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Rajesh D
Rajesh D
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Consider BV functions on a $1$-d torus(circle). The Fourier partial sum using the first $n$ coefficients will converge to the average of left and right limits of the function at every point of continuity, as $n\to\infty$. The convergence rate is $O(1/n)$. However, the total variation of the partial sum does not converge and grows unbounded as $n \to \infty$. The Cesaro partial sum does not have this problem, and its total variation converges to that of the function as $n\to\infty$. However, the convergence rate of the Cesaro sum, to the actual function (at points of continuity), is $O(\log(n)/n)$ which is slower compared to $O(1/n)$ which is the case for the Fourier partial sum.

I am trying to think of a method to construct the function using the first $n$ coefficients so that it does not have the above-mentioned drawbacks. Meaning, it converges to the average of left and right limits of the function at every pointpoints of continuity, at a rate $O(1/n)$, and its total variation also converges to that of the function.

Question: What would be the possible applications/usefulness of such a method to other areas of mathematics?

Consider BV functions on a $1$-d torus(circle). The Fourier partial sum using the first $n$ coefficients will converge to the average of left and right limits of the function at every point, as $n\to\infty$. The convergence rate is $O(1/n)$. However, the total variation of the partial sum does not converge and grows unbounded as $n \to \infty$. The Cesaro partial sum does not have this problem, and its total variation converges to that of the function as $n\to\infty$. However, the convergence rate of the Cesaro sum, to the actual function, is $O(\log(n)/n)$ which is slower compared to $O(1/n)$ which is the case for the Fourier partial sum.

I am trying to think of a method to construct the function using the first $n$ coefficients so that it does not have the above-mentioned drawbacks. Meaning, it converges to the average of left and right limits of the function at every point at a rate $O(1/n)$, and its total variation also converges to that of the function.

Question: What would be the possible applications/usefulness of such a method to other areas of mathematics?

Consider BV functions on a torus. The Fourier partial sum using the first $n$ coefficients will converge to the function at every point of continuity, as $n\to\infty$. The convergence rate is $O(1/n)$. However, the total variation of the partial sum does not converge and grows unbounded as $n \to \infty$. The Cesaro partial sum does not have this problem, and its total variation converges to that of the function as $n\to\infty$. However, the convergence rate of the Cesaro sum, to the actual function (at points of continuity), is $O(\log(n)/n)$ which is slower compared to $O(1/n)$ which is the case for the Fourier partial sum.

I am trying to think of a method to construct the function using the first $n$ coefficients so that it does not have the above-mentioned drawbacks. Meaning, it converges to the function at points of continuity, at a rate $O(1/n)$, and its total variation also converges to that of the function.

Question: What would be the possible applications/usefulness of such a method to other areas of mathematics?

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Rajesh D
Rajesh D
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Consider BV functions on a $1$-d torus(circle). The Fourier partial sum using the first $n$ coefficients will converge to the average of left and right limits of the function at every point, as $n\to\infty$. The convergence rate is $O(1/n)$. However, the total variation of the partial sum does not converge and grows unbounded as $n \to \infty$. The Cesaro partial sum does not have this problem, and its total variation converges to that of the function as $n\to\infty$. However, the convergence rate of the Cesaro sum, to the actual function, is $O(\log(n)/n)$ which is slower compared to $O(1/n)$ which is the case for the Fourier partial sum.

I am trying to think of a method to construct the function using the first $n$ coefficients so that it does not have the above-mentioned drawbacks. Meaning, it converges to the average of left and right limits of the function at every point at a rate $O(1/n)$, and its total variation also converges to that of the function.

Question: What would be the possible applications/usefulness of such a method to other areas of mathematics?

Consider BV functions on a torus. The Fourier partial sum using the first $n$ coefficients will converge to the average of left and right limits of the function at every point, as $n\to\infty$. The convergence rate is $O(1/n)$. However, the total variation of the partial sum does not converge and grows unbounded as $n \to \infty$. The Cesaro partial sum does not have this problem, and its total variation converges to that of the function as $n\to\infty$. However, the convergence rate of the Cesaro sum, to the actual function, is $O(\log(n)/n)$ which is slower compared to $O(1/n)$ which is the case for the Fourier partial sum.

I am trying to think of a method to construct the function using the first $n$ coefficients so that it does not have the above-mentioned drawbacks. Meaning, it converges to the average of left and right limits of the function at every point at a rate $O(1/n)$, and its total variation also converges to that of the function.

Question: What would be the possible applications/usefulness of such a method to other areas of mathematics?

Consider BV functions on a $1$-d torus(circle). The Fourier partial sum using the first $n$ coefficients will converge to the average of left and right limits of the function at every point, as $n\to\infty$. The convergence rate is $O(1/n)$. However, the total variation of the partial sum does not converge and grows unbounded as $n \to \infty$. The Cesaro partial sum does not have this problem, and its total variation converges to that of the function as $n\to\infty$. However, the convergence rate of the Cesaro sum, to the actual function, is $O(\log(n)/n)$ which is slower compared to $O(1/n)$ which is the case for the Fourier partial sum.

I am trying to think of a method to construct the function using the first $n$ coefficients so that it does not have the above-mentioned drawbacks. Meaning, it converges to the average of left and right limits of the function at every point at a rate $O(1/n)$, and its total variation also converges to that of the function.

Question: What would be the possible applications/usefulness of such a method to other areas of mathematics?

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Rajesh D
Rajesh D
  • 698
  • 9
  • 45

A question on convergence rates of Fourier series and strict convergence

Consider BV functions on a torus. The Fourier partial sum using the first $n$ coefficients will converge to the average of left and right limits of the function at every point, as $n\to\infty$. The convergence rate is $O(1/n)$. However, the total variation of the partial sum does not converge and grows unbounded as $n \to \infty$. The Cesaro partial sum does not have this problem, and its total variation converges to that of the function as $n\to\infty$. However, the convergence rate of the Cesaro sum, to the actual function, is $O(\log(n)/n)$ which is slower compared to $O(1/n)$ which is the case for the Fourier partial sum.

I am trying to think of a method to construct the function using the first $n$ coefficients so that it does not have the above-mentioned drawbacks. Meaning, it converges to the average of left and right limits of the function at every point at a rate $O(1/n)$, and its total variation also converges to that of the function.

Question: What would be the possible applications/usefulness of such a method to other areas of mathematics?