Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis

We study two classes of extension problems, and their interconnections: (i) Extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; (ii) In case of Lie groups, representations of the associated Lie algebras $La\left(G\right)$ by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space (RKHS) $\mathscr{H}_{F}$. Why extensions? In science, experimentalists frequently gather spectral data in cases when the observed data is limited, for example limited by the precision of instruments; or on account of a variety of other limiting external factors. Given this fact of life, it is both an art and a science to still produce solid conclusions from restricted or limited data. In a general sense, our monograph deals with the mathematics of extending some such given partial data-sets obtained from experiments. More specifically, we are concerned with the problems of extending available partial information, obtained, for example, from sampling. In our case, the limited information is a restriction, and the extension in turn is the full positive definite function (in a dual variable); so an extension if available will be an everywhere defined generating function for the exact probability distribution which reflects the data; if it were fully available. Such extensions of local information (in the form of positive definite functions) will in turn furnish us with spectral information. In this form, the problem becomes an operator extension problem, referring to operators in a suitable reproducing kernel Hilbert spaces (RKHS). In our presentation we have stressed hands-on-examples. Extensions are almost never unique, and so we deal with both the question of existence, and if there are extensions, how they relate back to the initial completion problem.Comment: 235 pages, 42 figures, 7 tables. arXiv admin note: substantial text overlap with arXiv:1401.478

No-Tillage Soybean Production Field Days Next Week

Iowa State University Extension and the Iowa Soybean Association (ISA) will host three field days on Sept. 8-10, highlighting soybean production under no-tillage conditions with emphasis on agronomic decisions in a no-till versus conventional tillage system. While the adoption rate of no-tillage practices in the northern Corn Belt has been slow compared to other parts of the world, it will likely increase in the future because of high diesel prices. However, no-tillage will not work consistently on all soil types in Iowa without yield loss, compared to a conventional tillage system. A key will be site-specific knowledge of the no-tillage system’s potential benefits

Flooding or drought?

Currently, we have the whole spectra of growing conditions in Iowa. Northern and central Iowa have received excessive amounts of rainfall during the past 14 days, whereas southern Iowa (especially southwestern Iowa) is getting really dry. Mother Nature\u27s actions are beyond our control, but a little bit too much rainfall is better than no rainfall at all

Use of strobilurins in Iowa

Iowa isn\u27t a state where we see a lot of foliar pathogens. Every year we see foliar pathogens in soybeans; however, many of these diseases typically have minimal or uneconomic impacts on yield. For that reason, use of foliar fungicides hasnt been necessary in Iowa. The recent discovery of Asian soybean rust, a potential threat to soybean producers in the United States, has increased the interest of using foliar fungicides as a preventive control method for managing Asian soybean rust

Looking back at the 2003 growing season

Now looking back at the 2003 growing season, who would have thought that it would turn out the way it did? I think that we all got challenged in many unexpected ways and we definitely learned a lot. One thing for sure, many do often underestimate mother nature but after the 2003 growing season, I think that we all can agree that no management practice can match her power. Another thing that we learned was that different seasons have different pests and pathogens. The interactions between a host, the environment, and a pathogen or pest couldn\u27t have been clearer than in 2003

Is It Time to Switch Soybean Maturity Group Varieties?

Many farmers were able to get back into the field this week and start finishing up planting. However, there are still many areas where fields remain flooded and it will take awhile before we can get back in and replant. Some fields need to be replanted and some don’t. It is important to accurately estimate a surviving stand and then evaluate the economics of replanting

Frost damage

Temperatures this morning (May 3) reached 25 degrees with many corn and soybean fields getting close to emerge. In a few fields, crops have just started to emerge and with the low temperatures many may wonder if replanting now is needed

Soybean seed inoculation

Much of the soybean plant\u27s nitrogen requirement is supplied through nitrogen fixation in which atmospheric nitrogen is converted into a usable form for the plant. Nitrogen fixation is critical for producing higher yield in soybean

Corn planting decisions in late May

After my first spring in Iowa, I wonder whether there are more cloudy days in Iowa than in Denmark! It definitely has not been an easy spring, and it’s only the third week of May. Based on USDA’s Crop and Weather report, we had a little more than half a day suitable for fieldwork from May 4 to May 11, compared with 4 days last year. However, with 5 days of reasonably warm weather (without any rain), there is a chance to get back into the field. Before this weekend (May 17 to 18), the USDA predicted that corn planting was 4 days behind normal with approximately 64 percent of the corn planted across the state, compared with 81 percent last year

Early planting IS important in Iowa

This winter there has been a lot of information in the media and on the web regarding soybean planting date. In part, the reason is that a paper was published in the Agronomy Journal by two researchers from University of Kentucky stating that there is no consistent advantage for planting soybean early in the United States, but it is first after late May and early June that yields start to drop off. That is NOT correct for Iowa and is not correct for many of the states adjacent to Iowa as well