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Walter R Bloom

Murdoch University, Mathematics and Statistics, Emeritus

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Emeritus Professor of Mathematics
Murdoch University
Honorary Curator in Numismatics
Western Australian Museum
Phone: +49 (0) 417 930 854

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Papers by Walter R Bloom

Wiener's theorem for positive definite functions on hypergroups

arXiv (Cornell University), May 19, 2014

The following theorem on the circle group T is due to Norbert Wiener: If f ∈ L 1 (T) has non-nega... more The following theorem on the circle group T is due to Norbert Wiener: If f ∈ L 1 (T) has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then f ∈ L 2 (T). This result has been extended to even exponents including p = ∞, but shown to fail for all other p ∈ (1, ∞]. All of this was extended further (appropriately formulated) well beyond locally compact abelian groups. In this paper we prove Wiener's theorem for even exponents for a large class of commutative hypergroups. In addition, we present examples of commutative hypergroups for which, in sharp contrast to the group case, Wiener's theorem holds for all exponents p ∈ [1, ∞]. For these hypergroups and the Bessel-Kingman hypergroup with parameter 1 2 we characterise those locally integrable functions that are of positive type and square-integrable near the identity in terms of amalgam spaces.

A SURVEY OF NUMISMATIC RESEARCH 2008 - 2013 OCEANIA Medals

The interest in medals in Australia and New Zealand continues to be strong, and as might be expec... more The interest in medals in Australia and New Zealand continues to be strong, and as might be expected, by and large the writers concentrate on the medals of their own country. In the period of this Survey there has been an intensive study of agricultural medals, which is reflected in the publications listed below. Furthermore, a major

Pseudo differential operators on local Hardy spaces on chébli-teimèche hypergroups

Journal of the Australian Mathematical Society, Apr 1, 2000

In this paper we consider pseudo differential operators on local Hardy spaces h p (0 < p < 1) on ... more

Contributions of Walter Bloom to Nomisma.org

Nomisma.org thesaurus of numismatic concepts, 2015

Local Hardy spaces on Chébli-Trimèche hypergroups

Studia Mathematica, 1999

Interpolation of multipliers of YpL

No abstract availabl

Computing, mathematics and statistics periodicals held at Kalgoorlie School of Mines, Murdoch University, The University of Western Australia, The Western Australian College of Advanced Education and the Western Australian Institute of Technology

No abstract availabl

7 Randomized sums of hypergroup-valued random variables

Infinitely divisible measures on hypergroups

Lecture Notes in Mathematics, 1982

No abstract availabl

Approximation Theory on the Compact Solenoid

Functional Analysis and Approximation, 1981

The compact solenoid Σ is the a-adic solenoid with a = (2,3,…). It is a compact connected metrisa... more The compact solenoid Σ is the a-adic solenoid with a = (2,3,…). It is a compact connected metrisable abelian group with dual the group of rational numbers. We give an analogue of the M. Riesz theorem on the boundedness of partial sums of the Fourier series of functions in LP(Σ), and use this to characterize the Lipschitz functions on Σ in terms of the rate of convergence of their Fourier series. In addition we prove a factorization theorem for these functions.

Characterisation of potential kernels of transient convolution semigroups on a commutative hypergroup

Lecture Notes in Mathematics, 1989

The work for this paper was partly carried out while the first author held an Alexander von Humbo... more

Convolution semigroups and resolvent families of measures on hypergroups

Mathematische Zeitschrift, 1985

Generalized Lipschitz Spaces on Vilenkin Groups

Mathematische Nachrichten, 1987

It is shown that, for certain choices of the defining indices, the generalized LIPSCHITZ spaces o... more It is shown that, for certain choices of the defining indices, the generalized LIPSCHITZ spaces on VILEWKIJ groups are incIudec1 in certain FIGA-TALAXAYCA spaces A,, and t h a t the FOURIER series of functions in the letter spaces converge uniformly. This result includes an extension of the classical DIN test, and endpoint versions of BERNSTEIN'S theorem on absolute convergence of FOURIER series. m =n ..exists x,€G,,\G,+~ such that Q),(x,,) =exp (2 7~i p ; ;~) , and each xEG has a unique representationb,x, where 0 s b, <pn + l. n =O *) Research partially supported by NSERC Grant # 4822.

