Application of periodic functions approximation to antenna pattern synthesis and circuit theory

Recently, mathematicians gave results on the approximation of periodic functionsf(x)by trigonometric sumsP_{n}(x). These results can be useful for antenna radiation and circuit theory problems. Rather than the least mean-square criterion which leads to Gibbs' phenomenon, it has been adopted that the maximum in the period of the error,| f - P_{n} |, is to be minimized. By linear transformation of the Fourier sum, aP_{n}sum can be obtained to give an error of the order1/n^{p}. The Fourier sum would give\Log n/n^{p}. Limitations on the maximum ofP_{n}derivatives are introduced allowing one to obtain the order of maximum error. Antenna power diagram synthesis is then looked at with these results. The power radiationv^{2}of an array ofnisotropic independent sources equally spaced can always be written under the form of aP_{n}sum. Thus it is possible to give general limitations for the derivatives ofv^{2}in the broadside case and the endfire case. These limitations depend upon the over-all antenna dimension vs wavelengtha/\lambdaand the maximum error. A practical problem of shaped beam antenna is examined. It is shown that, by using the mathematical theory, improvements can be made on the diagram from what is usually obtained. For circuit theory, physically evident limitations in timeTand spectrumFallow one to write the most general function under the form of aP_{n}sum, and thus to apply the mathematical results to that field. Formal analogy allows comparison of antenna pattern and circuit theories.