Convexity theorems for Fourier series

Some Stability Theorems for Nonharmonic Fourier Series

The theory of nonharmonic Fourier series in L2(-ir,tr) is concerned with the completeness and expansion properties of sets of complex exponentials {e'x"'}. It is well known, for example, that the completeness of the set {e'x"'} ensures that of {e'^"'} whenever 2 lA„ ~~ M»l < oo. In this note we establish two results which guarantees that if {elX"'} is a Schauder basis for l}(—n, it), then [e'^"...

Convexity Theorems

M(a, fi) S M'(ah ft)Jf^(a,, ft). This is M. Riesz's fundamental theorem. (See M. Riesz [5] and a different proof in Paley [4] ; see also a generalization of the theorem in L. C. Young [ i l ] . ) M. Riesz's argument proved the convexity only in the triangle O ^ a ^ l , 0 ^ / 3 ^ 1 , a+f3*zl. The extension to the whole quadrant is due to Thorin [9]. We shall not give the proof of the theorem her...

Bernstein Equiconvergence and Fej Er Type Theorems for General Rational Fourier Series Bernstein Equiconvergence and Fej Er Type Theorems for General Rational Fourier Series

Let w() be a positive weight function on the interval ;) and associate the positive deenite inner product on the unit circle of the complex plane by hf; gi w = 1 2 R f(e ii)g(e ii)w()d. For a sequence of points f k g 1 k=1 included in a compact subset of the open unit disk, we consider the orthogonal rational functions (ORF) f k g 1 k=0 that are obtained by orthogonalization of the sequence f1;...

Determination of a jump by Fourier and Fourier-Chebyshev series

‎By observing the equivalence of assertions on determining the jump of a‎ ‎function by its differentiated or integrated Fourier series‎, ‎we generalize a‎ ‎previous result of Kvernadze‎, ‎Hagstrom and Shapiro to the whole class of‎ ‎functions of harmonic bounded variation‎. ‎This is achieved without the finiteness assumption on‎ ‎the number of discontinuities‎. ‎Two results on determination of ...

Structural Theorems for Families of Fourier Hyperfunctions