A National Discipline-specific Professional Development Program in the Mathematical Sciences

Robert Israel “Bob” Jewett (1937–2022)

Notices of the American Mathematical Society, May 1, 2023

née Katz, was born in Providence to parents from Russia. In 1946, Bob's family, including his old... more née Katz, was born in Providence to parents from Russia. In 1946, Bob's family, including his older sister Rosalie, moved to Venice, California, a relocation partially motivated by Bob's problems with hay fever. Bob's much younger brother Phil was born in 1948. Bob attended the local public schools and was very interested, while at Venice High School, not only in physics and mathematics, but also animals and insects. In 1955, he started college at Caltech. There was no zoology major, so he focused on physics and mathematics, eventually the latter. In 1958, he received Honorable Mention for his individual performance in the nationwide Putnam competition in mathematics, helping the Caltech team to place third in the nation. Bob also stood out as a volleyball and track and field star. Bob graduated from Caltech in 1959 and went to the University of Oregon for graduate work in mathematics. Before this move, however, he worked for the summer at Caltech's Jet Propulsion Laboratory (JPL), in the coding theory section headed by Solomon Golomb. He was also employed there in the summers of 1960 and 1961. During that time he wrote several research papers, one containing the combinatorial Hales-Jewett theorem [HJ63]. Bob received his PhD in 1963, with a thesis on analysis on locally compact abelian groups written under the Communicated by Notices Associate Editor Emilie Purvine.

A religious medallion from the Gilt Dragon

A curious item was discovered inside a concretion of cannon balls salvaged from the Gilt Dragon…

Two 1921 New South Wales coin hoards

Within the last five years two New South Wales coin hoards fortuitously came intact on to the mar... more Within the last five years two New South Wales coin hoards fortuitously came intact on to the market. It was immediately recognized that these modern coins dating from the Victorian period through to just after World War I were not simply accumulations or coin collections, but must have come from hoards. This supposition was confirmed with the owners. Realizing the hoards’ importance, the authors recorded the coins by country (British and Australian), date, mint, and most importantly, their individual weights. This article is a report on these findings together with a statistical analysis of the weight loss of the coins during circulation. We show that for the Australian silver coins in the combined hoards, the mean percentage weight loss per year per coin by denomination follows a linear relationship, and that the smaller the coin, the larger the percentage weight loss.

World War 1 appeal medals of Western Australia

Appeal medals were produced in Western Australia over the first four years of World War I in supp... more Appeal medals were produced in Western Australia over the first four years of World War I in support of various worthy causes, including the Red Cross, Children’s Hospital and returned injured soldiers and widows of those who fought for Australia, the latter through the Ugly Men’s Voluntary Workers’ Association of Western Australia (Incorporated). There has been very little known about these medals, not even their manufacturer, but an analysis of the die characteristics together with an examination of newspapers of the day and other contemporary records leads us to their common manufacturer, Cumpston’s City Electric Engraving Works, then located in Hay Street, Perth.

Pure Mathematics in Australia

Vestes Australian Universities Review, 1975

Hardy spaces on Chébli-Trimèche hypergroups

No abstract availabl

Wiener's theorem for positive definite functions on hypergroups

arXiv (Cornell University), May 19, 2014

The following theorem on the circle group T is due to Norbert Wiener: If f ∈ L 1 (T) has non-nega... more The following theorem on the circle group T is due to Norbert Wiener: If f ∈ L 1 (T) has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then f ∈ L 2 (T). This result has been extended to even exponents including p = ∞, but shown to fail for all other p ∈ (1, ∞]. All of this was extended further (appropriately formulated) well beyond locally compact abelian groups. In this paper we prove Wiener's theorem for even exponents for a large class of commutative hypergroups. In addition, we present examples of commutative hypergroups for which, in sharp contrast to the group case, Wiener's theorem holds for all exponents p ∈ [1, ∞]. For these hypergroups and the Bessel-Kingman hypergroup with parameter 1 2 we characterise those locally integrable functions that are of positive type and square-integrable near the identity in terms of amalgam spaces.

A SURVEY OF NUMISMATIC RESEARCH 2008 - 2013 OCEANIA Medals

The interest in medals in Australia and New Zealand continues to be strong, and as might be expec... more The interest in medals in Australia and New Zealand continues to be strong, and as might be expected, by and large the writers concentrate on the medals of their own country. In the period of this Survey there has been an intensive study of agricultural medals, which is reflected in the publications listed below. Furthermore, a major

Pseudo differential operators on local Hardy spaces on chébli-teimèche hypergroups

Journal of the Australian Mathematical Society, Apr 1, 2000

In this paper we consider pseudo differential operators on local Hardy spaces h p (0 < p < 1) on ... more

Contributions of Walter Bloom to Nomisma.org

Nomisma.org thesaurus of numismatic concepts, 2015

Local Hardy spaces on Chébli-Trimèche hypergroups

Studia Mathematica, 1999

Interpolation of multipliers of YpL

No abstract availabl

Computing, mathematics and statistics periodicals held at Kalgoorlie School of Mines, Murdoch University, The University of Western Australia, The Western Australian College of Advanced Education and the Western Australian Institute of Technology

No abstract availabl

7 Randomized sums of hypergroup-valued random variables

Infinitely divisible measures on hypergroups

Lecture Notes in Mathematics, 1982

No abstract availabl

Approximation Theory on the Compact Solenoid

Functional Analysis and Approximation, 1981

The compact solenoid Σ is the a-adic solenoid with a = (2,3,…). It is a compact connected metrisa... more The compact solenoid Σ is the a-adic solenoid with a = (2,3,…). It is a compact connected metrisable abelian group with dual the group of rational numbers. We give an analogue of the M. Riesz theorem on the boundedness of partial sums of the Fourier series of functions in LP(Σ), and use this to characterize the Lipschitz functions on Σ in terms of the rate of convergence of their Fourier series. In addition we prove a factorization theorem for these functions.

