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By Euler L.

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Additional info for A solution to a problem of Fermat, on two numbers of which the sum is a square and the sum of their squares is a biquadrate, inspired by the Illustrious La Grange

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7) we have II. 5 of Chapter I), = F ( t ) K , ( t ) dt = a,(F, 0 ) . T T Whence limn-toolla,(f)-flip s limn+ma,(F, 0) = 0. 4 more precise and has important applications. 8 (Uniqueness). e. Proof. 7. 3. CHARACTERIZATION OF FOURIER SERIES OF FUNCTIONS AND MEASURES FejCr's theorem asserts that for continuous functionsf, a,(f,x) converges uniformly to f; it is natural to consider whether this statement admits a converse. More generally, is it possible to state a simple criteria that will enable us to identify among all trigonometric series those which correspond to Fourier series of continuous functions?

A subset E of H is called orthogonal if any two vectors in E are orthogonal to each other. , (x, y ) = 0 and (x, x) = 1 for all x # y in E. 2. If {x,},,A is an ONS, we associate with each x E H a complex function on the index set A by means of x ( a ) = (x, xm), (Y E A. 2) These are the Fourier coefficients of x with respect to the ONS {x,}. 3. If j s n, and llx112= C,n=llcjl2. = (x, xi), 1 Proof. 1. 4. If {xj}Fl is an ONS and { c j } is a complex sequence with x, then c, = x ( n ) , all n. CIcj12 < co, then C c p converges in H.

3. CHARACTERIZATION OF FOURIER SERIES OF FUNCTIONS AND MEASURES FejCr's theorem asserts that for continuous functionsf, a,(f,x) converges uniformly to f; it is natural to consider whether this statement admits a converse. More generally, is it possible to state a simple criteria that will enable us to identify among all trigonometric series those which correspond to Fourier series of continuous functions? or L P ( T )functions, or even measures for that matter? 1). 1. 1) is the Fourier series of a function E C ( T) if and only if the sequence {a,(t)} converges uniformly in T.

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A solution to a problem of Fermat, on two numbers of which the sum is a square and the sum of their squares is a biquadrate, inspired by the Illustrious La Grange by Euler L.


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