtained a Ph.

I always found that supervising dissertation studies was enjoyable and also stimulated my own research. Being the chairman did give me some satisfaction. I was able to provide computer equipment and travel funds to the faculty that they might not otherwise have had, and with the help of my associate chair, John Troutman, was able to do some good things for our graduate students as well.

During my term I made some notable appointments and, overall, I hired one quarter of the faculty. I note that I had steady research support until I reached Syracuse. What happened then illustrates the tendency in this country for research support to follow fashion instead of relying on the abilities and judgement of the researcher. It was widely cited and used.

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I had very pleasant letters from Orlicz and Chanturiya expressing their appreciation of this work. I am happy to see that the spaces of functions of generalized bounded variation I introduced are still the subject of study. Retirement has had both its good and bad points. I have been graciously received by the mathematics department at Florida Atlantic University. I miss my friends in Syracuse and also its superb library. Modern computer technology and interlibrary loan can compensate to some extent, but nothing can replace the sensation of walking into a library with hundreds of the latest issues of mathematics journals arrayed August 28, 6 WSPC - Proceedings Trim Size: 9.

Waterman on its shelves. In his text, Zygmund considered the partial sums of interpolating polynomials. I would like to estimate the degree of approximation of these partial sums to functions of various classes. In my discussion I have grouped my work by area.

Several papers could have been in more than one group. Papers will be referred to by the numbers in the publication list. References to papers of others will not be given; they are easily obtained from the cited papers. I cannot describe all of the papers in each group, but I will try to describe their main thrust as well as giving some extra attention to results that I am particularly fond of.

Of course there are some papers that fall into none of the principal groups and I will not discuss these. I can always recall the projects that I tried and failed to complete. One such project is very vivid in my memory. Nef concerning regular functions on the quaternions. It was very interesting, but I discovered a crucial measure-theoretic error. I was strongly attracted by the idea.

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When I studied Hilbert space with Stone, we did it over the quaternions and I now envisioned doing harmonic analysis over the quaternions. I wrote a detailed proposal and received a grant to pursue it. Unfortunately, my personal circumstances were such that I could not really undertake a project of this scope. I soon became captivated by the approach to Fourier series that we were pursuing and never returned to this project. Other results, with weights in the integrand, were also proved. We did not consider the case 0 1 has a subsequence whose C, 1 means converge strongly. Banach spaces with this property are said to have the Banach-Saks property.

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The reviewer MR misunderstood my statement and said that this was incorrect. Waterman for weakly convergent sequences in a uniformly convex Banach space the same conclusion holds. In [19], more general summability methods are considered. In [4] we proved similar results for functions analytic in a half-plane. The referee of [5] asked me to shorten the paper and I complied.

In [15], we considered another problem related to area. This enables us to reduce the problem to that for the circle. This has several corollaries such as an analogue of the Dini-Lipschitz test. Functions satisfying this property had uniformly convergent Fourier series. Continuous functions with this property are shown to have uniformly convergent Fourier series.

https://saltsancsizedang.gq An analogue of the Lebesgue test for continuous functions is proved. In [66], we proved a more general version of the Lebesgue test. In [17] we found the condition on a continuous function which ensured this. The idea behind this was based on a linearization of the Dirichlet kernel used by Salem to prove a theorem on uniform convergence of Fourier series. In [28], a similar result is proved for preservation of uniform convergence and another proof of this is given in [55].

Waterman the condition of [17] was also appropriate for functions whose only discontinuities were jumps, the regulated functions. The result of [64] with Jurkat is one of which I am very fond. Berezhnoi has shown that the generalization of the D-J theorem with HBV is the strongest test for uniform convergence that can be obtained with generalized bounded variation. As described for the previous result, this theorem cannot be improved.

There is, however, one to which I would like to draw attention. We have previously alluded to a result of Salem which was a test for uniform convergence of Fourier series. The most successful have been the results of Dyachenko. Among these is our joint [79]. Waterman bounded and converge Pringsheim at each point to the arithmetic mean of the quadrant limits.


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It is also shown that, in a certain sense, this convergence result cannot be improved. Recent work has been in the area of trigonometric interpolation [80]. Zygmund studied the convergence of partial sums of interpolating polynomials and proved a theorem resembling the Dirichlet-Jordan theorem.

Real Analysis Much of what we have done in this area is connected to Fourier series and so may be found in that section. There are a few things that were not related to that area and are of some interest. Denjoy introduced the approximately continuous functions in his work on derivatives. In [9] we showed that the approximately continuous transformations from an Euclidean space into a metric space are of Baire class 1, have a Darboux property and have separable images. A measurable set in En is homogeneous if it has metric density one at each of its points.

En can be topologized by taking the homogeneous sets as the open sets, yielding the d-topology. The approximately continuous functions are the continuous functions in the d-topology.

Topics in Classical Analysis and Applications In Honor of Daniel Waterman Topics in Classical Analysis and Applications In Honor of Daniel Waterman
Topics in Classical Analysis and Applications In Honor of Daniel Waterman Topics in Classical Analysis and Applications In Honor of Daniel Waterman
Topics in Classical Analysis and Applications In Honor of Daniel Waterman Topics in Classical Analysis and Applications In Honor of Daniel Waterman
Topics in Classical Analysis and Applications In Honor of Daniel Waterman Topics in Classical Analysis and Applications In Honor of Daniel Waterman
Topics in Classical Analysis and Applications In Honor of Daniel Waterman Topics in Classical Analysis and Applications In Honor of Daniel Waterman

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