Characterisation of potential kernels of transient convolution semigroups on a commutative hypergroup

Lecture Notes in Mathematics, 1989

The work for this paper was partly carried out while the first author held an Alexander von Humbo... more

Convolution semigroups and resolvent families of measures on hypergroups

Mathematische Zeitschrift, 1985

Generalized Lipschitz Spaces on Vilenkin Groups

Mathematische Nachrichten, 1987

It is shown that, for certain choices of the defining indices, the generalized LIPSCHITZ spaces o... more It is shown that, for certain choices of the defining indices, the generalized LIPSCHITZ spaces on VILEWKIJ groups are incIudec1 in certain FIGA-TALAXAYCA spaces A,, and t h a t the FOURIER series of functions in the letter spaces converge uniformly. This result includes an extension of the classical DIN test, and endpoint versions of BERNSTEIN'S theorem on absolute convergence of FOURIER series. m =n ..exists x,€G,,\G,+~ such that Q),(x,,) =exp (2 7~i p ; ;~) , and each xEG has a unique representationb,x, where 0 s b, <pn + l. n =O *) Research partially supported by NSERC Grant # 4822.

A National Discipline-specific Professional Development Program in the Mathematical Sciences

Robert Israel “Bob” Jewett (1937–2022)

Notices of the American Mathematical Society, May 1, 2023

née Katz, was born in Providence to parents from Russia. In 1946, Bob's family, including his old... more née Katz, was born in Providence to parents from Russia. In 1946, Bob's family, including his older sister Rosalie, moved to Venice, California, a relocation partially motivated by Bob's problems with hay fever. Bob's much younger brother Phil was born in 1948. Bob attended the local public schools and was very interested, while at Venice High School, not only in physics and mathematics, but also animals and insects. In 1955, he started college at Caltech. There was no zoology major, so he focused on physics and mathematics, eventually the latter. In 1958, he received Honorable Mention for his individual performance in the nationwide Putnam competition in mathematics, helping the Caltech team to place third in the nation. Bob also stood out as a volleyball and track and field star. Bob graduated from Caltech in 1959 and went to the University of Oregon for graduate work in mathematics. Before this move, however, he worked for the summer at Caltech's Jet Propulsion Laboratory (JPL), in the coding theory section headed by Solomon Golomb. He was also employed there in the summers of 1960 and 1961. During that time he wrote several research papers, one containing the combinatorial Hales-Jewett theorem [HJ63]. Bob received his PhD in 1963, with a thesis on analysis on locally compact abelian groups written under the Communicated by Notices Associate Editor Emilie Purvine.

A religious medallion from the Gilt Dragon

A curious item was discovered inside a concretion of cannon balls salvaged from the Gilt Dragon…

Two 1921 New South Wales coin hoards

Within the last five years two New South Wales coin hoards fortuitously came intact on to the mar... more Within the last five years two New South Wales coin hoards fortuitously came intact on to the market. It was immediately recognized that these modern coins dating from the Victorian period through to just after World War I were not simply accumulations or coin collections, but must have come from hoards. This supposition was confirmed with the owners. Realizing the hoards’ importance, the authors recorded the coins by country (British and Australian), date, mint, and most importantly, their individual weights. This article is a report on these findings together with a statistical analysis of the weight loss of the coins during circulation. We show that for the Australian silver coins in the combined hoards, the mean percentage weight loss per year per coin by denomination follows a linear relationship, and that the smaller the coin, the larger the percentage weight loss.

World War 1 appeal medals of Western Australia

Appeal medals were produced in Western Australia over the first four years of World War I in supp... more Appeal medals were produced in Western Australia over the first four years of World War I in support of various worthy causes, including the Red Cross, Children’s Hospital and returned injured soldiers and widows of those who fought for Australia, the latter through the Ugly Men’s Voluntary Workers’ Association of Western Australia (Incorporated). There has been very little known about these medals, not even their manufacturer, but an analysis of the die characteristics together with an examination of newspapers of the day and other contemporary records leads us to their common manufacturer, Cumpston’s City Electric Engraving Works, then located in Hay Street, Perth.

Pure Mathematics in Australia

Vestes Australian Universities Review, 1975

Hardy spaces on Chébli-Trimèche hypergroups

No abstract availabl